ON THE EQUIVALENCE OF
QUADRATIC FORMS
By
THOMAS ROSCOE HORTON
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
August, 1954
ACKNONLEDGMENTS
The writer wishes to express his sincere appreciation to Professor Edwin H. Hadlook, Chairman of his Supervisory Committee, for a generous contribution of time, energy, and helpful criticism throughout the preparation of this work and to Professors W. R. Hutcherson, F. W. Kokomoor, Z. M. Pirenian, and R. 0. Stripling, all of whom served as members of his Supervisory Committee. To Professor Hadlook is due a large measure of gratitude for sympathetic encouragement and guidance during the author's entire doctoral program. Finally the author wishes to acknowledge that without the help of his wife, Marilou Horton, his graduate studies could not have been begun and that without her continued confidence in him his graduate studies could never have been completed.
TABLE OF CONTENTS
ACKNOWLEDGMENTS . . . . . . . . . . .
LIST OF TABLES * * * * e e * . . * .
* . 0 0 * 0
* 0 * 0 0 0
Chapter
I. INTRODUCTION .. . . . . . . . . . . . .
Historical Remarks . . . . . . . .
Definitions * * * . . . . . . . .
Statement of the Problem . . . . .
II. CONDITIONS FOR EQUIVALENCE TO THE
FORM ft WITH b - O, b23 -Kb33
Examples #. ï¿½ . * * * * * III. CONSTRUCTION OF A FORM f EQUIVALENT TO SOME FORM f' WITH b13 b23 - 0 . .
Type A: bl1-b12 .
Type B: bll b h12, (bll, b12) - 1 .
Type C: b11 b 12, (bll, b12) e b>l Examples . . . . & * * . . * . 0 .
IV. CONDITIONS FOR EQUIVALENCE TO THE FORM f'
WITH b23 - O, b13 - N, AND RELATED FORMS
Examples . * . * . . . . . . a
V. CONDITIONS FOR EQUIVALENCE TO THE FORM f'
WITH b13 " 0, b23 - M, AND RELATED FORMS
iii
Page
ii
v
TABLE OF CONTENTS--Continued
VI. CqNDITIONS FOR EQUIVALENCE TO THE FORM
f WITH b -b15 -1 b - 0 . . . . . . . 76
VII. APPLICATIONS TO THE THEORY OF
TABULATION OF AUTOMORPHS . . . . . *.* 89
VIII. SUMMARY . . ï¿½ ï¿½ ï¿½ . . ï¿½ ï¿½ . * . ï¿½ . ï¿½ . 100 BIBLIOGRAPHY ï¿½ . . ï¿½ . . . . ï¿½ ï¿½ ï¿½ ï¿½ . . . . ï¿½ . . 106
BIOGRAPHICAL SKETCH * . . . . . . ï¿½ . . . . . . . 1 108
iv
LIST OF TABLES
Table Page
3
1. Automorphs of the Positive Form f -falixi 2 97 i-l
2. Summary of Conditions that a Form f of Determinant d j 0 Be Equivalent to a Form f ... 101
CHAPTER I
INTRODUCTION
Historical Remarks
The notion of equivalence is a basic concept in the study of the arithmetic theory of quadratic forms, which itself is a branch of number theory. The study of quadratic
forms may be said to have been initiated by Pierre de Fermat
1
in 1654. Since that time notable contributors have been, among others, Joseph Lagrange, Carl Friedrich Gauss, G. L. Dirichlet, G. Eisenstein, Henry J. S. Smith, and Leonard E. Dickson. The last-named has compiled an exhaustive history
of the field in his Quadratic and Higher Foms, Volume III of the monumental three-volume History of the Theory of Numbers.2 This work consists of a detailed, documented record of the results of research in the field of quadratic forms together with clear summaries of those results and in some cases sketches of the methods of proof of the results. As a field of mathematics number theory is unique in that such a precise, lucid, and thorough history is available to the research worker.
1L. E. Dickson, History of the Theoery of Numbers, Vol. III, Quadratic and Higher Forms, (Washington,) 1923, p. 1.
21bid ï¿½
A recent exposition en some of the modern aspects of the study of quadratic forms has been published by The Mathematical Association of America as a Carus Mathematical Monograph, namely, The Arithmetic Theory of Quadratic Forms, by Burton W. Jones.3 Number theorists engaged in research on forms owe much both to Dickson and to Jones for their successful and independent labors in systematizing and unifying their branch of mathematics. Although the results of researches into the properties of quadratic and higher forms have been extensive enough to fill, even in Dickson's concise style, three hundred pages of the History of the Theory of Numbers, Professor Jones remarks that "the theory of quadratic forms is rather remarkable in that, though much has been done, in some directions the frontiers of knowledge are very near.,,4
In this tercentenary year of the study of quadratic
forms, it seems to the author especially appropriate that such an elemental and salient notion as equivalence be studied anew; it is hoped that by such a study the frontiers of knowledge will have been in some measure extended.
Definitions
A fem is a homogeneous polynomial expression in two
3B. W. Jones, The Arithmetic Theory of Quadratic Forms, (Baltimore,) 1950.
41bid., p. viii.
or mere variables. A quadratic form is a form of the second degree. A form of the second degree in n variables is called an n-ara quadratic form (e., binary quadratic form, ternary quadratic form) and is thus a polynomial of the type
n
P(xI, x2, ..., xn) -aT aijxixj.
When aji- aij, ijul, 2, ..., n, then P is said to be a classic form. In this dissertation the major emphasis is on classic, ternary quadratic forms, i.e., only those quadratic forms which are in three variables and whose terms xixj, iJ, have even coefficients. Henceforth the word form will denote a classic, ternary quadratic form unless it is stated otherwise.
The form
3
f _ aijxixs, aji a aij, iJ-l,2,3, iJ-1
is the general classic, ternary quadratic form. The coefficients of f are said to have the matrix /all a12 a,,
(ajj) a (a12 a22 a23)
23 /
a 13 a23 a 3
With the form f there is associated the determinant d - laijILet Aij be the cofacter of aij in (aij). Then the form i(xl ,x2 ,x3), defined as
3
#(xI, x2, x3) - Aijxixj,
is called the adjoint form of f or the adjoint of the form f.
A linear integral transformation
x1 . CllY1 * c12Y2 + c1373 x2 ac21y1 + c22Y2 + c23Y3 x3 a c31Y1 + c32Y2 + c33Y3
is for brevity and convenience usually written as a matrix (Cjk), where
/Cll c12 0 13
(CJk) G21 E22 23)
\c31 032 c33)
When a form f is subjected to such a transfermatien (0jk) of determinant Icjkl - 1 (or -1), then the resulting form f' is said to be equivalent ( or improperly equivalent ) to f1. The form f' is a ternary quadratic form in the variables y, Y2' and y3.
Let T denote the g.c.d. (greatest common divisor) of the literal coefficients aij of f. Let T- denote the g.c.d. of its coefficients all, a22, a33, 2a12, 2a13, 2a23. Then if T - l, the form f is said to be p. Evidently - a1 or 2. When C- a, 1f is said to be properly primitive; when o - 2, f is said to be improperly primitive.
When xl, x2, x3 are integers, the value of f(xlx2,x3)
is some integer m, Then m is said to be represented by the form f. If the g.c.d. of the three xi is one, then m is said to be represented primitively by f or represented properly by f; the latter two terms are interchangeable, but throughout this study the former of the two will be used exclusively.
Statement of the Problem
The notion of equivalence is the central study of this dissertation. By the very definition of equivalence one may, given any form, produce an equivalent form by subjecting the original one to a unimodular transformation, i.e., a transformation of determinant one. Conversely, if the resulting form and the transformation be known, then the original one may be obtained by elementary means. A problem arises, however, if this question is posed: given two forms of the same determinant, find the unimodular transformation, if any, which sends one into the other. This question and even the question of determining whether such a transformation exists possess no general answer. In other words, given two forms, are they equivalent? To obtain a partial answer to this last query, the major problem of this dissertation is as follows: what are some necessary and/or sufficient conditions that a form
3
f aijxixj, aji - aij ,
of determinant d 9 0, be equivalent to a form
f . bskYsYk
a, k- 1
of a particular type? These types are based upon various restrictions which are placed upon the coefficients bsk of f'. These restrictions may be found in subsequent chapter headings, in the statements of propositions, and in the summary. The viewpoint of the writer in seeking the conditions for equivalence was that if enough conditions could be found, then applications of proper combinations of the results might serve as useful tests for equivalence of particular forms.
This conjecture proved to be true. During the course of the research several related questions presented themselves and were studied.
Results of the research are given in the form of
mathematical statements --- lemata, theorems, and corollaries, together with the proofs of these results. At the end of most chapters representative examples are offered.
CHAPTER II
CONDITIONS FOR EQUIVALENCE TO THE FORM
f WITH b130 0, b23w Kb33
When a ternary quadratic form is subjected to a linear integral transformation of determinant one, the coefficients of the resulting equivalent form may be computed by direct substitution (a tedious process), by matrix multiplication, or by the use of explicit formulae. These latter relations are well known and may be found in Dickson's studies in the Theory of Numbers.2 The information contained in Lemma 1 below may be found there in slightly different notation and is stated here as a lemma for convenience in later reference to it.
LEMA. 1. If (cjk), J,k - 1,2,3, is the matrix of a linear transformation of determinant Icjkl- 1, which takes f, with coefficients aji- ai, into the equivalent form f', then the coefficients bsk, sk- 1,2,3, of f' are given by
(1) bsk Xlkcls + X2kC2s + X~kO3s , bksmbsk v
i1bid., p. 2.
2L.E. Dickson, Studies in the Theory of Numbers. (Chicago,) 1930, p. 5, (18) and (19).
7
and where
(2) Xik " ail lk a12 2k 1ai3ck i-,2,3
in particular,
(3) bkk tk - f(Clk, 02k# 03k)"
PROOF. The coefficients bsk Of ft are given by
3
b sk - i, .iisaij'-Jk , jjk- 1,2,3.
Hence
bak a 1 Cis Z aijcjk -2 cisXjk i a i
which is (1). Also, bko w bsk.
The following result is due to E. H. Hadleck and is stated here without proof also for the purpose of later ref3
earence.
LEMMA 2. If f(xl, x2, x3) I aijxixj, aji- aij, reprei,J-l
sents primitively g or -g when xj - xj , J - 1,2,3, where g is the g.c.d. of the values of the three linear functions
3E. H. Hadlock, Terrna~ Quadratic Forms Equivalent to Forms with One Term of Type 2b1. .Y4. (Paper read at tWe four hundred nine ty-sixth meetng t t American Mathematical Society, Spartanburg, S. C., November, 1953). Abstract published in Bulletin of the American Mathematical Society, Vol. 60, No. 1 (1954), p. 47.
(4) X- a a51x1 &82x2 * asx3 , 8-1,2,3, associated with f, when xi- x, then g is an arithmetical invariant of f with respect to any linear integral transformation of determinant one.
LEMMA 3. A necessary and sufficient condition that the form
3
f ,a- laijxjxj ,aji - aij of determinant d 9 0 be equivalent to
ft blll2 + b22722 + b33732 + 2b12Yly2+ 2Kb33y273
is that f represent primitively g or -g, when xj - xj , J 1, 2, 3, where g is the g.c.d, of the values of the three linear functions
(4) X; - aslx * s2x2 * a,3x3 , s-1,2,3, associated with f, when xi - x , and where K is an arbitrary integer.
PROOF. Suppose that f is equivalent to f . Then there exists a linear transformation (0jk) with
(5) c12C12 * c22C22 + 032C32 - 1 in which the set of cofacters C12, 022, C32 of 012, 022, 032 respectively is a primitive set. By Lemma 1, the coefficients
kef f' are given by (1), where the k 1- 1,2,3, are defined by (2). The elements of the third column of (ajk) are 013, o23, 033 , which comprise a primitive set. By (2),
-X3 " allcl3 + a12c23 + &1303,
X23 - a12c13 * a22023 + a23*33 X33 - a13c13 * a23023 * a33c33
Not all of X3, X13, X33 are equal to zero, for then d - 0, contrary to the hypothesis d 9 0. Define XI3, X23, and X., by
(6) X i- il, 2, 3,
(6) Xi13 gXi3,
where
X1 X3, 3, X3)
All values, not all zero, of ol, c21, c31 for which
(7) X1311 ' X23021 * X33031 - 0 are given by the second order determinants of the matrix
X13 X23 X33)
a n k /
namely,
(8) 011 - X23k - X33n
O21" X33 a X13k 031 X13n - X23s
where the integral values of s, n, and k are chosen so that all# 021, and 031 will be a primitive set. By (1), (6), and
(7),
(9)
b13 - g(X13C11 + X23c21 * X33c31) a 0
Since f-'-1f', not only must (9) hold, but also
(10)
b23 a Kb33 *
Hence the system
(11)
g(X13 011 X23c21 * X33031) a 0
g(X1312 + X23c22 + X33c32) a Kf(c13, c23, c33) must be satisfied. By (3),
f(013,c23,e3 3) " b33 ,
by (1),
b3 a X I + X 3c33 b33 a 13 13 * 23c23 + X33 33
and by (8), g divides each of XI3, X'3, and X;3 . Hence g divides f3 - f (c13P 239 a33). Write
f3" - 4 ï¿½
Then (11) becomes, upon division by g,
(12) X13011 + X23c21 + X33c31 0 0
X13c12 * X2322 + X33032 - Kr41
a set of two independent equations, the first homogeneous and
the second non-homogeneous. The former has solutions, as shown by (8), and the second has integral solutions if and only if
(XI3, X23, X33) Kr4 ,
which is true since (X1, X23, X 33) - 1.
Let C5k be the oefactor of Ojk in (ejk). Then
(13) C12 - c31c23 - c21033
C22 011033 - a31013 C32 c21c13 11a23
which is, by (8),
(14) C12 M .(023X23 ' c33X33)a + 23X3ln * c33ki3k
0 22 ' o13X23s i(C 13Xk3 * '33X33)n + c 33X23 k c32 c 1333 + c23X33n "(c13X13 * '25X23)k
By (5) and (14)
c12(-c23X238 - c33X33s + c23Xlsn + o33Xisk)
022(013X23s - c13X13n - o33X33n + c33X23k)
* o32(cl3X33 * c23X33n - ol3Xiak - 023X23k) - 1, or
(15) Ole3(022X23 * 032X33) + c23n(c12X13 + 032X33)
* e33k(c12X13 * 022X23) - 012s(c23X23 + c33X33)
- 022n(013X13 * c33X33) - c32k(ol3Xl3 + 023X23) - 1
From the second equation in (12) three substitutions are obtained, namely,
c22X23 * c32X33 - K!4 - c12X13 C12X13 * c32X33 4 - 022X23 C12X13 * C22X23 a E4 - c32X33
and these values, when placed in (15), give
0 1S(Er4 - 012X13) ' o23n(Kf4 - '22X23)
c33 k(Er4 - 032X33) - o12s(c23X23 * 033X33)
-c22n (c13X13 * c33X33) - c32k(c13X13 + 023X23) - 1, which when multiplied by -g becomes
(16) (012S + 022n + 032k)b33 - Kf3(o13S + o23n + c33k) a -g
After factoring (16) may be written as
(17) f 3s(cl2 - Kc13) + n(C22 - Kc23) * k(o2 - Ko33)} -g.
Hence it is seen that f3 must divide g. But from (2) and (6) it is known that g divides f3" The first statement implies that If3 -g , and the second, that gi lf3j. Hence Jf3j * which is to say that f represents primitively, when x- xj I =-c j3 g or -g, where g is the g.c.d. of the values of the three
I I
functions Xs as defined in (4), when x- X
The condition is sufficient, for s, n, and k are arbitrary; so let s, n, k be a primitive set; then (17) is
satisfied, i.@., there exist integers 0i2 - Kci3, i-1,2,3, satisfying (17), since the g.c.d, of the three s, n, and k divides "g/f3 - . or -1. Relation (16), upon division by g, yields, by (6), (15), which retraces to (5). The values s, n, and k are placed in (8) to produce values of the Cll, 021, and 031 satisfying (7). From (5), ICjk(- 1. From (12), b23 s Kb33, and (7) implies that b13- 0. These are the explicit properties of f 1; hence f--f . Moreover (5) may be rewritten as
ClICII + c21C21 + c31C31 1,
and it may therefore be seen that (0ll, c21, c31) - 1. This completes the proof of Lemma 3.
COROLLARY 1.
If the form
3
3f laijxixj aji - aij of determinant d 9 0 represents one, than f is equivalent to
f - b11712 + b22y22
" b33y32 " 2b12Y1Y2 + 2Kb33Y2y3.
COROLLARY 2. form
A necessary and sufficient condition that the
3
f - i"iul ijij
aji m aij #
of determinant d j 0 be equivalent to
f' b ly2 + b22Y22 + b33Y3 + 2b12y1Y2
is that f represent primitively g or -g, when x- x1 j - 1,2,3, where g is the g.o.d. of the values of the three linear functions (4) associated with f when x - 1
-a a
This Corollary is a special case of, and follows directly from, Lemma 3. However, it has been proved independently by E. H. Hadlock.4
COROLLARY 3. A necessary condition that the form
3
f - aijxixj aji o aij '
of determinant d 9 0 be equivalent to the form
f' b lly2 * b22Y22 + by3332 + 2b12Y1Y2 + 2Kb33Y2y3 is that g - jb331, where g is the g.c.d. of the values of the three linear functions X; defined by (4).
PROOF. By Lemma 3, f3 - I g. By (3), b33 " f3" Hence bu M I g. But since g is a g.c.d., g must be positive. Therefore, g - Ib331.
This Corollary is applicable to the special case of K - 0, and for this case the reader is referred to the remark which
follows Corollary 2.
Corollary 4 below concerns a very special type of form, a form having but one cross-product. Although this
is a highly restricted type of form, its occurrence is frequent; many foxms of this type may be found in a table of
5
reduced forms. Moreover, in the next chapter it will be shown that such frms exist for every value of determinant d7O.
COROLLARY 4. If the form
f o al1x,2 * a22x22 + a33x32 * 2a23x2x3
of determinant d j 0 is equivalent to the form
f'= b11Y2 + b22Y22 + b33Y32 + 2b12Yly2 ,
and if (Cjk) is the matrix of the transformation of determinant one which sends f into f', then b33 divides al1 l3 ; in particular, if all - - t , t c1 is a multiple of b33. PROOF: By (2), X'3 - al1Cl3; also, g divides Xi3, i-1,2,3. Further, from Corollary 3, g - b33.1 Hence b33 j a11c13
THEOREM 1. A necessary conditien that the form
f i , aijxixj ,aj a aij
5L. E. Dickson, Studies in the Theory of Numbers, pp. 150-151; 181-185.
of determinant d ( 0 be equivalent to
f bllyl2 * b22Y22 * b3373 + 2b, 12
K 4p3 integer, is that f represent primitively, when xj - x a divisor of d. PROOF. Since by Theorem 4 of Dickson's Studies in the Theory of Numbers equivalent (n-ary) forms have the same determinant,6 then
bl1 bl2 0 d - b12 b22 Rb 3 0 E33 b33
which may be factored as bl bl 0 b11 b120
d b 33 b12 b22 K 0 D 33 1 But by (3), b33 a f3
Hence f3 divides d, the determinant of the form f.
In order that the general ternary form f with unrestricted coefficients aij be equivalent to fl, certain necessary conditions must be met. Further, given certain sufp . 7
ficient conditions, f will be equivalent to fI. Thus there are a number of necessary and/or sufficient conditions that f and f' be equivalent. These conditions evidently depend upon the values of the coefficients of the form f . Hence if fI is a highly restricted type of form, then the conditions for equivalence of f and f will be quite stringent; reciprocally, if the form ft is not so highly restricted, then the conditions for equivalence of f and f' will be less rigid. It should be noted that conditions for equivalence are usually expressed in terms of whether the given form f represents (or represents primitively) one or more integers.
Given any two forms f and f', one obviously necessary condition for equivalence is that their determinants be equal. Other necessary conditions might be written similarly for other arithmetic invariants, e , T and " . Clearly such conditions are not sufficient for equivalence.
An interesting necessary condition that the general
form 3
f - laijxixj , aji - aij
be equivalent to the form
- AyI 2 b22Y22 * b33Y3 2b12y7y2 + 2bl3yly3 + 2b232Y' where A is any integer, is that f represent A primitively. The proof of this follows directly from (3). For since fï¿½--f' then by Lemma 1, f(cli, c21, c 31 - bll - A, and the cil,
i - 1, 2, 3, constitute a primitive set, so that f represents A primitively. The converse of the statement holds, that if
f represents A primitively, then f is equivalent to a form having A as the coefficient of y12 and this converse is proved and stated as a theorem in Dickson's Studies in the
7
Theory of Numbers. Similar statements hold for equivalence to a form having A as the coefficient of 722 or y32, as evidenced by the three separate statements given in (3).
The form f' of Theorem 2 below is the particular f' form of Lemma 3, Corollaries 1 and 3, and Theorem 1. Conditions for equivalence of f and ft which were stated in those propositions pertained principally to representation ef some integer by the form f. The next condition relates to the adjoint of f.
THEOREM 2. A necessary and sufficient condition for the equivalence ef the forms
3
f 9 1 aijxixj aji a aij
of determinant d , , and
f' = b11Yl2 + b22Y22 + b33Y32 + 2b1271Y2 + 2Kb33Y2Y73
7Ibid., p. 12, Theorem 10.
is that g(X13,X23,X33) be equal to d or -d, where g is the g.c.d. of the values of the three linear functions XI asso2iated with f, when x xi , and the Xi, i - 1,2,3, are defined by (6), and where # is the adjoint of f.
PROOF. To prove the condition necessary, take as hypothesis
that f---f . Then by Lemma 3, f must represent, when x3 * ' Jc , either g or -g, where g is the g.c.d. of the xj - j3
values of the three linear functions Xs , s - 1,2,3, associated with f, when x3 x. . The double sign ( . ) in relation (18) is to be taken as either positive or negative,
not necessarily both.
(18) f3 a ! g
Multiplication of both members of (18) by d2 gives
ï¿½ d2g a d2f(c13,c23,c33)
and
(19) + d2g - f(do13,do23,dc33). By (2) and (6) there exists a set of values c13, c23, a33 with (l3, 023, 033) - 1 of Xl, x2, x3 such that
X3 all13 + a12023 + a13c33 - XI3g X!3 - a12013 + a22c23 * a23c33 = X23g
I a a13c13 4 a23c23 4 a33c33 a X33g ,
where (X13, X23, X33) - 1. Solving the above set of equations for del3, dc23, and do33, one obtains
(20) do13 - gN1
do23 - gN2
do 33"g
dc5 gN3,
where
(21) N1 - A11X13 * AI223 ï¿½ AI3+
N2 a A12-3 A22X23 A23X
N3 - A13X13 * A23X23 * A33X3 ,
and where the Aij are the cofacters of the elements aij of d. Substitution of (20) into (19) gives
d2g - f(gNI, gN2, gN3) ,
2 2
ï¿½ d2g _ g f(Nl, N2, N3) or
(22) +- d - 2= g(N1, N2, N3). By (4), which is
I
X8 " aslXI * a,2x2 * asZX3 , a - 1,2,3, use as index the letter i rather than a so that
i
ia ill i 2 2 #i 3 X3 1-l~
Then
f(Nl, N2, N3) aXIN X~2 + X3N3
x4- xj(N1, N2, N3), i - 1,2,3. By (21)
Xj(Nlp N2, N3) = %X3 Therefore, by (21),
f(Nis N2, N3) a dX 3N1 + dX23N2 + dX N3
or
(23) f(Nl, N2, N3) - dj(X13, X23, X33)
where #(xl,x2,x) is the adjoint form of the form
3
a -a
f (x ,x2,x3 ) w, a jx xj, aj i a a ij
Substitution of (23) into (22) gives
ï¿½ a2 gd$(X13, x23, X33) or
(24) d a g(x13t, X230, X33)
The relation (24) states that g (X13, X23, X.3) equals the determinant of f or its negative, which was to be proved.
The sufficient condition follows readily by retracing the steps (18) to (24). Then since (18) holds, application of Lemma 3 guarantees that f-,-f o This completes the proof of Theorem 2.
Throughout this chapter conditions were sought that
a general form f be equivalent to a form ff' of a specified type. Several conditions were obtained; some of these are useful in particular applications, while others are rather unwieldy. In the practice of determining whether two forms are equivalent one must, since the specific transformation is not known, resort to all possible means for testing equivalence. It will be shown that certain of the conditions for equivalence developed in this chapter do serve in some cases as useful tests for equivalence.
Examples
Several representative examples are given here to illustrate the preceding results. Consider the form
f - -37x12 + 9x22 + 3x52 -30xlx2 + 52xlx3 + 56x2x3
of determinant d - -590. One may apply to f the transformation
Xl " Y1 " Y2 " 73
TS x2 a-yI +2 y2 + y3
x ay1 - 72
whose matrix is written
(ajk) - ( 1
(i -1 0
and which is of determinant one, to obtain an equivalent form f'. First Lemma 1 will be applied in order to obtain
the coefficients bak Of f
X1 1 (-37)(
S ( (26)( x .(-37) (
X2 *
X'- ( 26)( 1 (-37)( X - (-15)(
x I- ( 26)(
1) 1) 1)
-1)
-1)
-1)
-1)
-1)
-1)
explicitly. By (2),
+ (-15)(-1 + ( 9)(-l
* ( 28)(-1 + (-15)( 2 + ( 9)( 2 + ( 28)( 2 + (-15)( 1 + ( 9)( 1 + ( 28)( 1
Placing these values in (1),
( 4)( (-19)( (22 )(
(-19)(-1 22)(-l 22)(-1
. ( 4)(-l) + ( 5)(-l) + (24)(-l)
* ( 5)( 2) + (24)( 2) " (24)( 1)
+ ( 1)( 1) - 1 + (27)( 1) -3
* ( 2)( 1)- 0 + (27)(-l) - 2 + (2)(-l) -24 + ( 2)( 0) n 2 .
Also, by (3),
b 22 f(-l, 2,-1) - 2,
and b - f(-l, l, 0) - 2.
26)( 28)(
3)(
26)(
28)(
26)(
28)( 3)(
0) 0)
0)
0i) 0i)
-4
-4
- 1
--19
- 5
- 27
- 22
- 24
-2.
'C
bll b12 " b 1
b13"
b22 b23 " b 33
t
Thus the form f may be written
" 2 + 2Y22 + 2y32 + 6Y172 + 48Y2Y3
The form f' just obtained may be considered as the fI of some of the preceding propositions, namely, Lemma 3, Corollaries 1 and 3, and Theorems 1 and 2, for b23 - Kb33, or 24 - 2 K, so that K - 12. By Lemma 3, a necessary and sufficient condition that fl-'f is that f3 equal g or -g. Since f3 2 and g - 2, this condition is satisfied.
Corollary 3 is illustrated in that g - )b331-121" 2. By Theorem 2 g#(X13,X23,X33) must equal d or -d. The adjoint of the form f is computed and is found to be
(xIxx3) - -757x,2 -787x22 -558x32 +1546xlx2 -1308xlx3 +1292x2x3
By (6), the values of X13, X23, and X33 are found to be 11, 12, and 1 respectively.
Then
J(lll2,l) - -91597 -113328 -558 *204072 -14388 +15504 --295. Hence
g#(X13, X23,X33) - 2(-295) - -590.
Since gj(XI3,X23,X33) - -590 - d, the condition of Theorem 2 is satisfied,
Theorem 1 is illustrated by the form
f - -18x12 + 7x22 + 6x32 - 16xlx2 + 26xlX3 + 36x2x3
26
and the equivalent form
f yl2 + 2y22 * 5y32 + 2y1Y2 + 0y2y3
which is obtained by subjecting the form f to the transformation T of determinant one. By Theorem 1, f3 must divide d. By computation, f. - f(-l, 1,, 0) w 5. Since 5 divides d - -235, the theorem is illustrated.
CHAPTER III
CONSTRUCTION OF A FORM f EQUIVALENT TO
SOME FORM f WITH b13 - b23 - 0
In the preceding chapter the form
f w bllYl2 + b 22Y22 + b33y32 + 2bl2yly2 + 2Mb33y2y3 was considered. A special case of f occurs when K - O. By Corollary 3 a necessary condition that a form
3
I -,jflaijxixj, aji 0aij
of determinant d ( 0, be equivalent to a form f' with K - 0 is that g - jb331. In order to exhibit this property one should have a method of obtaining a form f which will be
I
equivalent to f . It might be wondered whether such forms exist for every value d for determinant. In this chapter this last question is answered affirmatively, and an explicit method for constructing such forms of determinant d, where d is any non-zero integer, is given.
Consider the form
f l b11Y * b2222 * b33Y3 2 2b12Y1Y2
All such forms f' are classified into three mutually exclusive types type A, any form fl with bll - b12 ; type B, a form f' with bll j b12 and (bllb12) - 1; and type C, a form If' with bll ( b12 and (b11 b12) - b>l. The construction of a form f equivalent to f' will be accomplished separately for the three different types of fl.
Type A: bll U b12
Given any integer d ' 0 and a f orm f' of type A, if such exists, of determinant d, then application of any unimodular transformation will yield a form f equivalent to f'. Since determinants of equivalent forms are equal, then the value d of the determinant of f is given as follows:
bll b12 0
d- b12 b22 0 b33(bllb22 - b122
0 0 b3112
Since bll - b12 , then d - b33bll(b22 - b1l) Given any integer d ( 0, factor d into two factors, not necessarily prime or relatively prime,
d - b33bll .
Assign to b22 the value b - b 1 1. Then a form f having such coefficients is of determinant
b bl1 0
bll b11+1 0 - b33bll+l)bll-b112]- b 33b - d.
0 0 b33
Let cll, 021, 31and C130 C23, C33 be two primitive sets such that
a11013 + a21C23 + 031C33 - O1
Then there exists a primitive set o12, 022t 032 for which
C13 a 021032 - 031022 C23 " a31012 - 011032
C33 ,011C22- 021'12,
by Theorem 9 of Dickson's Studies in the Theory of Numbers. Let 013, 023, 033 be any three integers satisfying
013C13 0c23C23 a 33C33 -.
Then the matrix (ajk)# jk - 1,2,3, represents the matrix of a linear transformation of determinant one. Consider the transformation (a' ) which is the inverse transformation of (Ck). The inverse transformation is given by
iIbid., p. 11.
I22Ca33- C23 032 013 a32- 12 a33 012 023- 13 c22
(25) (Ol 023 0310 21 0 33 0 110 33 31 13 a 13 C21- 011023 21)
0 21 32 022031 0 12c31"0 11032 C11C22 0 21C12
If the transformation (c'k) is applied to the form
f'b y2* 2, 2,
f bl17l2 b22Y22 b33Y32 2b11Y1Y2 then f is carried into some equivalent form
f 4.aixix, a ji " aij
Thus there exists a form f, as given above, which is carried into f by the transformation (Ojk).
1 1
Define Yik similarly to the Xk given in (2) by
(26) Yik a bilClk + bi2C2k + bi3Ok , is 1,2,3. Then by Lemma 1. if the transformation (o) is applied to the form f', the coefficients ask of the equivalent form f are given by
(27) ask Ylkols +' Y2ks + Yko~a
Applying relations (25) and (26) to the particular form f with b , b22 - b1 + 1 gives
Yi " b11C22C33 " bli023 c32 b11C23c31 - bllc21c3
y' b c c -b a a +b a
21 112233 1123 32 1123 31
-b1121c33 + 02331 - c21033
!
Y31 b33c 21032 YI2 1 i13 32 aL b 11o13032 Y2 I bi1a1c5 Y32 " b33c12c31 y13 " b11c12c23 2 b11 12023
Yi3 b3311 22
-b33o22 a 31
-b 11012 C33
-b11 012033
- b1131 013 b 33 11 32
-11 C13 22
-b11c13C22 b 11biliC23
-b33c12021
b11c11C a - b103113 + b1111c33 a 11a33 - a31a13
+b11c13C21 - b l11li23 + b11C 13021 + 013021 -C11C23
By (27) the first coefficient all of the form f is computed.
all " (bli0220133 - bla23032 + b11023c31- b11021a33)(022033
a1 aa (b11a22ob3 ab + -b 0
232 + (b11i22 33 - bliC23C32 ,11 23 C1 -11 bi21 33 + c23 31 - a21a33)(c23c31 - c21 33)
+ (b33c21032 - b3a22c31)(c21c32 - 022a31).
Carrying out all indicated multiplications,
22
all a b110222 033 - bliO22023o32c33 *b11022023031033
- b11210220332 . bl122233233 + blC23 2322
- b1232031032 + blie 2123032033+ b11C22C230 31033
- b11C232031c32 * b1iC232c51 - b1121o23'31033
+02320312~ 2 + - 021 23 31 33 - b11021 22 33
+ b11C21a233233 - b11212331C33 + b110212 332
- 121a23a31a33 + e21 233 2 b33021 2322
- b3321022031o32 - b33021C22o31C32 b 3302220312
Upon collection of like terms this reduces to
a b 0C20 2 + bliC2 2 2 + b 0 2052 + b 0 20 2
11 11 22 33 * 11o23 032 11 21 33 33 21 32
2 2 2 2
" b33022 031 + b,123 031 - 2b11 022023032033
+ 2blio22c23c31c33 + 2b1C21C2332C33 - 2b11C21023031033
- 2b 3021c22031a32 - 2b11c21022C33 -2b11c232 032
33212 3 2 -21c31
-2021023c31c33 3 2 031 +021 2033 2
After appropriate factoring, one obtains
a1 .~~o3o c 2 co )2
alI (a1c23 02133 b11 022c33 023032 )2
+ b11(023a31 - 021033 )2 b33 (021032 022031)2
2b11 (22C033 c23032 Me23031 - c21033).
Now denote by Cjk the cofacter of the element jk in (ojk). Then the above equation may be written as
(28) al -11 C122 + b11(C11 + C12)2 + b33C132
In the same manner each aij, ij- 1,2,3, may be computed. The result of this computation is as follows:
a 11 C122 b11(C11 + C12)2 + b33C132 a22 - 22 b11 (C21 + C22) + b33C23 a33 C32 23 bl ( 31 C2 2 b33C332
a12 - 12C22 b 11b(C 1 C 12 (C21 C 22 b 33C13C23
a - CC b (C C 0)(C .C 5).b C
13 12 32 11 11 12 31 32 33 13 33
a 23 " 22032 b11 (O21 C 22 )(C31 C 32 + b33C23C 33
The foregoing disoussien provides a method by which, given any integer d O, a form f of determinant d may be construoted whioh is equivalent to some form f t of type A.
THEOREM 3. Given any integer d j 0, there exists a form
,2 2 2
f # bllyI * b22Y2 2 b33Y3 * 2b11YlY2
of determinant d - b11b33 . The coefficients of a form f equivalent to f' are given by
(29) a3ij - Ci2C2 + bll(Cil + Ci2)(CJ1 + CJ2) + b33Ci3Cj3, and in particular
(30) aii - C122 + b11(Ci + C12)2 + b 33Ci3 2 where the Cj are cofactors of the elements of any matrix (Cjk) of determinant one.
Evidently such a form f as obtained by the use of the above method must satisfy the results of Corollary 3, i.e., it must be true that g - b . To demonstrate this compute the three X13 by (2).
XI3 - allol3 * a12023 * a13c33
f b( +C)2 +b C2]
13 013 1CI2 1 bl 11 12 33 13
0 o23[C12C22 + bll(Cll + C12)(C21 * C22)
+ b33C13C23] * c33[C12C32 + bll(Cll
+ C12)(C31 + C32) + b33C1C33
X13 a C12(013C12 + c23C22 * c33C32)
" b11C11(b3 C11 * 23C21 + c'WC31) " bl11C(013C12 +023C 22 c33C32) Sb11 C12(c13C11 c23 C21 + 033C31) bIC12(C13C12 + c23 C22 + 033C32) " b33C13(a13C13 + 023 C23 + c33C33)
Since 13 Cls + 023 C2s + c33C3a equals 1 for s - 3 and 0 for a -1 or a a 2, the above equation may be written as
3 b33C13
Similarly, X23 - b33C23 and X - b33C33 . Since a13G13 + c23C23 + C33C33 1, then (C130 C23' C33 * . Therefore
g *(X13, X23, X3) (b33C13, b33C23 b3333)
- (b 3, b3, b)33 lb331, as expected.
This completes the discussion of the constructed form of
type A.
Type B: bll , b12, (bl,b12) - 1
Consider a form f' of type B. Any form f equivalent
I
to f is of determinant
bl b2 0
b11 b120
d" b12 b22 0 " b 33B33
0 0 b33
Thus b divides d and B.3 a d/b33 ï¿½ For any given integer d 0, write d b33B33 so that B33 is positive. Then b33 is positive or negative according as d is positive or negative. Now
(31) B 3 b1lb - b 12
may be written as the congruence
(32) b12 2 -B33 (mod bll)
If congruence (32) is solvable in integers, then it must be true that
(33) . 133 +.
Then by the quadratic reciprocity law,
B3 l1b(34 Nb 1 .
(3) (_l) 2
Therefore, given any integer d 9 O, let B33 be any odd prime factor of d. Then by relation (34) and Dirichlet's Theorem, obtain an odd prime b lie In case d contains no odd prime as
a factor, let B - 1 and by (33) obtain an odd prime bll. This guarantees the existence of an integer b12 satisfying
(32). Then b22, as defined by (31), will be an integer. Finally b - .d/B33 Hence the coefficients have been obtained for some form
,2 2 2
f - blly1 2 b22y2 * b33y3 * 2b12y1y2
of type B and of a given determinant d - b 3B33 9 0.
In the same manner as in the discussion of the construction of a form of type A, let 0l1, c21, c31 and c12, 22P a32 be any two primitive sets such that the g.c.d, of the cofactors C13, C230 and C33 is one. Choose 013, o239 and a33 to be any three integers satisfying
c13C13 4 c23C23 033C33 * 1.
Then (ejk) is the matrix of a linear unimodular transformation. The inverse transformation (ck) sending f' into some form f is given by
Co 11C21 C3
( 012 02 32
C13 C23 C33/ ï¿½
Y
Since a f- C where C is the cofactor of a in (Cjk)
jk kj 3 jhj
the relations (35) and (36) of type B, which correspond to relations (26) and (27) of type A are given by
(35) Yik bilCkl * bi2Ck2 + bi3Ck3
and
(36) ask - YlkCsl Y2kCs2 + Y3k*s3 The a sk are, by Lemma 1, the coefficients of a form f equivalent to fl. Computation of the values of the nine Ylk results in
bllCll + b 12C12 b1lC21 + b12C22 b11C31 + b12C32
b12Cll + b22 C12 b12C21 + b22C22 b12C31 + b22C32
t
Y? b C 31 b33C13 Y32" W 23 Y33 b33 33
The nine explicit
!
Yik values are substituted into (36) to yield
values of the six coefficients of f.
all - bllCll2 + 2b12C11C12 + b22C122 + b330132 = f'(Cll, 012 013) a22 1 b1C212 + 2b12C21C22 + b22C 22 2 b330232
-f I(C21, C22, C23) a33 " bllC1312 + 2b12C31C32 + b22C322 + b33C332
- f (C31, C32P C33)
iiY12 ' YI"
4i3
y2Y3 2
Y23"
a12 - bl1liC21 * b12C1IC22 + b12C12C21 b22 C12 C22 + b33 13 C23
a13 b b11C11C31 + b12C11C2 + b12C12C31 + b22C12C32 S C33 13 C33
a23 b b11021031 * b12021032 * b12022031 + b22C22032 b33 C23 33
This completes the proof of Theorem 4.
TBEOREM 4. Given any integer d Y 0, there exists a form
f - ablll2 + b22Y22 + b33Y32 + 2b12Y1Y2
of determinant d - b33B33 and with (bl1, b12) - 1. The coefficients aij of a form f equivalent to f' are given by
(37) aij a bl CCjl * b12 CilCJ2 + b 2Ci2CJ, + b22Ci2CJ2 + b33 iC3 ï¿½3
In particular.
aii - bllCil2 + 2b12CiiCi2 + b22Ci22 + b33i32
M ft (CilCI2I Ci3),
where the Cjk are the cofactors of the elements ojk of any
matrix (cjk) of determinant one.
A numerical example of the application of this theorem is given at the end of this chapter.
Type C: b1l j b12 , (bllpbl2) - b> 1
The construction of a form of type C is accomplished
similarly to that of the form of type B.
Let b11 and b12 be any two distinct integers such that (bll,bl2) - b> 1. Define b{l and b 2 by
(38) b n bt b, b b ' b
11 11 12 12
Then (bl,bl2)- 1. From the expression for the determinant d of the proposed form , 17l2 b 2 *b32 2bl f' b- y b22Y2 33y3 YlY2
(39) d b 3B , where B,3 - bllb22 - b122
hence by (38) and (39), b divides B33. Thus define B3 by B3 B33b. Hence from (38) and (39),
33
(39) B' -blb - b b' 2
33 22 12
A necessary and sufficient condition that (39') have integral solutions in b2 and b22 for assigned values of B I b' and b is
(40) b b12 - -B53 (mod bl) Define N by
b N 1 (mod b1l *
Then (40) becomes
b = -B N (mod b'
12 - 33 11
Hence for (40) to have a solution it is necessary that
(.n&B ) ( -B N2)(Bb )
or
(4 1 ) (l l 1b )
BI33
Take - 1. Then given any integer d 0, let B33- b be Take B3 any odd prime factor of d. Since B13 - 1, then (41) and Dirichlet's Theorem guarantee the existence of an odd prime bll j b. By (38), b is now fixed in value. Since B ' 1
1111 33
and (41) is satisfied, then by (40) there must exist an integer b2 satisfying
b b{2 -1 (mod bll)
and hence (b'l, bI2) - 1. Relation (38) fixes bl2 in value, and (39) gives b22. Finally, b33 is given by (39). Hence all of the coefficients bll, b12, b22, and b33 of f have been determined. Let Cjk be the cofactor of 0jk in a unimodular transformation (cjk). Then the coefficients of a form f equivalent to f' of determinant d 9 0 are given by (37).
THEOREM 5. Given any integer d 9 0 which contains as a factor an odd prime, there exists a form
,2 2 2
f I bllyI 2 b22Y2 * b33y3 2 2b12Y1Y2
of determinant d - b33B33 and with (bll, b12) - b>l. The coefficients of a form f equivalent to f' are given by (37), where the Cjk are the cofactors of the elements Cjk of any transformation (cjk) of determinant one.
Examples
Let the given determinant be d - -20. Forms f equivalent to forms f of each of types A, B, and C will be constructed. For all of this work the transformation (cjk) will be taken as
(o jk ( i 7 (-14 -2 -15
Then the matrix of cofactors Cjk of the elements cjk is
(Cjk) - 1 -2
1 0 -1
The method preceding Theorem 3 is employed for the first construction. Then
-20 - b11b33 o
Since b is any divisor of -20 (there are twelve choices), one might take bll - 5. Then b -b - 5P b22 - 5 + 1 - 6, and b33 - -4. By (29), a12 0 97, a13 *30, and a23a 7, and by (30), a11 " 229, a22 a 30, and a33 - 1. Hence
f . 229xI + 30x2 + x3 + 194x1x2 + 0x1X3 + 14x x.
The above form f is the constructed form which is equivalent to a form
f . Y1 2 + 6y22 -4y32 + 10YY2 of type A.
For the construction of a form of type B of the same
determinant d - -20, B33 is taken as the odd prime factor B 33 5 of d. Then (34) is
( 1b) a (-) 2
t5
which is, when b is taken as b - 3,
Then (32) gives
b
12 5 (mod 3),
which is satisfied by b 12 i. By (31), b 22- 2. Finally b33 a d/B33 - -20/5 - -4. Then the coefficients of ft have all been determined. Using the same transformation as in the construction of the form of type A, (37) gives the coefficients of the form f equivalent to f . The result of this computation
is the form
f - 87xi2 + 2x22 + 7x32 + 42x1x2 + 8x1x3 -x2x3 which is equivalent to the form
f 2 2 2
f Yl * 22 4y3 + 2Y1Y2
both of which are of determinant d = -20.
For the construction of a form of type C, B33 a b - 5, an odd prime factor of d - -20. Relation (41) is now
Take b 7, and then by (38), bll - (7)(5) - 35. There must exist an integer b'2 satisfying
5 b'2. -1 (mod 7).
12
Take b12 - 2. Then b12 - 10, by (38). By (39), b22- 3. Finally b33 - -4. Thus the form
f'I, 35y12 * 3Y22 - 4Y32 + 20yLy2
is an example of the form of type C, with (bli, b1) - (35, 10) - 5 - b>l. The desired form f equivalent to f' can be found by (37) in exactly the same manner as the two previous types.
CHAPTER IV
CONDITIONS FOR EQUIVALENCE TO THE FORM f' WITH
b23 " Op b13 " N, AND RELATED FORMS
Before conditions for equivalence can be obtained, a preliminary lemma concerning a Diophantine equation must be proved.
LEMMA 4. All solutions in integers of the non-homogeneous
linear Diophantine equation
(42) a x + b y + o z - d, where (a,b,c) d, are given by
(43) x - x0 + bk - on
y a yo + es - ak
z - ze + an - bs,
where xe, y@, ze is any particular solution of (42) and where s, n, and k are arbitrary integers. PROOF. Since x*, Yo, z is any particular solution of (42), then every solution x, y, z of (42) must satisfy
(44) a(x - x0) + b(y - y0) + c(z - ze0) - 0.
But by Dickson's Studies in the Theory of Numbers all of the
solutions of (44) are given by the second order determinants of the matrix
(45) /a b 0
(S
namely
(46)
n k)v
xo a bk - cn Yo o cs - ak z 0 an - be
and hence by (43). Moreover, (43) satisfies (42), since
a(xe + bk - on) + b(y0 + cs - ak) + c(ze + an - be) (axo + by0 cz0) + a(bk-on) + b(cs-ak) + c(an-bs) - d. Since this argument follows for every solution x, y, z of
(42), then all solutions of (42) are given by (43).
The information contained in Lemma 4 is to be used in
the proof of Lema 5, which in turn will be employed as a means to the proof of subsequent theorems. LEMMA 5. A necessary and sufficient condition that the form
f - _ aijxixj aji- aij
iIbid., p. 24, Theorem 26.
X Z-
46
of determinant d 9 0, be equivalent to the form
f a bllyI2 * b22Y22 + b33y32 + 2bl2y1Y2 * 2Nyly3 , where N is any preassigned integer, is that the g.c.d. of the values of the three constants
(47) X[ 3k No,
{f3[X'3k - X 53s - 0214 N02,311
a{ f 3IN3' - Xj3n - 03], Noc33}
divide g, where g is defined by (6), s, n, and k are arbitrary,, and where oil " ci , i - 1,2,3, is a p ar r solution of
(48) x ' '
X13011 + X26021 + X33 31
PROOF. Assume that f-,--f '. Let a13, 023, 035 be the third column of the transformation (ojk) carrying f into f '. Then (C13, 0230 c33) - 1, for otherwise ajk 1 1, and then ff, contrary to hypothesis. By (2), not all of X'3, 42, and X'3 are equal to zero, for then d - 0, contrary to hypothesis that d j 0. Again define X13, X23, and X., as in (6) so that
g * (X13, X23, X33).
By (1), (48) holds, and therefore g I N is a necessary condition that fl*.'f W rite
N - gNI .
(49)
If l - Cl, e21 - a2, and c31 - 03 is a particular solution of (48), then by Lemma 4 all solutions of (48) are given by
(50) el+ + ' - X'
1 X23 33
c -ca + Xv 8 -X Ik
21 2 33 - .3 C -c + Xl3n - X3s
c31 c3 1 2'
where s, n, and k have arbitrary integral values. By (13) and (50), write the values of the cofactors C12 of ai2 in (cjk)P 1- 1,2,3.
(51) C12 - 03c23- 2c33
-(XI3c23 + X13c33)s +(X{3023)n +(13033)k
C22 - 01033 - 03013
+(' (X0 + 3c,,33)k 2(x 3o13)s -(X{3013 X333)n (333
C32 - 2a13 - C1C23
*(o13)s +(X;3023) 1.X3c13 4 3023k
Since
.1cJ1 0c12012 * 022022 + 032C32 1 1, then
(52) 01203023 - 01202c33 -(X23023 + X13033)'12s
+(Xi3o23)22 n +(k3c33)'12k + '22c1033 - '22'3013 (313)c228 -(X!313 + Xi3c33)c22n *(X 3c33)c22k
+032'2013- c32cl23 + (X~313)'32' +(X323)c32"n 1(113c13 + k3023)032k - 1
or
(53) "(N3o23 + X;3'33)c12s -(X{3c13 * X3c33)c22n
-313 * X 3'23)c32k * c13s(N322 * X;3c32)
+ 023n(X13c12 * X 332) + o33k(X!3c12 + X!3c22)
1 12c3c23 + c12c2033 0 2201033 + c22c313 a 32a2c13 + a32a1023
Since f ~f % then it must be true that X23c22 'X;3c32
(54) X!12+X622+X32 0 Use (54) to make three substitutions into (53). Then
-(43023 * X13033)012s - (Xj3013 + X'3o33)c22n
- (X{3o13 + X43023)o32k - 013S(X13012) - c23n(X3o022)
-c3k( 3332) -1 - 0123023 4 01202033 - 022clO33
S0220303 - 03202013 + 032cl023 ï¿½
The above equation may be multiplied by -1 and factored as
o12s(C13X{3 + c23X 3 4 c33X3) + c22n(c13x3 023X923
+ 033X) c32k(o13X{3 + 023% + c33XI3) - c1203023
- 012c2c33 + 02201033 - 02203013 + 03202013 - c3201023 -i Then relation (3) is employed to obtain
(55) f3(o12S + 022n + a32k) = 01203023 - 01202a33 + a22c1033
- 0223013 + 032c2013 - 0320123 -1 .
Dividing (54) by g gives
X13*12 * X23022 + X33c32 ' 0 all solutions of which are given by
(56) 012 - X23Y- X 33
022 - X330(- X13
ï¿½32" X13 -X23 ,
where o( (5' , and t are arbitrary except that their values shall not cause (a12, 0220 032)>1. Substituting (56) into (55), one obtains
f 3-(X23 X3356)s (X33o - X13Y)n + (X13/- X234)k
- (X23 - X33/*)o3c23 - (X23Y - X33'O)'233 " (X33 - X13( )1'C33 - (X33o1- X135)c313
" (k3 - X23"<)c2c13 - (X139 - X23'7'1c23 - 1. This last equation may be refactored as
f3{ (X33n - X23k) *#(XI3k - X33s) + ((X23s - Xie)f
' [1('023X23 0 33X33) - c13(c2X23 + '3X33)]
+ 02 [c2(c13 x13 c33X33) - c23(c1X13 * "30
+ r [c(013X13 ' 23X'23) - o 33(c1X13 * '2X23)] which, af ter three substitutions from (48), results in
f3(X3 n- X23k) * t4(Xl3k - X33s) * )"(X238- X3n7}
-o( [c(c23X23 * '33X33) + 013(X301 -'Y]
,[o2(c1.3X., * c33X33) * c23(X23o2 - Yp]
3(l.313* '23X23) * 033(X33"3 - Nl)J- 1
Multiply the above equation by g to obtain
f3 (X33n - X'3k) +, (X{3k - X13s) + ((Xs - Xlsn)} 0[(41(x1X'3 23k3 33X33) - 13N] P [Y'(13X43 * '23XL3 * '3X')-'2] + Y [03(0133 + 023X23 + '33X;3) - c3N] g which, by (3), is
t~f((X'3n- X'3k) + p$ (X I k - X3s X 1(48 )q, n))
. - 1.3NI 90 - 023N]
or
01'1
(57) {f~ ~~3 k - X' k - i * c3No
{[X13 X33 -21 + 023NIA
+3T38 - Xt3n - '3] + c33Nr "
Now (57) is a non-homogeneous, linear Diophantine equation in O1, l , and * A necessary and sufficient condition that integers v( , , and I exist satisfying (57) is that the g.c.d. of their coefficients divides g. Thus the condition stated in the Lemma is necessary. Moreover, the condition is Ouffiolent, for the steps (48) through (57) may be readily retraced.
COROLLARY 5. A sufficient condition that the fonm f be equivalent to the form
, 2 2
f - bllYI + b22Y2 + b 5Y3 * 2b12y'12 + 2Nyly3,
where N is ny preassigned integer, is that the g.c.d. of the three constants, Nc3 - f3 cl, No23 - f3c21, and Nc33 - f3c31' divide g, the g.c.d, of the values of the three linear functions XJ. defined by (2).
PROOF. Since s, n, and k are arbitrary, take s - n - k - 0. Then by (50), Oil - ci, i - 1,2,3.
It is of interest to compare the results of Corollary
5 and Lemma 3. For by interchanging the variables yl and Y2 of f and the columns ail and c12 of (cjk), Corollary 5 states that a sufficient condition that f be equivalent to
f' f b11y12 + b 22Y2 + b33Y32 + 2bI2y1y2 * 2NY2y3
is that the g.c.d. of the constants, No 3- f3c12, N023- f3 22 and Nc33 - f3c32, divide g. If this be taken as hypothesis, then there exist integers v< , / , and ( satisfying (NOI3 - f3c12)0( + (N023 - f3022) 4 + (No33 - f3c32)Y - -g, In the special case of N - Kb33 , the above becomes (Db33c13 - f3012)o( + (Kb35c23 -f3022
+ (Kb33c33 - f~c32)f - -g
and since by Lemma 1, f3 * b33 , the latter is rewritten as
f- c02)0' + (Ko23 - 022)/9 + (Kc33 - c32)Y - -g
Hence f3 divides g. But g divides each term of f3 b 331 by
(2) and (6), and hence divides f30 Thus f3 equals g or -g, exactly the condition of Lemma 3. Thus in the special case of N - Kb33 the results of the two separate statements are seen to agree. Therefore Lemma 3 may in a sense be considered as a
special case of Corollary 5.
Throughout this study the three linear functions (4), namely,
1 Xs(xI, x2, x3) o aslxI + as2X2 * as3x3, s- 1,2,3, have been of great value in obtaining conditions for the equivalence of two forms. The functions X' are said to be associated with the form
3
f = Z a_ a j - ai
i.e,, given any form f, the functions Xs are defined by (4). A complete change of viewpoint yields the following statement: given any three linear functions Xs , as defined by (4), there is associated a form
3
f - Jiai i
or, in other words, the form is said to be associated with the functions. This novel viewpoint is of some interest and is found to be helpful in the proof of further theorems later in this chapter, notably Theorem 6. The idea of associating a form with a set of three given functions Xs raises one diffi-
culty, however, namely, that a form f so associated with the functions * may not be a classic form, i.e., it may not be true that aji - aij, i j, ij - 1,2,3. Up to the present point in this dissertation only classic forms have been studied. Hence a few definitions and minor lemmas concerning
non-classic forms are required and are now presented.
In Chapter I the divisors T and r of the classic form
3
f Z: i~ aijxixj, aji " aij
were defined by
' - (all,a12,a13,a22,a23,a33)
and - (all,a22,a332a12,2a13,2a23)ï¿½ If - 1, then f is a primitive form, and then ( - 1 or 2. When - 1, f is properly primitive, and whenG-= 2, f is improperly primitive.
Analogous definitions for the non-classic form are 'F - (al1, a12, a13, a21, a22, a23, a3l, a320 a33) and T_ M (al1, a22, a3, a12 a21, al3. a1, a23. a32). It is seen that the second set of definitions includes the first set as a special case. If 7% 1, f is primitives then the value of T" is not restricted to 1 or 2, as in the classic case. If T - 1 and r > i, then f is improperly primitive. If " - i, then 'V- ., and f is said to be p
primi tive.
In Dickson's Studies in the Theory of Numbers Theorem 6 states that any properly primitive (classic, n-ary quadratic) form represents primitively some integer prime
2
to any assigned integer a. A similar statement holds for ternary non-classic forms.
LEMMA 6. Any properly primitive classic or non-classic form q represents primitively an integer prime to any assigned integer m.
PROOF. The statement for the classic case is proved by
3
Dickson. The proof given by him suffices with slight modiE
fioations for the non-classic case. A shorter proof will be presented, however. Since q is a non-classic properly primitive form, then f - 2q is a classic improperly primitive form. By Theorem 7, Dickson's Studies in the Theory of Numbers f represents primitively the double of an odd integer prime to any assigned integer m.4 Let 2N be the integer so represented by f and such that (m, N) - 1. Since 2q - f represents 2N primitively, then q represents N primitively.
LEMMA 7. If a classic or non-classic form f represents A
primitively, then f is equivalent to a form having A as the
2
coefficient of yl .
2Ibid, p. 8.
3!bid. 4Ibid
5
PROOF. The classic case is given by Dickson, and the non6
classic case is mentioned by Sagen. Both cases are proved
here. Since f represents A primitively, there exists a primitive set such that f(c11,C21,c31) a A with (cli, 0219 c31) a 1. By Theorem 123, Modern Elementary Theory of Numbers, there exists a determinant with integral elements having the value one and having oll, 021, c31 as the elements in the
7
first column. Denote the matrix of this determinant by (Cjk), jk-= 1,2,3. Apply to the form f the unimodular tranaformation associated with the matrix (Cjk). The resulting equivI
alent form f has coefficients bsk , where, by (3), which holds for classic and non-classic forms, the value of bl1 is b 11 f(c11, 021, '31) a A.
The forthcoming Theorem 6 has been proved, with slightly different hypothesis, for n linear functions in
n + m indeterminates by H. J. S. Smith.8 It is stated in this study for the sake of clarity and for its application to the theory of forms. Smith's method of proof is used in
part of Lemma 8 below.
51bid.. p. 12.
60. K. Sagen, The Integers Represented by Sets of Positive Ternary Quadratic Non-classic Forms, (Chicago,) 1936, p.7.
7L. E. Dickson, Modern Elementary Theory of Numbers., (Chicago,) 1939, p. 172.
8j. W. L. Glaisher (ed.), The Collected Mathematical Papers of Henry John Stephen Smithj, (Oxford)19, 39293
LEMA 8. Let . denote the g.c.d. of the literal coefficients Aij of the adJoint j of the properly primitive classic or non-classic form f of non-zero determinant d. If the leading coefficient all of f is relatively prime to the integer d/1l , then there exists a primitive set x', x,, xI of values of the indeterminates xl, x2, x3 for which the g.c,d. of the three linear functions
1
(58) X, o allxI * al2x2 + alsx3
X2 - a2x a22x2 + a23x3
Xt "a ~x.a32x2 * a,3x3,
associated with the form f, is one. PROOF. First it will be shown that the values of xl, x2, and
can be so chosen that the g.c.d. of the three X i - 1,2,3, is prime to some given integer M, provided only that M is prime to all . For assign to xl, x2* x the values x 1 a, x 2 - , x - M. When these values are placed in (58), the value of-X{
3 X is prime to M, for the first term of X' is prime to M, and the second two terms are multiples of M. Since (X',' M) a 1, then the g.c.d. of the values of the three Xi is prime to M, for, otherwise, a divisor of Xf is not prime to M, a contradiction.
A method has been outlined to obtain a set of values
019 02 03 of Ij Xt, X' respectively, satisf ying
(59) allx1 alex2 + al3x3 C1
a211 a22x2 + a23x3 02 a31x1 + a32x2 + a 33x3 -"
and such that (Cl, C2, 03) - C is prime to some assigned integer M, provided only that (all, M) - 1. A similar result follows if the hypothesis is taken that any aij is prime
to M, but this latter result is not required for the purpose of this lema.
Lot Aij be the cofactor of aij in (aij )..n has been defined as
n - (All,, A12, A,3, A21, A22, A23, A31, A32, A33). It is well known that
d
whore R and A are invariants of the form f. Hence . Write
M d
Then there exist integers xl, x2, x3 satisfying (59) and for which the g.c.d. C of CI, C2, C3 is prime to d/fl
Define dl, d2, and d3 by
O1 a12 a13
dI 0 2 a22 a23 C3 a32 a33
all C1 a13 d2 a a., C2 a23 a31 C3 a33 a11 a12 C1 d3 a a21 a22 C2 a31 a32 C3
It has been shown explicitly that the system (59) is satisfied in integers. But by the Theorem of Heger, a necessary and sufficient condition that (59) ban integral solutions is that d divides each of dl, d2, and d3; i.e., the value of the determinant d must divide each of the three "augmented" de9
terminants. Write
d1 A11C1 + A21C2 + A31C3 d-2 A12C1 + A22C2 + A32C3 d3 A 13AC1 + A23C2 + A33C3
The g.c.d. of the nine Aij islA, and the g.c.d. of the three Ci is C, so that -C must divide each term of dl, d2, and d V and hence nC divides each of the di, i - 1,2,3. Therefore there exist integers @1, 02, and 03 such that
dl - 1 C fl
d2 - 92C f
d3 - 03C .
91bid.. p. 387.
The necessary and sufficient condition that (59) has integral solutions, namely, that d divides each of dl, d2, and d3, can now be stated as
(60) d I QiCf, i - 1,2,3, and since -L divides d, then (60) is tantamount to
(61) d Oic i - 1,2,3.
It has been shown that (59) has integral solutions and that by the Theorem of Heger a necessary condition that (59) has integral solutions is that (61) holds. Hence (61) is necessary. But (d/Tl , C) - 1, so that
(62) dIei i - 1,2,3,
which is
(63) d , d/C , i - 1,2,3. But this last statement, that d divides each of dl/C, ddC, and d3/C, is, by the Theorem of Heger, precisely the condition that the system
(64) a11x1 + a12x2 + a13x3 - CI/C
a21x + a22x2 + a223 - C/C a 11 a32x2 + a x - C3/0
be satisfied in integers. Therefore there exist integers x1 Xl, x2-xx3 t which satisfy (64), and thus, since
(Cl/C, C2/C, C3/C ) - 1, it may now be said that there exist integers xi - xi for which the g.c.d. of the values of the three Xi, defined by (58), is one, Moreover, the set Xl, x2, X is itself a primitive set, for if (x, 4. 0X') " h>l, then by (58), h divides each of X contrary to the proven fact that (Xl, X2, X3) " I.
THEOREM 6. If (all# a22, a33, a12+a21, a13+a31, a23+a32) - 1, then there exists a primitive set x1 " x1 2 -xa , xs - x for which the g.c.d. of the three linear functions 58)
X1 " allxl * al2x2 4 a13x3 X2 - a21xl * a22x2 + a23x3
X - alx1 * a32x2 * a3
is one.
PROOF. Associated with the set (58) of linear functions is the not necessarily classic form
3
f - aijxixj
Compute d/11 as defined in Lemma 8. If (a11, d/1 ) - I, then by Lemma 8 there exist integers for which the g.c.d. of the three Xj(xj, x2, x1). i - 1,2,3, is one. Also by Lemma 8 the xi constitute a primitive set. The only other case arises when (all, d/f ) i 1. By hypothesis, T- - 1, and hence f is properly primitive. By Lemma 6, f primitively represents an
integer prime to d/f) . Denote this primitively represented integer prime to d/ft by N. By Lemma 7, f is equivalent to a form ft whose first coefficient is N. Associated with the form ff' is a set of three linear functions defined by
(65) Yj - bllyI + b12Y2 * bl3Y3
Y b21Y b22Y2 * b2373
Y5 " b3lyl * b32y2 * b3Y
where b l - N is prime to the arithmetic invariant d/. The transformation (Cjk) sending f into f is given by
(66) x - yllY1 * c12Y2 + '13Y3
x 2- a2171 * c22Y2 + 023y3
X 3 " cl3I7+ c32Y2 + '33Y3 ,
where Iejkl- 1. By Lemma 8, since ft is properly primitive and since (bll, d/ft ) 1 , there exist integers yl - IP y2 0 72' 73 ' 73 for which the g.c.d. of the values of the three linear functions Yj , i a 1,2,3, is one. A well known relation between the functions Xi of (58) and Yj of (65) is
(67) " I X202s X33 S n 1,2,3.
By (66), obtain the values x1, x, xf corresponding to the yj values. Then the three integers xi so derived cause the g.c.d. of the values of the three X1 to be one. For assume that (X{, 4, X3) - g >1 with the values xi, i - 1,2,3. Then
by (67), g divides each corresponding Y{, Y2, and Y' a contradiction that their g.c.d. is one. Therefore given any set of functions (58) with the restrictions of the hypothesis, then there exists a primitive set x1, xi, xv for which the g.c.d. of the values of the three linear functions (58) is one,
It is worthy ef note that although the entire proof
of Theorem 6 is based upon properties of quadratic forms, the statement of the Theorem itself includes no reference to forms. In other words, the concept of quadratic forms has been completely divorced from the three linear functions (58), and Theorem 6 is simply a theorem concerning a system of linear Diophantine equations with certain restrictions on the nine coefficients.
THEOREM 7. Every ternary quadratic form
3
f a . aijxixj , aji - aij
of determinant d ( 0 is equivalent to some form f'. b11712 + b22y2 + b33Y32 + 2b12Y2 2b2323
PROOF. Given any general ternary quadratic form f of nonzero determinant, a method will be given whereby a transfermation (cjk) ef determinant ,jklm 1 can be obtained which
sends f into a form f' with b13 - 0. Let c13, 023P 033 be particular primitive set 013 - c23 - c33 1 1. Then by (2), not all of the three linear functions Xi1, i n 1,2,3, are equal to zero, for then d a O, contrary to hypothesis. Use of (6) gives the value of g and the three numerical values Xi3, i a 1,2,3. If f is to be equivalent to f' then by (7),
X13C11 + X23c21 + X33c31 a 0
must hold, and all values, not all zero, of Cll, c21, 031 for whioh (7) holds are given by (8), namely,
0 11 X23k X 33n c21 aX . X3 k
031 X13n - X238 ï¿½
where the s, n, and k are arbitrary. In the proposed transformation (jk) the eofactors C12 of the elements 012 are given by (13), (14), and, since 013- 023 * c 33 l, by
(68) 012 a (X13)k - (X23 * X33)S + (Xl3)n
C22 - (X23)k * (X23)s - (X13 * X33)n C32 *-(Xi3 * X23)k + (X33)s + (X33)n
Now the three cofactors given by (68) may be considered as three linear functions in k, s, and n, for the X13 are fixed numbers while the k, s, and n are completely arbitrary. By Theorem 6, since (XI3, X23, X33) - 1, then there exist inte-
gers k - , s - s , n - n for which the g.c.d. of the three cofactors C12, C22, and C32 is one, Moreover the values k', o', and n' when placed in (8) give a primitive set of values of all, a21$ and a31, for, if not, then by (13) the cofactors C12, 022, and C32 are not a primitive set, an obvious contradiction.
The values a13, a23, c33 were chosen as I, 1, 1; the of 0l1, 0219 C31 are now fixed by the choice of s, n, k as s', n' k', respectively. It was shown that the cofactors C12, C22, C32 given by (68) form a primitive set. Hence there exist integers 012, 022, 032 satisfying (5), which is
012C12 + c22C22 + a32C32 1.
Moreover, 0129 a221 c32 is a primitive set by (5).
A transformation (Cjk) of determinant one has been obtained satisfying (7). Hence if (ojk) be applied to f, then f is sent into a form with b13 " 0, which is precisely the identifying feature of fl.
Lemma 9 concerns the non-homogeneous linear Diophantine equation of Lemma 4.
LEMMA 9. If (a,b,c) - 1, then there exists a primitive solution x , y , z of the non-homogeneous linear Diophantine equation (42),
a x + b y + c z - d,
for any integer d 9 0. PROOF. When d - 1, the statement is trivially true. Hence consider d 1 1. Since (a,b,c) - 1, then there exist solutions of
(69) a x + by z - .
Let one such solution of (69) be x - xo, y - Y0' z - zo. Since (a,b,c) I d for any integer d, there exist solutions of
(42). In fact one such (non-primitive) solution of (42) is x - dxop y * dye, z - dzo. By Lemma 4, since dxo, dy0, dzo is a particular solution of (42), then all solutions of (42) are given by
(70) x - dxo + bk- cn
y- dyo + cs - ak
z - dz0 + an - be
where s, n, and k are arbitrary. Define UI, U2, and U3 by
(71) UI - x - dxo
U2 a y- dyo U3 - z - dz0
Then (70), after rearrangement, becomes
(72) U1 - ( b)k + ( O)s + (-c)n
U2 a (-a)k + ( c)s + ( O)n U3 - ( O)k + (-b)s + ( a)n
By Theorem 6, since (a,b,c) a 1, there exist integers s - s', n - n', k a k' for which the values of the Ui i - 1,2,3, form a primitive Set. Place these integers s', n, and k' into
(70). Then by Lemm 4 the resulting values x', y', z' of x, y, and z are solutions of (42). Moreover, x ,y ,z is a primitive set, for if (x',y',z') - h>l, then by (42), h Id, and thus by (71) h divides each Ui, a contradiction. Hence there exists a primitive solution x', y I zI of (42). CONJECTURE: Any properly primitive form
3
f ; laijxixj, aji aij of determinant d ," 0 is equivalent to some form
f,2 2 2
f bllyI 2+ b22Y2 + b33Y3 * 2NY1Y2 * 2bl3yly3 * 2b23y2Y3, where N is any preassigned integer. REMARKS. Since f is properly primitive, then the three linear functions
(73) X12 - all12 * a12022 a13c32
22- a12c12 + a22c22 *a2332
2 a13c12 *a23022 a3c32
associated with the form f and defined by (2) possess coefficients which satisfy the hypothesis of Theorem 6, which is to say that r'- 1. Hence there exist integers c12 c{2, 022
022, 032 * c2 for which the g.c.d. of the values of X1
022 ,2-0' 12P X 1' and X2 is one. Also, by Theorem 6, (c2, c22' c52
1. Consider the equation
(74) X n I I * -N.
A sufficient condition that (74) have solutions is that (X{2,
22 X32) * 1. Moreover, by Lemma 9, since (X!2, X2, X32)
- 1, (74) possesses a primitive solution a' f c1 Now 11' C210 31*
if the three cofactors C13, C23, and C defined by
(75) C13 021032 - 3122
C23 - cle1i2 - cilc02 033 " ciloL - chOc2
comprise a primitive set, then there exist integers c13, 0239
such that Icjkft 1. From the infinitude of possible
choices for the sets Ci1 and c12, i - 1,2,3, there seems to be nothing which would preclude obtaining a primitive set of cofactors (75), However, it has not been proved that such a choice is in every case possible. The writer has found no counter-example. The problem of attempting to show that such
a choice is always possible (and hence that any properly primitive form f of non-zero determinant is equivalent to some f ) is of considerable complexity.
LEM 10. The form
2 2 2 +x
f - x 1 22l2 2a13xlx3 + 2a23x23
of determinant d ( 0 is equivalent to some form
2 2 2
f' blly1 2+ b2272 + 3 + 2b12y172 + 2NyIy3
where N is any preassigned integer. PROOF. Assign to c13, 023, 033 the values ol1 - 1, c23 Ma, ando M 0. Then by (2), 3- , XI *a12, X63 a so that by (6), g - 1, and X1 - X, i - 1,2,3. Define by (49), N1 - N/g a N. By (3), f3 - f(l, 0, 0) - 1. Assign to ell, 0210 031 the values ell - N- al3, c21 - 0, and c.1 - 1. Then (ell, 021s 031) - 1. The cofactors C.2 of 0i2 are
(76) C12 M 0
C22 C32 - 0 ,
a primitive set. The values of the three expressions, Nci3
- f5eil, i - 1,2,3, of Corollary 5 are
(77) Nc13 - f3 - N (1) 1 (N -al) - a13
No23 - fac21 a N (0) - 1 (0) - 0
No33 - f3a31 - N (0) - (1) - -1,
and since their g.c.d. is one and hence divides g, then, by Corollary 5, f is equivalent to a form having b23 - 0 and b13 IR. By (3), b33 a f3 - 1. Hence f^.--f' . Lemma 10 is now
proved. However it may be desired to have at hand an explicit transformation carrying f into f' for purposes of later reference. Thus by (57) and (77),
(78) (a 1)0( + ( o)3 (-)/ --" and a suitable set of values satisfying (78) is
(79) C- a, Ot - 0o ," 1-.
By (79) and (56), 012 - al2, 022 - -1, and c., - 0. Thus a transformation sending f into ff is given by
(80) Na3 a12 1
(Cjk) - -1 0
0 0.
That transformation (80) is of determinant one is evident. Moreover, by computation, (48) and (54) hold. THEOREM 8. An properly primitive form f of determinant d ( 0 which primitively represents one is equivalent to
, b1y2 * 22 2,
f Ia bllY + b2272 2 +y3 2 2bl2yly2 + 2Nyjy3 where N is an preassigned integer. PROOF. By Lemma 7, f is equivalent to a form
f* Xl2 , a22x2 a.3x32 + 2al24lX, + 2al+lX3 X2a 23xx3
By Lemma 10, f*,. f'. Therefore, ,,f f.
COROLLARY 6. Any properly primi tive form f of determinant d 1 0 which primitively represents one is equivalent to the form
f"- b11y12 * b 22Y2 + 2b12yly2 +2yly3
PROOF. By Theorem 8, f'-af' with N any preassigned integer. Let N equal one.
Examples
The form of the title of this chapter, i.e_. the form f' with b - N, b23 - 0, possesses rather complex conditions that a general form f be equivalent to fL. These conditions as given in Lemma 5 are quite explicit but nevertheless cumbersome. However, if the relations (47) through (57) be considered as a set of working equations whereby given any form f, a form fl equivalent to f can be obtained, if such a form f exists, then Lemm 5 is quite useful. Consider the reduced, positive form
- 2 3x22+ 5x32 + 2xlx2 + 2xlx3 + 2x2x3
of determinant d - 22. If there should exist a form f# with b13 a N, b23 - O, which is equivalent to f, where N is arbitrary, then such a form f' can be obtained by Lemma 5. Lot N - 17. By (2),
L3 2013 c23 033
X26 - 013 + 302 3+ 033
X33 0 13 0 23+5033,
and by Theorem 6 there exist values of 013, 023, 033 such that g - 1. Such a set of values is 013 - 5v o23 a 4, and c33- 3. Then by (2) and (6), XI3- 17, 13" 20, Xf324, g- 1, X13 - 17, X23 - 20, and X33 24. By (49), N1 , 17. Relation (48) now becomes
17011 + 20c21 + 24031 - 17,
one solution of which is 01 - 1, 02 - 0, 03 - 0. The values of the three constants (47) are
237 (24n - 20k -1) + 85
237 (17k - 24s) + 68 237 (20s - 17n) + 51
and, taking s - n - k - O, the expressions (47) have respective values -152, 68, and 51, whose g.c.d. is one. By (57),
-152( + 68 + 51. Y - -1 ,
one of whose solutions is Ol a -i, /I - 0, and -3. By
(56), 012 - -60, 022 - 27, and c32 - 20, Hence the transformation
(i -60 5
(0jk) - 0 27
0 20 3
of determinant one sends f into the desired form f'. By matrix computation,
(1 0 2) (0 1 1 1 -60 5
(f -0 27 2 1 3 1 0 27 4 E5E 4 $ 1 5 0 2 3)
73 41 27 4)- 3 6827 0
(17 20 24o( 20o 0 237)
so that the desired form f' is
f' 2 12 + 6827Y2 2 + 237y3 - 146y1y2 + 34yly3
No examples are given here of Theorem 7 since its application is similar to that of Lemma 5, which was just illus tra ted.
Lemma 10 has an interesting application, For given
any form f of the type described in the statement of the Lemma, not only does there exist a form f' equivalent to f and with b33 - 1, b13 - N, b23 - 0, but one can find the form f immediately by applying to f the transformation (80). Take as the form f
2 2 2
if - x,1 * 3x2 * 5x3 + 6xlx2 + 2XlX3 + l0x2x3,
and lot N - 11. Then by (80),
73
(a jk ( -1 0 (1 0 0 . Applioation of (ojk) to f gives the form
'K *J 22 y2,
f 1257I " + Y3 " 4y72 + 22y173
of the desired type.
CHAPTER V
CONDITIONS FOR EQUIVALENCE TO THE FORM
f WITH b13 - Op b23 - X9 AND RELATED FORMS
If one is given a form f and the problem to determine whether f is equivalent to
f I b1l112 * b22Y22 + b33Y32 ï¿½ 2b12y1y2 * 2My2y3 ,
one should first determine whether M is a multiple of b33. If M - Kb33, then conditions for equivalence of f and f' may be found in Lemma 3. If M is not a multiple of b,3, then apply to f the transformation
(81) (o 1 o)
0 0 -1
to obtain the equivalent form
f" =b22z12 + b11z22 + b33z32 + 2b12Z1z2 . 2Mz1z ,
which, after suitable changes of notation, is treated in Lemma 5 and Corollary 5. In both of these separate oases, ie. Lemmas 3 and 5, necessary and sufficient conditions for equivalence of f and fl are given. Throughout this dissertation various necessary conditions and sufficient conditions for equivalence are given separately.
74
Given a form f as above, the problem of determining
whether f is equivalent to some form ft is far from a routine matter. In the summary of this study a reference table will be furnished whereby the student may locate applicable tests for equivalence more readily. If no theorem, lemma, or corollary of this dissertation seems to apply to the problem at hand, suitable interchanges of variables, as accomplished by
(81), may yield results.
Two rather obvious but sometimes overlooked suggestions are made: (1) if testing whether f and f' are equivalent seems hopeless, interchange f with f', i.e., attempt to determine conditions that ft be equivalent to f rather than that f be equivalent to f'; (2) if the problem is still unresolved, then compute the adjoints and ' of f and f' respectively, for f f if and only if
CHAPTER VI
CONDITIONS FOR EQUIVALENCE TO THE FORM f WITH b12 " b13 - b23 -0
Professor E. H. Hadlook will present to the American
Mathematical Society the following theorem, which is stated here because of its usefulness in the further development of the theory of equivalence.
THEOREM. A necessary condition that there will exist a
linear transformation (Cjk) with IojkJ- 1, j,k-l,2,3,
which will take the form
f a aixixj aji " aij
into the equivalent form
'2 2 2 f - b1ly1 2 b22Y2 * b33Y3
is that f represents primitively a divisor f3 of d, where
f I X and where f3 equals g (X3' X2, X3),
X3 - ail1l3 * ai2O23 + ai3033 , i - 1,2,3.
A necessary and sufficient condition for the equivalence
of f and f' is that there exist integral values of 0
0
(i) u - U(C P 10(2,# 03) "jki jA'PO kj1 - ki j divides each of the three linear functions Y1 Y2' Y3 orï¿½f , CxS, O(3 whoere_.
(ii) Y i n kij j' kji a kij v i- 1,2,3, where the determinant associated with U is equal to zero, where all values of O(I, ' are excluded such that U( 2 -0, and where kij, ij - 1,2,3, are given
2i0 X2
(iii) kc, 2 a00 X" ax2 - 2a X,,X.
J
k3 k 23 k33
-a12 x33 2 a33X13X23 + al13X23X33 + a23X13X33
-a13 232 - a22X133 + a~2X33 + a.3X13X2
- ~32 2-2a 1 ~3 ï¿½ x ~
allX332 + a33X13 - 2alX13X33
-alX23X33 - a23X132 a 12X13X33 a3X13X2 a11X232 + a22X13 2 - 2a 12X13X23 ï¿½
Any indefinite form is excluded if i k2 a '" aS 33 0, or-any set X13, X23, X33 if (X13, X23# X33) - 0, Whore
is the adjoint form of f. Moreover, if (0Jk) exists, with IcJkI 1,, then c12, c22, c32 are given by
(iv)
012
022 c32
X23Y y x 33o - X-13,
X13 X230.
are integral solutions of a pair of
where o,
the equations
(v)
h K h2 a * h3 Y - i Bil0( + B12p + B13 - 0 , i - 1,2,3.
Also hl, ho, h3 are given by
(vi) hl " X230(3 - X33 0(2
h-2 X33 1 - Xl3C 3 h3 - - X23"(1
Further, Bj, j - 1,2,3, are the elements of the matrix
(vii) (hlki hkYc h3k22-h2kl2 23h2kl,
Mm jb:k,3-h3k11 h1c23-h.3k12 h1k 33-h3k13)
h2kl3-h3kl2 h2k23-h3k22 2k33h 3k23/ Finally , c21, c31 are given by (viii) Cll. X23k X33n
S21 " X33s - 13k
0 31 Xl3n- X23',
where a, n, k are integral solutions of the system
(ix) A s B n + C k - 0
C12S c22n + c32k - -1,
and where
(x)
A kllC + k12 + k133 B-a k2"( + k22/ + k 23f 0 - l30( " k23p " k 33Y"
I
Although the above theorem was designed to determine
whether a particular form f is equivalent to some general form f having no cross-products, it can also be applied to the problem of ascertaining whether two particular forms are equivalent, and even more important, if they are equivalent, to the problem of finding the transformation sending one into the other. L. E. Dickson gives two indefinite forms
f - -x12 + 2x 2 2 16x32 + 2x2x3 and f' -12 - y22 + 33y32 , of determinant -33, and states that a transformation sending f into f# was found.1 Suoh a transformation was sought for the purpose of tabulation of reduced, indefinite forms. Use of the preceding Theorem gives the result explicitly and with a minimum of work of the trial-and-error variety. By (3), f3 n b 33 so that the values of c13, c23, c3 must cause f to represent 33. Take the values of the ai3 as 33, 29, and 8 respectively. Then -1x-33, - 66, and X' --99 so that
spc ivly TenX3 X23 X3
g - 33 - b33, which satisfies Corollary 3. Hence by (6), X 13"-1, X23 - 2, and X 33- 3* By (iii), kl1 a -4, 2 "
-29, k13r -8, k -25, k- -7, andk -2. Then
u -W -34 2 25 W22 - 2 <32 _ 58 C 1 < 1 - l - 14 2 .
By (ii), YI - 34 " 29 W2 - 80(3
p. 149.
1L. E. Dickson, Studies in the Theory of Numbers,
Y2 - -290(i - 250(2 -7% Y 3 "-8ï¿½ .1 7W2 -20 3
The set of values U- I, O( - lo 0(% - 1 gives U- -34 -25
-2 +58 +16 -14 - -1, or 1UI - 1. Then YI -3, -3, Y3 a
-1. Thus jUI divides each of YI Y20 and Y3" But this is sufficient that f^/ft; hence the elements of the transformation (cjk) sending f into ff' can be determined. By (vi),
h." ( 2)( 1)- (-3)( 1) 5 h 2 (-3)(-')- (-')( 1)- 4 h3- (-l)( 1) - (2)(-l) - 1.
The elements of the first row of the matrix M of (vii) are
B - ( 5)(-29)- ( 4)(-34) a -9 B 2- ( 5)(-25)- ( 4)(-29) - -9 B3- ( 5)(- 7) - ( 4)(- 8) - -3. Then the system (v) of equations is
Employing Cramer's Rule,
-2 0 -1, 3 ,3.
The ref ore,
Ain2cx 3, Ys +3,
and by (x),
A - -34 ( C) -29(-2O4 -1) -8(3oe +3) a 5 B a -29( < -25(-2 o -1) -7(3C< +3) - 4 c a - 8(W ) - 7(-20< -1) -2(3 I+3) - 1. The system (ix) is then
5s+4n+k-O
3 s + 3 n + k - -1,
to which Cramer's Rule is applied, yielding n - 1 - 2s, k - 3s - 4, a = s, where s is arbitrary. By (iv) the three ci2 are computed and found to be
012 ( 2)(30( +3) (-3)(-2 (-1) -3
0 -22 (-3)( 0( - (-1)(3 + )
0 32 (-l)(-2K-i)- ( 2)( )<7Finally the first column of (cjk) is given by (viii).
o011- ( 2)(3s -4) - (-3)( 1- 28) = -5 021 - (-3)( a ) - (-1)(3s -4) - -4 o31 - (-1)( 1-2s) - ( 2)( s ) -1 Hence the transformation
(0 jk) -4 29 (-l 1 8
carries f into f'. This is not, however, the exact transformation given by Dickson. Postmultiplication of (Cjk) by the automorph
0 -1 01
1 0 0
0 0 1
of fI gives the transformation given by Dickson, namely,
3 5 33/
3 4 29
1 1 a8 .
Certain other changes of sign and interchanges of columns are possible, due to the nature of the form f', in that f' has no cross-product terms and has two coefficients equal.
In the process of finding the transformation (ajk)
sending f into fl, a crucial point is the choice of the elements c13, 023, c33 of the transformation. In the transformation obtained above, the elements ci3 used were those given by Dickson. This choice is not at all arbitrary, for the ci3 must satisfy, by (3), f3 a b33' or in this case
(82) -c132 2o232 -16c332 +2e c -33 - 33. Furthermore, Corollary 4 must be satisfied, so that c13 must be a multiple of b33, since a11 equals -1. By (82), c13 cannot be even. Hence o13 is an odd multiple of b33. The two conditions just mentioned are necessary conditions but are not sufficient; another condition is that there must exist columns Cil and ai2P i - 1,2,3, for which f1 and f2 equal b11 and b22 respectively. Finally, the determinant I , cJki must equal one.
Now if the choice one makes for the elements of the third column gives the third column of an actual transformation sending f into f t then the preceding Theorem will give the other elements of the transformation. Finding the third column elements remains largely a problem of trial and error.
An example is given here of a set c13, c23P c5 for which f3 equals b33, yet where there exists no possible choice of columns cil and ci2 causing f to be sent into f'.
Apply to
f, 2 2 2 f "Y1 - 2 + 33Y32 the transformation
0 0O
0 1 0
-1 0 )
to obtain an equivalent form "U 2 2 2 f 33Y1 Y22 Y3
Now if there exists a transformation (djk) sending f into f then there exists a transformation
( 0 0 1)
(cjk) - (djk) 0 0
which sends f into f". Conversely, if there exists a transformation (Cjk) sending f into f', then there exists a transformation
(djk) , (0jk) 0 1 0 i 0 0)
which carries f into f for
0 0 1 0 0 I1 0 0.()
0 1 0 0 1 0 0 10
the identity transformation. One must seek, then, a transformation (CJk) sending f into f". The elements of the third column must satisfy f3 " b33' by (3), or
(83) -o13 2 2023 - 160332 + 2c23c33 --1 An obvious choice of integers satisfying (83) is 013 1, 023 - 033 -0. Then by (2),
,5 alll " -i
X23 a12013
X33 a13c13 " 0
so that g a 1, and therefore, X3 * Xi5, i - 1,2,3. By (1),
b13 ' (-l)(cll) + (0)(c21) + (0)(031) 0
If this last relation is to be satisfied, then al. - O. But since, by (3), bll must equal fl, or
12 + 2 212 16a312 + 2e21c31
then an even integer equals an odd integer, a contradiction.
This illustrates that extreme care must be taken in choosing the elements of the third column so that the choice does not go beyond the realm of possibility.
Another pair of equivalent forms given by Dickson is given below, and the transformation will be found as before.2
2 2 2
f - xI1 + 2x 2 2- 28x 3 2+ 2x 2x 3
- Y22 + 57y32
Again, by Corollary 4, 013 is a multiple of b - 57, and since 013 is odd, take c13- 57. Then since g - 57 by Corollary 3, compute
I -57
X3- 2c23 + 033
43 c23 - 28c33 , and use the resulting congruences
2c23= -c33 (mod 57)
023_28c33 (mod 57)
By (3), f(57, c23, c33) - 57, so that
2c 23c3 - (57)2 + 57 or 0232 + c23c55 - 14c32 - 1653 from which it is easily determined that c must be odd and c33 even. The pair c23- 149, c33 - 44 is discovered to sat2Ibido, p. 149.
isfy all the above requirements. Now XI - -57, X '3 = 342, f 13 23
X1. a -1083, so that g a 57, and by (6), X13 - -1, X23 6,
33
and X 33-'-19. By (iii), kll =-58, k 2"--149, k13"a-44P k22 a -389, k23 - -115, k33 - -34, and by (i),
-U 58'W12 +389 " 2 +.34O( 2 + 298 1) + 880( 0(+230q q 1 2 3 12 103 2 3 By (ii),
- YI a 58i1 + 149 D2 + 44D(3 " Y2 " 149c I + 389 0(2 + 1150(3 - Y3 - 44Pi + 1150(2 + 34 0<3
U( 0, 3, -10) - (-1); Y1(O, 3, -10) - 7, Y2(0, 3, -10) - 17, and Y3(0, 3, -10) o 5. Hence by the Theorem of this chapter, since integers 0iW 0, 0<2= 3, < 3- -10 exist for which IuI divides each Yi' i - 1,2,3, then the transformation (cjk) of determinant one exists which carries f into f' . Then by (vi),
h_ ( 6)(10) - (-19)(-3) - 3 h2 (-19)( 0) - (-1)(10) -lO h2 (-1)(-3) - ( 6 )( 0) - 3 By (vii),
B 11 133, B12" 323, B -95. The system (v) becomes
3 ï¿½+ 17,e +3 5 --1 0 179 +s'( - 0.
Solving by Cramer's Rule gives the three values
0
Then Y -17 (mod 19), or - 19k + 2, where k is arbitrary. Hence
to(- k 10, -6k - i, l 9k + 2. By (iv), e12 --7, 022 -17, a52 - -5. By (x),
A - - 58(k + 1) - 149(-6k - 1) - 44(19k + 2) - 3 B a -149(k + 1) - 389(-6k- 1) -115(19k + 2) - 10
C - - 44(k + 1) - 115(-6k - 1) - 34(19k + 2) - 3 , and the system (ix) is
3 s + 10 n 3 k- 0
7 a + 17 n + 5 k 1, which by Cramer's Rule gives, finally,
s - m + I, n a -6m - 3, k -19m + 9.
By (viii), oil - -3, C21 a -10, c31 - -3. Then the transformation
-3 -7 57\
i0 -17 149
-3 -5 44
carries f into f t. In obtaining this particular transformation, the values used for O i 2 and o( were 0, -3, and -10, re1' 2'3
spectively. The choice of 0, -3, and 10 would have been equally appropriate and would have resulted in the slightly different transformation
88
(3 7 5
10 17 149
3 5 44/. It is the latter transformation which is given by Dickson.
CHAPTER VII
APPLICATIONS TO THE THEORY OF TABULATION OF AUTOMORPS
Given any form f there exists at least one transformation sending the form f into itself. Such a transformation is called an automorph of f. In the study of whether a form f is equivalent to some form f', it is often quite useful to know the automorphs of the form f or of the form f'. This is
true because given one transformation of f into f , one can find all such transformations of f into f'. if all the auto1P
morphs of f (or of f') are known.
In studying the automorphs of the form f' of Chapter
VI, an immediate corollary of the Theorem of E. H. Hadlock of that chapter is useful.
COROLLARY. A necessary and sufficient condition that the f f"allXl2 * a22 2 * a33x32 and I blY2 *b22y22
2
+ b33y3 be equivalent is that there exist integral values of NJ I, 02, 0(3 for which 1u1 divides each Yi' i - 1,2,3, where IUI is defined by (i) Yi by (ii). and ki , i, j a 1, 2, 3, ?
1L. E. Dickson, History of the Theory of Numbers III, 210.
(84) kll - a22X332 a33 X232
k12 - -a33Xl323 k13 - -a22X13X33
k22 " allX32 32
kc23 a -a 11 x223X33
k33 0 allx232 + a22X32
Consider the positive form f - a11x12 + a22x22
a33x32, where 0<<33g '22< all In computing the automorphs of f, one seeks all transformations (cjk) sending f into f' - f, so that aii - bii, aij - bij - O, i ( J, ', J 1,2,. It is necessary, by (3), that f3 - b33 - a33' and all sets of integers c13, c231 033 such that f(c13, c23, 033) a33. if a33
When c13 O, 023 O 33 1 then by (2), X3
a., ; thusg=a33 3m=23m0, X33 . Therefore, by (84), kll- a22# k22 - all, and k12 a k13k23" k5 a 0. By (i) and (ii),
U " a2212 + a 11(22
and Y1 .a220(l # Y2 - all<2 Y3 -0. It is required that each Yi/U be an integer. Since Y3 is the integer zero, then the only requirements are that YI/U and Y2/U be integers. This last condition is true if and only if either 0(l ' a C (2 a 1l or C( -ï¿½i -+ll 2 - 0. or if
IPte1 > 1, tohn112 a22 W12> a220 and hence
a22( 2 + a :110( 22> a22jD(4l so that a22( 1
a22K12 + all K2
is not integral. Similarly, ifj(Q21>l, then Jul does not divide Y2, Therefore it is observed that when c13 0 , 023 = O, c33 ' 1, then either of two cases occur. Case (1) comprises the values CI O 0, P .5 and case (2), the values COI= -ï¿½I, (2 ' 0, 0(3. 0(3.
By (vi), case (1) gives hI = 1, h2 - h3 - 0. By
(vii), B11 * O, B12 + all, B13-= 0. The system (v) yields C< al, u0, f= (. Then the values of A, B, C, by
(x), are A l - 1, B 0 0, C - 0. The system (ix) is s - 0, n I l, kk, so that c12 - 0, a22 1, C32 - 0. Finally, ell " l, 0 21 = ac31 = O.
Case (2) gives, by the same method, the values hl 0, h2= 1, h3 - 0; B11- ; a22, B12 = B13 = O - O, - 0 1, Y ( ; c12 - , 022 a 032 - O. Continuing this process, 011 - 0, 021 = ,31 = 0. The transformations just obtained, namely, those of case (2) are not automorphs of f - a11x12 +
2 2by3,f
a22x2 + a33x3 , a,> a22>a33, for by (3), f2 o a l a22 "
Taking the second set of values for the ci3, i - 1, 2, 3, i.e., 130, c23UO, c33 -I, then 3 O, X23 O,
X'3 -a3. Hence g = a3>0, so that XI3 0 O, X23 a O, X.33
-1. By (84)0 k - a22, k22 a all, k12 0 k13 .23 0 '33 * O. The only values for 0(i' ' and % are, case (3), IC 0, 1,.ol 0 ( , and, case (4), 0( 1 0, .
2 a 3 1(2 3
Case (3) yields a11 - + 1, 021. a 31 - 0; a22 = ! i, 012 = c32 a 0. The transformations made up of those elements constitute automorphs of f, as can be established by subjecting f to the transformations.
Case (4) yields no automorph of f, for the values obtained, 011 -, 021 " ; i, '31 0, c12 - ;, 022 - c32 a 0 give transformations which, by (3), give f2 n a119 a22 "
Hence all sets c13 have been obtained satisfying f3
B
b33,. all sets o'i were found satisfying u I Y,
33~ Y~,i -m 1,2,3; thus all transformations sending f into itself have been exhibited. They may be written in matrix form as follows. From case (1), two such automorphs are
Al: 1 0 A2: 0 -i
0 0 1 0 0 1)
p
From case (3), two
by
A3: tE
additional
0
0
automorphs A3
1 A4: 0
0
and A4
are given
O
0
-1 )
When transformations AI, A2, A3, and A4 are applied to the form f, they are found to be automorphs of f.
The four transformations obtained which are not automorphs of f, when a11> a22>a 3, namely, those of case (2),
A5:5 1 0 0 and A6: 1 0 0
and those of case (4),
0i o1 00 1 o0
A7: 0 0 and A8: ( 0 0
(0 0 -1i 0 0 -I1,
send f into the equivalent form f 2 2 2 f- a22yI 1 ally2 * a33Y3
Theref ore in the case of a form f having a22 ' all, for example the form f - 2x 12 + 222 + x32 , then f has the additional four automorphs A., A6, A7, and A8.
Consider now the form f with all>a22 a a33 - Then
f3 & b33 is satisfied by the sets c13, a239 c33 respectively, 0, 0, 1; 0, 0, -1; 0, 1, 0; and 0, -1, 0. The former two sets of values have already been employed to obtain automorphs A,# A2, A., and A4. The third set of values, 0, 1, and 0, will be investigated next.
When o13 0,c23 l, c33 0, then XI3 0X3 a - 0, and g - a By (84), k1l a33, k33 a11, a22,1 X33 - 22. 33 a all
and k12 - k13 -k22 k23 .0. Then
U- a 3312+ all 3 2 Y1 " a33 *' PY2 0, Y3 - all(3" The two resulting sets of values for the C(i' i -12,3, are given by case (5), 0(l- 0, C- 2'( 0( - 2 1, and case (6),
2 2 3
1 t C" 2 X 2 9 , 3M 0.
Case (5) gives the following values: h1 - 1 1, h2 h3 -O;B21 -B22-O, B23--l, c12 -22-O, c32 -.l, c 11 i, c21 0,and c - 0.
Case (6), after similar computation, yields all - 0, c21 - 0, c31 - ; i ,c12 - ; I, c22 - 0, and c32 - 0.
The transformations given by case (5) are
A9: 0 1 and A0: 0 0 i
0 1 0) 0 -1 0,
and those of case (6) are
All: 0 1 and A12: 0 0 1
"1 0 0 )1 0 0)
Only A9 and A10 are automorphs of f when all> a22 - a33, for All and A12 send f into
f - a33Y1 + a11Y2 + a22Y3 ï¿½
When 013 - 0, 023 - -1, and c33 - O, four new transformations arise as follows:
113 0 -i 111 0 -0 0 0 ,0 -1 0
A15 0 0 -1 A16: 0 0 -1
1 0 0 -1 0 0ï¿½
Of these only the first two are automorphs of f when al1>a22 a33* Hence when f is so restricted, all automorphs of f are given by A1, A2, A3, A4, A8, A10, A13, and A14.
Finally the form f will be considered when all of its non-zero coefficients are equal, i.e. , al1 - 22 - a33* Then other possible values for the ai3 are 1, 0, 0 and -1, 0, 0.
When c13 i, 023 33 0, then I5 all' X13
X3 -0;g-a -a22 - a33, X13 - l, X23 - 0, and X33 - 0. Then ki * kl2 k13 a k23 " 0; k22 * a33' k33 " a22 ' so that either o nl- e' 2" 0' o - -1 1, or Cx'I- 0<1, 01" + IV 0< - 0. These cases yield four automorphs of f, namely,
A17: 1 0 0 ) 18: - 0 0
0 1 0), 0 -1 0)
A19: 1 A0: 0 -1 0
1 0 0)'V 1 0 0).