THE MINIMUM OF h BINARY QUARTIC FORM (I),
B y C. S. D A V I S
TRINITY COLLEGE~ CAMBRIDGE.
I. I n t r o d u c t i o n .
1. T h e q u e s t i o n of t h e lower b o u n d or m i n i m u m of an algebraic f o r m ~ ( X 1 . . .
Xm)
for i n t e g e r values, n o t all zero, of t h e variables x I . . . x m is an i m p o r t a n t one a n d has a t t r a c t e d a g r e a t deal of a t t e n t i o n for m a n y years. T h e p r o b l e m is a difficult one, h o w e v e r , a n d r e l a t i v e l y few results are k n o w n .
Confining o u r a t t e n t i o n t o t h e case of f o r m s with real coefficients in t h e t w o variables x, y, t h e results for t h e b i n a r y q u a d r a t i c f o r m are classical. If
~(x, y) : a x 2 ~ b x y ~ - c y ~
is a b i n a r y q u a d r a t i c of d i s c r i m i n a n t D z b2--4ac, t h e result is t h a t t h e r e exist integers x, y, n o t b o t h zero, such t h a t
(1.1) [~(x, Y)I ~-- k~/IDI,
where k = ~ w h e n D < 0 ( t h a t is, w h e n t h e q u a d r a t i c has c o m p l e x roots1), a n d 1 ]c = ~ w h e n D > 0 ( t h a t is, w h e n t h e q u a d r a t i c has real roots). W h e n D = 0 t h e 1 lower b o u n d is t r i v i a l l y f o u n d to be zero. T h e s e results are best possible, in t h e sense t h a t t h e i n e q u a l i t y (1.1) is no longer t r u e for all f o r m s of t h e t y p e specified if t h e c o n s t a n t k is r e p l a c e d b y a smaller n u m b e r .
E s t i m a t e s for t h e lower b o u n d of a b i n a r y cubic f o r m were g i v e n m a n y y e a r s ago b y ARNDT ( l ) a n d HERMITE ( 2 ) , b u t t h e best possible results were o b t a i n e d o n l y r e c e n t l y , b y 1VIORDELL (3). If n o w
1 B y t h e " r o o t s " of a b i n a r y f o r m ~(x, y) we m e a n t h e r o o t s of ~(x, 1) = 0.
264 C.S. Davis.
q~(x, y) = ax3 + b x 2 y + c x y 2 + d y 3 is a b i n a r y cubic with d e t e r m i n a n t , or n e g a t i v e d i s c r i m i n a n t ,
D = 27a2d ~ - 18abcd--b2ce+4ac3-4-4db a , t h e result is t h a t
(1.2) I~( z, Y)I ~ t~lDI t
for integers x, y # 0, 0, w h e r e k = 2 3 - t if D > 0 a n d k --~ 49 -'~ if D < 0. These cases c o r r e s p o n d to cubics w i t h just one a n d t h r e e real r o o t s respectively. Again t h e lower b o u n d is zero if D = 0, w h e n t h e cubic has a r e p e a t e d root.
T h e p u r p o s e of this i n v e s t i g a t i o n is to p r o v i d e e s t i m a t e s for t h e lower b o u n d s of b i n a r y q u a r t i c forms. E s t i m a t e s of this sort h a v e been given b y HEnMITE (4) a n d Jt~LIA (5), b u t t h e i r results are in general v e r y crude. A few best possible results are k n o w n , these being d u e t o MORDELL (6, 7), DERRY (8) a n d MAHLER (9). These results will be m e n t i o n e d l a t e r in t h e a p p r o p r i a t e places. W e find here t h e best possible results in m a n y cases, including all q u a r t i c forms with four distinct c o m p l e x roots, a n d an infinity of (in f a c t " m o s t " ) f o r m s with f o u r d i s t i n c t real roots. I n all cases e s t i m a t e s are given which are, with one e x c e p t i o n , b e t t e r t h a n a n y p r e v i o u s l y k n o w n .
2. Much light is t h r o w n on these p r o b l e m s b y considering t h e m f r o m a r a t h e r m o r e general p o i n t of view. I n s t e a d of seeking d i r e c t l y t h e lower b o u n d of q(x, y) for integers x, y, we m i g h t i n v e s t i g a t e its lower b o u n d for points x, y of a lattice L of d e t e r m i n a n t A defined b y
(2.1) x = ~ + f l ~ , y = ~ + ~ ,
where ~, fl, y, ~ are real n u m b e r s , d -- I ~ d - f l ~ l :~ 0, a n d ~, V t a k e all i n t e g e r values.
Considering all possible lattices, t h a t is all values of ~, fi, y, b, w i t h given A, we are led to seek a c o n s t a n t k such t h a t
(2.2) Iq~( x, Y)I ~ ( k + s ) Alt'~
for some point x, y 4= O, 0 of e v e r y lattice of d e t e r m i n a n t A, w h e r e n is t h e degree of t h e f o r m ~(x, y) a n d s is a n y positive n u t , bet.
N o w for a given linear t r a n s f o r m a t i o n (2.1), we m a y c o n s i d e r t h e f o r m ~(x, y), of d i s c r i m i n a n t D~ ~: 0, say, as a f o r m ~(~, ~]) in t h e variables $, ~, with d i s c r i m i n a n t D ~- A~(~-I)D~. T h e n t h e result (2.2) m a y be e x p r e s s e d in t h e f o r m : T h e r e exist integers ~, ~, n o t b o t h zero, such t h a t
T h e M i n i m u m of a B i n a r y Q u a r t i e F o r m (I). 2 6 5
1
(2.3) I~~ ~)l ~< ( k + e ) \ D ~ / '
where ~o(~, V) is a n y f o r m of d i s e r i m i n a n t D o b t a i n a b l e f r o m ~(x, y) b y a real linear t r a n s f o r m a t i o n . U n d e r these c i r c u m s t a n c e s we shaI1 say, for b r e v i t y , t h a t ~v(~, r]) a n d 7(x, y) are transformable into each other, a n d we shall refer to ~v(x, y) as a standard form for t h e class of f o r m s ~(~, ri). 1 Clearly a s t a n d a r d f o r m is n o t u n i q u e ; we m a y select a n y c o n v e n i e n t m e m b e r of t h e class of f o r m s t r a n s f o r m a b l e into each other.
W e r e m a r k here t h a t if t h e f o r m y:(~, V) possesses a n o t h e r i n v a r i a n t , say I of i n d e x l, so t h a t I = zlzIe, we m a y , if I~ 4= 0, replace t h e r i g h t h a n d side of (2.3) b y
n
( k + e ) ( I / I ~ ) 2 t . I n t h e following we shall c o n t e n t ourselves w i t h s t a t i n g t h e results in one or o t h e r f o r m .
I n t h e case of b i n a r y q u a d r a t i c s we m a y t a k e xy or x2--y 2 as s t a n d a r d f o r m when D > 0, a n d x2@y 2 w h e n D < 0. F o r b i n a r y cubies we m i g h t select xa~-y a w h e n D > 0, a n d xy(x4-y) w h e n D < 0. T h e facts are necessarily m o r e c o m p l e x for a b i n a r y quartic, since t h e possibilities r e g a r d i n g r e a l i t y a n d e q u a l i t y of r o o t s are t h e n m o r e n u m e r o u s . F u r t h e r we will find it necessary in some cases to use a s t a n d a r d f o r m c o n t a i n i n g a p a r a m e t e r .
Our p r o b l e m is t h u s r e d u c e d to one in t h e g e o m e t r y of n u m b e r s , n a m e l y to find a c o n s t a n t /c, p r e f e r a b l y best possible, such t h a t t h e r e exists a p o i n t x, y ~ - 0 , 0 of e v e r y lattice L of d e t e r m i n a n t A, with
(2.4) If(x, y)] < (lc+~)J 2 ,
where f(x, y) is a n a p p r o p r i a t e s t a n d a r d f o r m . W e consider t h e region ~ defined b y If( x , Y)t <-- l ,
a n d seek t h e lower b o u n d A* of t h e d e t e r m i n a n t s of lattices admissible with r e s p e c t to c)~, t h a t is ones w i t h no p o i n t o t h e r t h a n t h e origin as an i n n e r p o i n t of ~)~. T h e n t h e best possible v a l u e of/c, say k*, is 1/zl *z, a n d if t h e lower b o u n d is a t t a i n e d b y some admissible l a t t i c e with a p o i n t on t h e b o u n d a r y of ~1~ we m a y p u t s = 0 in (2.4).
An admissible l a t t i c e w i t h A = A* is called a critical lattice, a n d t h e c o r r e s p o n d i n g f o r m ~0(~, ~) a critical form. If we c a n n o t d e t e r m i n e A*, b u t can show t h a t e v e r y lattice of d e t e r m i n a n t A' (say) has a p o i n t o t h e r t h a n t h e origin in ~)~, it follows t h a t ]c* _< 1/A '~.
1 N o t e t h a t a canonical f o r m , in the u s u a l algebraic sense, need n o t be a s t a n d a r d f o r m ; for it m a y n o t be o b t a i n a b l e b y a real linear t r a n s f o r m a t i o n .
266 C.S. Davis.
3. Consider a b i n a r y q u a r t i c
~( ~, ~]) ~-- a~4-~ 4b~3~-~ 6c~2~2 ~ - 4 d ~ 3 ~-e~4
w i t h real coefficients a, b, c, d, e. This f o r m has t w o irreducible i n v a r i a n t s ~ a n d ~ , of indices 4 a n d 6 respectively, given b y
=- a e - - 4 b d ~- 3c 2 ,
~ = a b
b c c d T h e d i s c r i m i n a n t of t h e f o r m is
C
d
e
~_ ace ~- 2 b c d - - a d 2 - - e b 2 - - c 3 .
= ~ 3 - - 2 7 ~ 2 .
W e m u s t e x a m i n e t h e n a t u r e of t h e r o o t s of ~v(t, 1) --~ 0, a n d for this p u r p o s e we need to i n t r o d u c e t h e f u r t h e r expressions
2 ~
I t a p p e a r s difficult t o give an e x a c t reference to t h e results we need. T h e q u e s t i o n was dealt with in full detail b y ARZ~DT (10), b u t he does n o t e m p l o y t h e i n v a r i a n t s a n d his n o t a t i o n differs f r o m ours. BURNSIDE a n d PANTON (11) m a y also be con- sulted. F o r t h e s t a n d a r d forms, a n d a general discussion of t h e algebra, of t h e quartic, reference m a y be m a d e to t h e works of SALMON (12) or ELLIOTT (13). I n giving s t a n d a r d f o r m s we suppose, as we m a y f o r o u r p u r p o s e , t h a t a > 0.
W e s u m m a r i s e below t h e results we need.
@<0.
9 > 0 .
9 = 0 .
T w o d i f f e r e n t real roots a n d t w o c o m p l e x roots. S t a n d a r d f o r m x ~-~ 6 m x 2 y 2 - - y 4 .
(a) ~r~ > 0, ~ > 0. F o u r d i s t i n c t real roots.
(b) Otherwise. F o u r d i s t i n c t c o m p l e x roots. S t a n d a r d f o r m in each ease x 4_~ 6mx2y2_~ y4 .
I f ~ = 0, yJ has t h r e e equal r o o t s a n d so is of one of t h e f o r m s a(~--~o~) 4, a($--~0~)3($--c0'~]), co =~ ~0'. W e n e e d n o t distinguish b e t w e e n these cases.
I f ~ # 0 se~;eral cases arise.
(a) ~ - - 0. T h e n ~ = a ( ~ - - c o ~ ) 2 ( ~ - - ~ ' ~ ) 2, o~ :~ r (i) ~ > 0. to, w' real.
(ii) ~/~ < 0. co, ~0' complex.
T h e M i n i m u m of a B i n a r y Q u a r t i c F o r m (I).
(b) ~ 4= O. T h e n ~v = a(~--o~y(~--~o'rl)(~--o/qT), where o~, co', ~o"
different.
(i) ~ f > O. o/, co" real.
S t a n d a r d f o r m x2(x2--y2).
(ii) ~ < O. ~o', ~ " complex.
S t a n d a r d f o r m x2(x2§
267 are all
II. Quartics w i t h ~ > 0 (four c o m p l e x roots).
4. We consider first t h e case of forms with f o u r d i s t i n c t complex roots. W e find t h e best possible results for this case; later (in w 22) we discuss t h e results p r e v i o u s l y k n o w n .
Here we m a y t a k e t h e usual canonical form
(4.1)
f(x,
y) = x ~ 4- 6mx2y 'z ~ y4as t h e s t a n d a r d f o r m ; t h e p a r a m e t e r m is a f u n c t i o n of t h e i n v a r i a n t s ~ a n d ~ .1 T h e i n v a r i a n t s I a n d J a n d t h e d i s c r i m i n a n t D of this s t a n d a r d f o r m are given b y (4.2) I - - l + 3 m 2, J - - m ( 1 - - m " ) , D = ( 1 - - 9 m 2 ) ~.
T h e n a t u r e of t h e roots being u n c h a n g e d b y a real linear s u b s t i t u t i o n , t h e roots of f(t, 1) = 0 will be c o m p l e x a n d distinct 9 N o w f(t, l) = 0 gives
(4.3) t 2 = - - 3m + ~ / ( 9 m 2 - 1 ) ,
a n d so this q u a n t i t y m u s t be n e g a t i v e a n d t w o - v a l u e d , or complex. H e n c e 9m 2 # 1.
I f 9m 2 < l, t 2 is complex, while if 9m 2 > 1, t 2 < 0 if, a n d only if, m > 0. T h u s
~ n > - - ~ , m ~ = 1 89
I f we a p p l y t h e s u b s t i t u t i o n
x = z ( X - - Y ) , y = :~(X + Y) ,
where z 4 = 89 -~, which is a real s u b s t i t u t i o n for m > --~, we o b t a i n f ( x , y) = X ~ + 6 M X 2 Y 2 + y 4 ,
1 - - m
where M - - 9 N o w if m > 1 ~, we find - - 3 l < M < ~, so it is e n o u g h to t a k e 1 1 ~- 3m
~ < m < ~ .
1 F o r g i v e n ~ a n d ~ , m in (4.1) m a y t a k e in g e n e r a l a n y one of six v a l u e s ; b u t o n l y t w o of t h e s e arise f r o m real t r a n s f o r m a t i o n s .
268 C.S. Davis.
A l t h o u g h t h e r e d u c t i o n of a q u a r t i e to its c a n o n i c a l f o r m is discussed a t l e n g t h in t h e references (12, 13) cited, a n d elsewhere, it seems i m p o s s i b l e to give a reference to a n y s t r a i g h t f o r w a r d m e t h o d of f i n d i n g t h e p a r a m e t e r m. ~ H e n c e we give here a s i m p l e m e t h o d of c a l c u l a t i n g t h e m we h a v e d i s t i n g u i s h e d a b o v e .
I f A is t h e d e t e r m i n a n t of t h e s u b s t i t u t i o n t r a n s f o r m i n g f ( x , y) into V($, ~?), it is k n o w n t h a t A 2 m - z, say, satisfies
4 ~ = - J z + ~ = o .
F u r t h e r ~ = - . J 4 ( l @ 3 m 2 ) , so A 4 = ~ ) - - 3 z 2. T h u s we h a v e
Z 2 Z 2 X 2
m 2 - -
zj 4 - - ~Z]__3z 2 - - 3(1__x2 ) ,
3 z 2
w h e r e x 2 = 7 - satisfies
I f we write
4 x 3 - - 3 x 4 - = 0 .
we h a v e x = cos 0, w h e r e 0 ~ ~,1 89 a n d so m 2 = 89 cot 2 0. (This e x p r e s s i o n e x h i b i t s t h e six values of m.) N o t e , h o w e v e r , t h a t we r e q u i r e [m r < 89 a n d also t h a t t h e n , f o r a real t r a n s f o r m a t i o n , t h e signs of m a n d ~X m u s t be t h e s a m e , since
~ A%n(1--m3). T h e a p p r o p r i a t e v a l u e of m is t h u s g i v e n b y
(4.5) m = ~ 3 cot ,
w h e r e ~ is chosen t o s a t i s f y (4.4) a n d x < ~ < 2 m
5. }Ve n o w define a region ~j~ b y If(x, Y)I ~< 1, a n d i n v e s t i g a t e its shape. Since t h e r o o t s of f a r e c o m p l e x , we h a v e f ( x , y ) > 0 for real x, y, a n d so ) ~ is b o u n d e d b y t h e c u r v e (C g i v e n b y f ( x , y) = l. Since
f ( x , y ) = f ( - - x , y) = f ( y , x) ,
we see a t once t h a t ~ is s y m m e t r i c a l a b o u t t h e lines x ~ 0, y = 0, y z ~ x , a n d so also s y m m e t r i c a l a b o u t t h e origin O.
1 A n e x p l i c i t f o r m u l a f o r m 2 is g i v e n b y Fa)~ DE B~UNO (14), b u t it is in a n i n c o n v e n i e n t f o r m f o r o u r a p p l i c a t i o n .
2 T h i s d e t e r m i n e s a r e a l ~0, s i n c e ~ ~ ~ 3 - - 2 7 ~ 7 2 > 0.
The Minimum of a Binary Quartic Form (I). 269 I f ( x , y ) is a p o i n t on c5 a n d we write t = y / x , ~ e = x 2 + y 2 , Q > O , w e h a v e x4f(1, t ) - 1 a n d
l @ t 2
~ 2 _ _ v f ( ~ 1 , t) "
Since f ( 1 , t) ~ 0 for real t, ~ is e v e r y w h e r e finite. I t is also a s i n g l e - v a l u e d f u n c t i o n of t. I t follows t h a t ~ is a b o u n d e d s t a r d o m a i n . A brief c a l c u l a t i o n gives
d~ ~ = ( 1 - - 3 m ) t ( 1 - - t 2 ) f - ~ .
Since 1 - - 3 m > 0, clearly t = 0 gives a m i n i n m m a n d t = + 1 a m a x i m u m v a l u e of ~, a n d we d e d u c e
( 5 . 1 ) ~ < 0 2 < .
T h e c u r v e (~ h a s no d o u b l e points, so its p o i n t s of inflection are g i v e n b y its i n t e r s e c t i o n s w i t h t h e H e s s i a n
t h a t is, w i t h t h e lines w h e r e 2 is a n y r o o t of
m x 4 @ (1 - - 3m2)x2y 2 ~ - m y 4 = 0 ,
y = + 2 x , y = •
I f m : / 0, t h i s gives
(5.2)
m ) , 4 @ ( 1 - - 3 m e ) X 2 - ~ m 0 .
2m2,2 = 3 m 2 - - 1 4 - ~ / { ( 1 - - m 2 ) ( 1 - - 9 m 2 ) } ,
a n d 2 is real only if - - 1 < m < 0. I f m 0, we h a v e 2 = 0, b u t t h e p o i n t s of i n t e r - section, e. g. (0, 1), are t h e n p o i n t s of u n d u l a t i o n , n o t inflections. T h u s t h e region
is c o n v e x if 0 < m < ~, a n d n o n - c o n v e x if - - 1 < m < 0.
A t y p i c a l n o n - c o n v e x ~j~ is i l l u s t r a t e d in Fig. 1 (for m = - - ~ ) .
W h e n t h e region is n o n - c o n v e x , we shall call a n arc of (6' lying b e t w e e n t w o c o n s e c u t i v e p o i n t s of inflection, a n d including a n i n t e r s e c t i o n w i t h one of t h e axes, a " c o n c a v e " arc, a n d t h e r e m a i n i n g ares " c o n v e x " ares.
6. W h e n t h e r e g i o n ~ is c o n v e x , t h e t h e o r y of MI~I~OWSKI (15) concerning c o n v e x regions s y m m e t r i c a l a b o u t t h e origin m a y be applied. T h e p r o b l e m is t h u s r e d u c e d to t h a t of f i n d i n g t h e p a r a l l e l o g r a m of m i n i m u m a r e a which h a s one v e r t e x a t O a n d t h e o t h e r t h r e e v e r t i c e s on c6~. T h i s p r o b l e m was solved f o r m = 0 b y MORDELL (7), a n d l a t e r for 0 < m _< 1 b y DERRY (8). W e n o w s h o w t h a t t h e m e t h o d
270 c . S . Davis.
of M i n k o w s k i m a y be generMised to a p p l y to t h e regions u n d e r c o n s i d e r a t i o n , e v e n w h e n t h e y are n o n - c o n v e x .
W e r e q u i r e first
L e m m a 1. Let P : ( x , y) be a point on ct~ with x > O, 0 < t < 1, where t = y/x, and let s be the slope of the tangent to ~ at the point P. Then 1/s <_ t, with equality only when P is the point (1, 0).
F o r we h a v e
1 d x y y(y2~-3rnx2) y y ( l + 3 m ) ( x 2 - ~ y 2)
- - - t - - - - - - < 0 ,
s dy x x(x2@3my 2) x x(x2-~3my ~)
since x2-~-3my 2 >_ ( l ~ - 3 m ) y 2 _> 0. E q u a l i t y occurs o n l y if y = 0, a n d t h e n x = 1.
W e n o w p r o c e e d to p r o v e t h e f u n d a m e n t a l
L e m m a 2. A* is the lower bound of the determinants of admissible lattices with six points on ~ .
I t suffices to p r o v e t h a t a n y a d m i s s i b l e l a t t i c e m a y be d e f o r m e d i n t o a n o t h e r w i t h six p o i n t s on ~,, a n d w i t h a n e q u a l or s m a l l e r d e t e r m i n a n t . Since t h e region is b o u n d e d , a d m i s s i b l e l a t t i c e s c e r t a i n l y exist. L e t O A B C be a cell of an a d m i s s i b l e l a t t i c e w i t h no p o i n t on ~ , a n d i m a g i n e t h e l a t t i c e d e f o r m e d so t h a t A m o v e s a l o n g OA t o w a r d s 0, whilst A B r e m a i n s parallel to OC, u n t i l a p o i n t of t h e l a t t i c e first a p p e a r s on ~(~. Call this p o i n t A ' ; t h e n b y s y m m e t r y its i m a g e in 0 also lies on ~ . E v e r y p o i n t of OA' is in ~)~, a n d so OA' c o n t a i n s no l a t t i c e p o i n t s o t h e r t h a n 0 a n d A ' . H e n c e t h e r e is a p o i n t B ' such t h a t (A', B') is a basis of t h e d e f o r m e d lattice.
R e p e a t t h e a b o v e p r o c e d u r e w i t h t h e p o i n t B ' in place of A. I n this w a y we a r r i v e a t a n admissible l a t t i c e of s m a l l e r d e t e r m i n a n t w i t h f o u r p o i n t s on c(~.
W e n o w s h o w t h a t , if no o t h e r p o i n t of t h i s l a t t i c e is on c(~, we m a y a g a i n d e f o r m it to b r i n g a t h i r d p a i r of p o i n t s on to c6, a t t h e s a m e t i m e r e d u c i n g its d e t e r m i n a n t still f u r t h e r . L e t P a n d Q be t w o i n d e p e n d e n t s p o i n t s of t h e l a t t i c e on ~ . Such p o i n t s exist, since o n l y t w o l a t t i c e p o i n t s c a n be eollinear w i t h 0. Com- plete t h e p a r a l l e l o g r a m OPRQ; its a r e a S is a m u l t i p l e of A, t h e d e t e r m i n a n t of t h e lattice. W e will a s s u m e , w i t h o u t f u r t h e r r e p e t i t i o n , t h a t tile process to be d e s c r i b e d below is d i s c o n t i n u e d as s o o n as a f u r t h e r p o i n t of t h e l a t t i c e a p p e a r s on ct~. I f e i t h e r P or Q, s a y P , lies on a c o n v e x are, it m a y be m o v e d a l o n g c(~ in such a w a y t h a t its
1 T h a t is, n o t c o l l i n e a r w i t h O.
T h e M i n i m u m of a B i n a r y Q u a r t i c F o r m (I). 271 distance f r o m OQ, a n d so also A, decreases, u n t i l it lies on a concave arc a t a p o i n t w h e r e t h e t a n g e n t is parallel to OQ. H e n c e we m a y suppose t h a t b o t h P a n d Q lie on c o n c a v e arcs, a n d if P a n d Q are t h e points (xl, Yl), (x2, Y2) respectively, we m a y a s s u m e w i t h o u t loss of g e n e r a l i t y t h a t t h e y lie in t h e first q u a d r a n t , w i t h yl < x2, i . e . t h a t 0 <_ tl <_ lit 2 _< 1, where t~ = y l / x i a n d t 2 ~ y2/x~. N o w if t I < 1/t 2 w e
m a y d e c r e a s e A b y m o v i n g P along c(~ u n t i l t I = 1It 2. For, if sl is t h e slope o f t h e t a n g e n t a t P , we h a v e s I > 1/t~ > t2 b y L e m m a 1, since s~ > 0, a n d so d u r i n g this process P m o v e s t o w a r d s OQ. T h u s we m a y suppose h e r e a f t e r t h a t t~ = 1/t2, i. e.
2 2
x2 = Y~, Y2 ~ x~. T h e n S = x l y 2 - - x 2 y ~ = x ~ - - y 1, a n d d S / dXl Ylxl )
2X 1 | - - ~ 0 ,
\ d Y l - -
b y L e m m a 1. F u r t h e r , S ---- 0 w h e n x I z yl, a n d it follows t h a t we m a y r e d u c e A steadily to zero b y m o v i n g P a n d Q s i m u l t a n e o u s l y , keeping t I : lit 2, i. e. x 2 ~ Yl.
N o w A clearly has a p o s i t i v e lower b o u n d , since ~ contains t h e circle x 2 d - y 2 <_ 1, b y (5.1), a n d so t h e process of d e f o r m a t i o n m u s t be t e r m i n a t e d at some stage b y t h e a p p e a r a n c e of a n o t h e r pair of lattice points on cg. This p r o v e s t h e l e m m a :
W e r e m a r k here t h a t t h e lower b o u n d 3 * c a n n o t be a t t a i n e d b y a lattice unless it has six points on ~ . This follows i m m e d i a t e l y f r o m t h e a b o v e p r o o f if it is n o t e d
d S
t h a t , b y L e m m a 1, ~yy~ < 0 unless Yl z 0.
Our p r o b l e m is t h u s r e d u c e d to t h a t of finding t h e lower b o u n d of t h e d e t e r m i - n a n t s of all admissible lattices w i t h six points on (6'. Since, however, these six points m a y in general o c c u p y a v a r i e t y of different positions in t h e lattice, we h a v e still to solve a n u m b e r , p e r h a p s a large n u m b e r , of m i n i m u m problems. As these p r o b l e m s are s o m e w h a t t r o u b l e s o m e , we f i n d it c o n v e n i e n t to deal with t h e question b y this d i r e c t m e t h o d only w h e n t h e n u m b e r of m i n i m u m p r o b l e m s i n v o l v e d is n o t m o r e t h a n two. This will be so if t h e dimensions of ~j~ are sufficiently small; t h e r a n g e of values of m t h a t we consider in this w a y is d e t e r m i n e d b y t h e following l e m m a . L e m m a 3. L e t P ~ u A d - v B be a p o i n t o f a lattice L w i t h a reduced basis 1 ( A , B ) . S u p p o s e the p o i n t s A , B are not i n n e r p o i n t s o f the region ,~flc, w i t h m >_ m o ~-- --0.259. T h e n i f lu] >_ 2 or lvl > 2, P is not a p o i n t of ~ .
1 E v e r y l a t t i c e h a s a t l e a s t one reduced basis (A, B), w h i c h h a s t h e p r o p e r t y t h a t t h e a n g l e 0 b e t w e e n t h e v e c t o r s OA a n d OB s a t i s f i e s 60 ~ _< 0 _< 120~ see BAC~MAZUN (16).
272 C.S. Davis.
L e t OA = a, O B = b, a n d let 9o be t h e m a x i m u m v a l u e of t h e radius v e c t o r 9 of (:(~7'. T h e n
9(~ = < 3 .
Since 0 ~ 1 b y (5.1), we h a v e a ~ 1, b ~ 1. If lul > 2,
O P 2 > u2a~+v2b2--uvab = (vb-- 89 2 ~_ ~u 2 > 3 > 9~;
a n d t h e same result is clearly t r u e if Ivl ~ 2.
If m >_ m0, i t follows f r o m L e m m a 3 t h a t a n e c e s s a r y a n d sufficient c o n d i t i o n for a lattice L with a r e d u c e d basis (A, B) to be admissible is t h a t t h e points u A + v B With ]u I < 1, Ivl _< 1, (u, v) :~ (0, 0) should n o t be inner points of ~ . F u r t h e r , if L is admissible a n d has six points on ~ , t h e n luI _< 1, Ivl < 1 for each of these points. N o w t h e r e are j u s t eight points of L with lul < 1, Ivl ~ 1, e x c l u d i n g t h e origin u = v ---- 0. These eight points are t h e vertices a n d t h e m i d - p o i n t s of t h e sides of a p a r a l l e l o g r a m f o r m e d b y f o u r cells of t h e lattice which m e e t a t O. H e n c e t h e r e m a i n i n g t w o points are e i t h e r
(I) opposite vertices, or
(II) m i d - p o i n t s of opposite sides
of this parallelogram. Correspondingly, t h e six points of L on ~ are of t h e f o r m P , Q, P - Q a n d t h e i r images in O, or P , P + Q , P - Q a n d t h e i r images in O, where in each case P , Q is a basis (not necessarily reduced) of t h e lattice. W e note, h o w e v e r , t h a t t h e points P , Q, P - Q in t h e first case m u s t include a r e d u c e d basis. W e shall refer t o admissible lattices whose points on c6~ are of t h e t w o f o r m s m e n t i o n e d a b o v e as lattices of t y p e I a n d t y p e I I respectively.
T h u s our p r o b l e m , for m > m0, becomes t h a t of finding t h e lower b o u n d of t h e d e t e r m i n a n t s of lattices of t y p e I a n d t y p e I I . A l t h o u g h this provides us d i r e c t l y w i t h t h e answer to o u r original p r o b l e m o n l y w h e n m >_ m 0, we r e q u i r e some of t h e results in a n y case. I n consequence, we will n o t suppose in t h e following t h a t m ~ m 0 u n t i l this is e x p l i c i t l y s t a t e d .
7. W e n o w consider a l a t t i c e L of d e t e r m i n a n t A defined b y
(7.1) x = ~ + f l ~ ] , y = y~+&], A = ~xd--fiy,
a n d h a v i n g t h e points P : (~, y), Q: (fl, d) and P - Q = R: (~--fl, y--d) o n ( 6 ~. W r i t e
(7.2)
f ( x , y) = ~($, ~t) ~- a ~ ' + 4 b S a v + 6 c ~ e v e + 4 d ~ t a + e v ' 9T h e M i n i m u m o f a B i n a r y Q u a r t i c F o r m ( I ) . 273 On m a k i n g t h e s u b s t i t u t i o n , we find
( 7 . 3 ) a = o r 4 ,
(7.4) b : o ~ 3 f l + y s ~ + 3 m ~ x ~ , ( a 6 ~ - f l y ) ,
( 7 . 5 ) c =
a2fl~+~262+m(o~262+4o~flyO-~f12y2),
(7.6) d = o ~ f 1 3 ~ - ~ , 6 3 ~ t - - a m f l b ( ~ x O ~ - f l y ) ,
(7.7) e = f14+6mf1252+(34.
We p r o c e e d to d e d u c e a r e l a t i o n b e t w e e n A a n d t h e p a r a m e t e r c, 1 W e m i g h t , of course, o b t a i n a r e l a t i o n b e t w e e n A a n d some m o r e obvious p a r a m e t e r , e. g. y/~, b u t t h e p r o c e d u r e we follow has t h e g r e a t a d v a n t a g e of yielding simple criteria for deciding w h e t h e r or n o t a g i v e n l a t t i c e p o i n t is a p o i n t of ~ .
T h e points (2, ~) - - (1, 0), (0, 1) a n d (1, --1) lie on ~ , a n d so
(7.8) V(1, O) = V(0, 1) = F(1, --1) = 1 .
F r o m (7.2) a n d (7.8) we find (7.9)
(7.10)
F r o m
(7.11)
(7.12) Using (7.13) (7.14)
a - - e = l ,
4 ( b + d ) == 6c-~1 . t h e i n v a r i a n t s in (7.2) we h a v e
I A 4 = a e - - 4 b d + 3 c 2 , J A 6 = a c e ~ - 2 b c d - - a d 2 - - e b 2 - - c 3 .
(7.9) a n d (7.10), these b e c o m e
I A 4 = 1 - - 4 b d + 3 c ~ ,
J A 6 = 2 b d ( c + 1) + c - - c 3 - 1 . ( 6 c + 1) 2 . E l i m i n a t i n g b d b e t w e e n t h e last t w o e q u a t i o n s , we derive (7.15) 8c 3 - 1 2 c 2 ~ - 1 2 c + 7 - - 8 ( c + 1 ) I A 4 - - 1 6 J A 6 = 0 .
W e write d ~ = A here, a n d l a t e r use t h e same n o t a t i o n w i t h suffixes. T h e n (7.15) becomes
(7.16) q~(A, c) = 0 ,
w h e r e
(7.17) ~b(A, c) = 8c 3 - 1 2 c ~ - 12c-~ 7 - - 8(c-~ 1)IA 2 - 1 6 J A 3 .
1 O u r m e t h o d i s a d e v e l o p m e n t of t h a t u s e d b y MORDELL (7) f o r t h e c a s e m = 0 ( a n d s o J = 0).
18 - 642138 Acta mathematica. 84
274 C.S. Davis.
W e suppose now t h a t m ~ m 0, a n d find conditions for t h e lattice L to be of t y p e I. As n o t e d above, t h e points P , Q a n d P - Q = R include a reduced basis (A, B). Hence a necessary a n d sufficient condition is t h a t the points u A q - v B with tu] < l, Iv] G 1, (u, v) # (0, 0) should n o t be inner points of ~ . We i d e n t i f y these points in t e r m s of the basis ( P , Q). We h a v e
~ P + ~ Q = ( ~ + ~ ] ) Q + ~ R = ( ~ + r l ) P - - r l R ,
so if a n y two of !$1, 1~71, I$-~vtl are > 2 it follows t h a t lul > 2 or lvl > 2. _ _ _
If I$1 ~ 3 we have 15+~1 ~ I$]--Iv]l ~ 3--1~1 , so either I~] ~ 2 or [~+~71 ~ 2.
Hence ]~1 --~ 2, a n d we m a y suppose ~ _~ 0. Taking ~ --~ 0, 1, 2 in t u r n , a n d rejecting the values (~, ~) : (0, 0), (0, + 1 ) , (1, 0), (1, --1) which correspond to t h e origin a n d t h e points on (~, we find t h a t our condition is t h a t t h e points (~, ~]) ~ (1,1), (1, --2) a n d (2, --1) m u s t n o t be inner points of . ~ . I t follows t h a t L is of t y p e I if (7.18) ~p(1, 1) >_ 1, ~fl(1,--2) >_ 1, F ( 2 , - - 1 ) >_ 1.
Using (7.9) a n d (7.10), these conditions become
(7.19) 6c > --1, 3(c--b) < 1, 3(c--d) < 1 .
W e r e m a r k t h a t e q u a l i t y signs in (7.19) correspond to those in (7.18) a n d so occur only if the corresponding lattice point is on c~.
8. We t u r n our a t t e n t i o n n o w to with a basis (a, y), (fl, 6) a n d with t h e on (~'. Again m a k e the t r a n s f o r m a t i o n N o w the points ($, r/) = (1, 0), (1,
(8.1) a - - 1, b ~ - d
lattices of t y p e I I . Consider t h e n a lattice points (a, y), ( a + f l , y + d ) a n d ( a - - f l , y - - d )
(7.1) a n d write f ( x , y ) = F(~, ~7), as before.
1) a n d (1, - - l ) lie on cC, a n d we deduce
~ 0 , e = - - 6 c .
If t h e lattice is of t y p e I I the point (fl, 6), i. e: ($, V) = (0, 1), is n o t an inner point of .~)~, a n d so F(0, l) ~ 1. This gives the condition
( 8 . 2 ) 6c < - 1 .
Following t h e same procedure as t h a t we a d o p t e d previously, we find
(s.3)
(8.4) a n d thence (8.5)
I A 4 ~ 4 b ~ - - 6 c + 3 c 2 , j A 6 : b2(4c _ 1 ) _ 6 c 2 c 3,
~ ( A , c) : 1 6 c a - - 3 c 2 - ~ 6 c ~ - ( l - - 4 c ) I A ~ + 4 J A 3 = 0 .
T h e M i n i m u m of a B i n a r y Q u a r t i c F o r m (I). 2 7 5 9. W e are n o w in a position to solve o u r m i n i m u m problems. W e first t a k e t h e case -- < ~ m < ~, ~ a n d we will assume, u n t i l t h e c o n t r a r y is s t a t e d , t h a t this r e l a t i o n is satisfied. W e m u s t consider lattices b o t h of t y p e I a n d of t y p e I I ; we c o m m e n c e with t h o s e of t y p e I.
F r o m t h e s y m m e t r y of t h e region oj~, c e r t a i n lattices of t h e t y p e i n v e s t i g a t e d in w 7 will o b v i o u s l y h a v e m a x i m u m or m i n i m u m values of A. I n p a r t i c u l a r , this will be so for t h e lattice, which plainly exists, w i t h t h e points (a, a), (/3, 6) a n d (--6, --/3) on ~ , where ~ = f l + 6 > 0.
If t h e d e t e r m i n a n t of this lattice, s a y L~, is A1, we h a v e
Since (a, a) a n d (/3, 6) lie on ~ ,
(9.1) 2 ( 1 - ~ - 3 m ) a a = 1, f14~-6m/325e-~6'= 1 .
N o w 2/3 = / 3 ~ - 6 + / 3 - - 6 = a - - e ) , where o~ = 6--/3. Also 26 = a + w a n d A 1 = ao).
T h e n , b y (9.1),
( a - - e)) 4 ~- 6 m ( a i - - o~i): ~- (a-~ ~o) a=-- 16, 2 ( a 4 - t - - 6 9 2 o ) 2 - ~ - w 4 ) - ~ - 6 m ( a 4 - - 2 o c 2 o ) 2 - t - w 4 ) : 1 6 , 2(l ~- 3m)~4 ~-12(1--m)a2~o2 ~- 2(l ~- 3m)a4--16 = O,
i~ e .
i . e .
which gives
(9.2) 4 ( l + 3m)2A~ + 1 2 ( 1 - - m ) A ~ - - 1 5 = O . Hence, r e m e m b e r i n g A~ > 0, we o b t a i n
(9.3) A~ = A~ - -
2(1-~3m) 2 { 2 ~ / [ 6 ( 6 m 2 + 3 m + 1 ) ] - - 3 ( I - - m ) } .
NOW L 1 has t h e points (fl, 6), ( - - 6 , - - f l ) a n d (/3+6, 6 + f l ) , i . e . (a, a), on o(~.
P u t (9.4) a n d write (9.5)
B y (7.5), w e h a v e
f (x, y) = ~o( ~, ~) = ~' + 4bo~a~ + 6co$2~e + 4do$~3 + V ' 9
C o = 2 f 1 2 6 2 - t - m ( f l a ~ - 4 f 1 2 6 2 - / - 6 4 )
: ~ , + 6 m ~ 2 6 ~ + 6 , - ( ] - m ) ( / 3 ' 2~26~+64)
= 1--(1--m)A~ ,
276 C.S. Davis.
t h a t is
(9.6) c o ~- 1 - - ( I - - r e ) A 1 .
Since ~o(~, ~) = f ( x , y) = f ( - - y , - - x ) = YJo(~, ~), w e h a v e b o = do, a n d so (9.7) bo = do = 89 = ~ ( 6 c o § ---- ~ { 7 - - 6 ( 1 - - m ) A i } .
W e n o w r e q u i r e s o m e e s t i m a t e s f o r t h e n u m b e r s c o a n d A I .
L e m m a 4. I f - - ~ < m < 1 then -- 88 ~ , ~ C o ~ 12 .
F r o m (9.6) we h a v e ( 1 - - m ) A 1 ---- 1--co, so, u s i n g (9.2), 4(1-~3m)~A~ = 1 5 - - 1 2 ( 1 - - % ) = 3 - [ - 1 2 % . T h u s we h a v e t o p r o v e 1 < 2 ( i - ~ 3 m ) A 1 < 3. I f we p u t m - -
s a y . T h e n
2(1 + 3 m ) A 1
3 - - K 3 - ~ 3 K : ~ / ( K 2 ~ - 1 5 ) - - K : ~ ( K ) ,
K d m 4
A ' ( K ) - - ~ / ( K ~ 1 5 ) 1 < 0 , d K = 3 . 1 + K . ~ < ( ) 0 ,
in (9.3), we f i n d
so t h a t ~ is a s t r i c t l y i n c r e a s i n g f u n c t i o n of m. F i n a l l y , w h e n m = - - ~ , K : 7 a n d X(K) = 1; while w h e n m = 89 K = 1 a n d X(K) = 3. T h i s p r o v e s t h e l e m m a .
L e m m a 5. I f - - ~ < m < 1, then A 1 < 1 .
W i t h t h e s a m e n o t a t i o n as in t h e l a s t l e m m a , we f i n d
s a y . T h e n
f f ' ( K ) =
A~ = I ( l q - K ) ( 1 / ( K ~ - k l 5 ) - - K } = # ( K ) , 1
8//(-K2 q - 15) { / / ( g 2 q- 1 5 ) - - K } {1/(K2 ~ - 1 5 ) - - ( 1 q - g ) } ,
a n d # ' ( K ) = 0 o n l y w h e n K = 7. T h e first t w o f a c t o r s in t h e e x p r e s s i o n f o r / ~ ' ( K ) a r e e s s e n t i a l l y p o s i t i v e , a n d t h e l a s t f a c t o r , w i t h K = 7-~s, is - - l s q - O ( e 2 ) . H e n c e K = 7 is a m a x i m u m . B u t K = 7 w h e n m - - ~, a n d / ~ ( 7 ) = 1 ; t h e r e s u l t follows.
N o w f r o m (9.7) a n d L e m m a 4,
co--bo = co--do = c0--~(6co-t-1) - - 88 ~ < 0 ,
The Minimum of a Binary Quartic Form (I). 277 a n d 6% > - - 1 . T h u s t h e conditions (7~19) are satisfied b y t h e l a t t i c e L~, which is t h e r e f o r e of t y p e I.
10. We n o w find t h e o t h e r roots of t h e cubic O ( A ~ , c) = 0 g i v e n b y p u t t i n g A = A~ in (7.17). T h e lattice Lx has t h e points (~, c~), (fl, 8) a n d (c~--fl, c~--5), i. e.
(8, fl), on <~. P u t
( l O . 1 ) x = ~ ' , + f l ~ ' , y = ~ ' , + @ ' ,
a n d write
(10.2) f ( x , y) = ~o~(~', rl' ) = ~'4-{-4b1~'3"el'@6c~'2~]'e@4d~'~"a-]-~}'4.
N o w if we p u t
(10.3) ~ = $ ' + ~ ' , ~7 = - - ~ ' ,
we find
x = ~ ' + f l v ' = ( / ~ + 0 ) ~ ' + f l ~ / = / ~ ( ~ ' , { , ~ ' ) , { , ~ ' = f l ~ - @ ,
a n d similarly y = ~ ' + @ ' = d~--fl~. T h u s (10.1) is e q u i v a l e n t to (9A) a n d (10.3), a n d it follows t h a t
W~(~', ~') = ~o(L 7) 9
S u b s t i t u t i n g (10.3) in (9.5) a n d c o m p a r i n g w i t h (10.2), we find
(10.4) b~ = 1 - - 3 b o + 3 c o - - d o = 1+3Co--4bo = 1+3Co--1(6Co+1) = 89 (10.5) cl = 1 +Co--2b o = ~{1 ,{,2(1-- m ) A i } ,
(10.6) d~ = 1--b 0 = l { l + 6 ( 1 - - m ) A ~ } .
I t follows f r o m t h e a b o v e t h a t ~b(A~, c) ---- 0 for c = Co, c~. I f t h e t h i r d r o o t of this e q u a t i o n is % we h a v e , b y (7.17),
(10.7) Co+C1+C 2 = ~ ,
giving
c 2 -~ ~ - - C o - - C 1 = 8 8 = e l .
T h u s
(10.8) q ) ( A 1 , c) = 8 ( C - - C o ) ( c - - c l ) 2 .
F u r t h e r , since c o < 89 (10.7) gives cl > 1 > Co.
W e will p r o v e t h a t no lattice of t y p e I has a d e t e r m i n a n t A < A1, a n d t h a t t h e v a l u e 31 is a t t a i n e d only b y L1, a n d b y its image in t h e y-axis, say L'I. F o r this, we first show t h a t we m a y suppose c > Co. F o r suppose c < co. First, A =# A1, since q~(A1, c) < 0, b y (10.8). S u p p o s e n e x t t h a t 0 < A < A,. F r o m (7.13) a n d (9.7) we h a v e
2 7 8 C.S. Davis.
I A ~ = 1--4bodo+3C~o: 1@3c~--• 2 1(; o = ~ { ( 1 - - C o ) 2 + l } - Since c o < 1, 1 _ c > 1 % > 0 a n d so
(b--d) 2 = (b+d)2--4bd = iG(6c+ l)2--(l + 3c~)+IA2
2 3 1 2
= I A 2 - - ~ { ( 1 - - c ) 2 @ l } < I A l - - ~ { ( g - - C o ) @1} = O, which is i m p o s s i b l e .
W e m a y w r i t e
~(A, c) = q)(A~, c)+q)(A, c)--q)(A1, c)
= q~(A~, c) + ~b(A, Co) - - q)(A1, Co) + 8(c - - Co)I(A ~ - - A 2) , t h a t is, using (10.8),
(10.9) (/)(A, c) =- 8(c--c0) { ( C - - C l ) 2 @ I ( A ~ - - A 2 ) } @ q D ( A , Co) .
N o w q~(A, Co) > 0 f o r 0 < A < A r For, firstly, q~(A1, co) = 0. T h e n f r o m (7.17) we find, for c ---- Co,
~A - - - - 16A { 3 J A + ( % + 1)I} ;
we s h o w t h a t this is n e g a t i v e . W e n o t e I ---- l + 3 m 2 > 1 a n d J = r e ( l - - m 2 ) . I f n o w m > _ O , J > _ O , so
3 J A + ( c o + l ) I >_ Co+1 > 0 ,
l - - c ~ b y (9.6), so b y L e m m a 4. Also, if - - ~ < m < 0 , J < 0 , a n d A 1 -
1 - - m
3 J A + ( c o + 1)I > 3 J A l + ( c o + 1)I = 3m(1 + m ) ( 1 - - C o ) + ( 1 +%)I
> 3(--~-)(~)(1--Co)@1@c o = 1 ( 7 @ 1 7 % ) > 0 .
T h u s ~qS/~A < O, a n d so qs(A, Co) > r Co) = 0. I t follows t h e n f r o m (10.9) t h a t q ~ ( A , c ) > O f o r c > c o , O < A < A 1 .
This e s t a b l i s h e s t h a t /Jl is t h e m i n i m u m d e t e r m i n a n t of a lattice of t y p e I.
11. W e m u s t n o w consider l a t t i c e s of t y p e I I . I f m > 0, t h e region ~ is c o n v e x a n d no s u c h l a t t i c e s c a n exist, f o r t h e y w o u l d h a v e t h r e e collinear p o i n t s on t h e b o u n d a r y (C, which is impossible. S u p p o s e t h e n - - ~ < m < 0. W e h a v e b 2 >_ 0 a n d , b y (8.2), c _< - - ~ . T h e n , f r o m (8.3), it follows t h a t I A 2 > --6c+3c 2 _> l + ~ . B u t I = l + 3 m 2 < 1 + ~ , so A > I > A 1 , b y L e m m a 5. T h a t is, t h e r e are no l a t t i c e s of t y p e I I w i t h d e t e r m i n a n t A _< d l.
The Minimum of a Binary Quartic Form (I). 279 12. W e h a v e p r o v e d t h a t A* = A 1 w h e n --~- < m < g,1 a n d , f u r t h e r , t h a t t h e d e t e r m i n a n t d 1 is a t t a i n e d o n l y f o r l a t t i c e s of t y p e I g i v i n g c = c o, Cl. W e n o w c o n s i d e r w h e t h e r t h e r e a r e a n y o t h e r critical l a t t i c e s t h a n L1. W e r e q u i r e t h e fol- l o w i n g l e m m a .
L e m m a 6. F o r _ 1 < m < 1, the o n l y t r a n s f o r m a t i o n s o f f ( x , y) into itself, that is a u t o m o r p h i s m s of ~/( , are the reflections i n the axes of s y m m e t r y a n d i n the origin, n a m e l y , x ~ • y ~ + ~ a n d x - - ~ ] , y ~ +_~, where all the signs are i n d e p e n d e n t of each other.
C o n s i d e r a t r a n s f o r m a t i o n of t h e t y p e (7.1) g i v i n g f ( x , y) = ~v($, V) = $4-~6m$2~72-~-r/4.
Since t h e i n v a r i a n t s a r e u n c h a n g e d , we h a v e
(12.1)
N o w f r o m (7.5) we find, u s i n g (12.1),
t h a t is, (12.2) H e n c e
o~2 f12-~ y2~2 + 6 m ~ fly6 - - O .
( ~ f l - - y6)z + 2(1-}- 3m)~xfly6 = O, a n d so ~xfly6 < 0, since 1 + 3 m > 0. S i m i l a r l y
a n d so ~fly5 > O, since 1 - - 3 m > 0. I t follows t h a t af17(~ ~ O, a n d so a t least o n e of ~, fl, y, 6 m u s t be zero.
S u p p o s e , e. g., a = 0. T h e n , f r o m (12.2), y($ z 0, while, f r o m (12.1), fly = • a n d so ~ ~: 0. H e n c e (~ ~ 0 a n d , b y (7.3) a n d (7.7), ~4 = fit z 1. T h e r e f o r e a --~ 0, fl ~ + 1 , y = ~ 1 , 6 = 0, w h e r e t h e signs a r e i n d e p e n d e n t , g i v i n g x ~-- ~ ] , y z + $ . T h e o t h e r r e s u l t s arise s i m i l a r l y o n a s s u m i n g fl ~-- 0, a n d t h i s c o m p l e t e s t h e p r o o f of t h e l e m m a .
W e n o w f i n d all t h e c r i t i c a l l a t t i c e s . G i v e n d a n d c, t h e v a l u e s of b a n d d a r e d e t e r m i n e d , e x c e p t f o r o r d e r , b y (7.10) a n d (7.13). T h e n , h a v i n g f o u n d o n e l a t t i c e w h i c h g i v e s t h e s e v a l u e s of b, c, d, all s u c h l a t t i c e s will he g i v e n b y t h e t r a n s f o r m s of t h i s o n e b y t h e a u t o m o r p h i s m s of .+1~. I n o u r p r o b l e m we h a v e d ~ A 1, c = c 0, c 1,
2 8 0 C . S . D a v i s .
and these conditions are satisfied by the lattice L 1. From the s y m m e t r y of L~, the only other lattice produced by the automorphisms of ~ is its reflection in t h e y-axis, L' 1.
We have thus proved the following result.
T h e o r e m 1. I f --~ < m < 3,1 there is a point x, y, other than the origin, of every lattice L of determinant A, such that
As Ix4 ~-6mx~y2~-Y41 ~ A--I '
where A 1 is given by (9.3). This is the best possible result, the equality sign being required if, and only if, L is proportional 1 to one of the lattices LI or L' 1.
We m a y take as a critical form, corresponding to the lattice L~ or L' 1, the form YJ~($, 9) defined by (10.2) with (10.4), (10.5) and (10.6); all other critical forms, e.g. ~0(~, ~), are equivalent to this. Writing
A = 4d~ : 89 (10.5) gives 6c I ~ A ~ - l , and so
Substituting from (9.3), we find 1
(12.3) A -- ( l + 3 m ) ~ { a ( 1 - - m ) d i 6 ( 6 m ~ + 3 m + 1 ) ] - - 4 ( 1 - a m ) } .
Again, using (9.6) and L e m m a 4, we find 2 < A < 4, and, further, these bounds are best possible in the sense t h a t any such value of A corresponds to an m in the range considered.
Recalling our introductory remarks, and effecting a trifling reduction, Theorem 1 leads immediately to
T h e o r e m 2. Let ~f(~, 9) be a binary quartic form with real coefficients and ~ > 0, and either . ~ ~_ 0 or ~ ~ O. Further, let 3 5 ~ 3 > 393~ 2 i f ~ < O, so that m given by (4.5) satisfies --~ < m < 89 Then there exist integers ~, 9, not both zero, such that
1 B y a l a t t i c e " p r o p o r t i o n a l " t o t h e l a t t i c e x = a $ - b f l v l , y = ? ~ + 6 ~ 7 , w e m e a n o n e d e f i n e d b y x" = l x , y ' ~ t y , w h e r e 1 is a r e a l n o n - z e r o c o n s t a n t .
The Minimmn of a Binary Quartic Form (I). 281 1~(~, U)[ ~ 4 ~ / { 6 ( 6 m 2 @ 3 m @ l ) } + 6 ( 1 - - m ) ~ 9
15(1 9m2) 1/3
This is the best possible result, the equality sign being required if, and only if, ~(~, ~) is equivalent to a multiple of the form
with 2 < A < 4. The value of A as a function of m is given by (12.3).
13. W e now investigate t h e case m ~ -~-. W h e n m < --{- t h e lattice L~ is no longer admissible, for t h e proof of L e m m a 4 would t h e n give c o < --~, in contradiction to t h e conditions (7.19).
W e first p u t the last two inequalities in (7.19) in a different shape, n o t involving b a n d d explicitly. We note t h a t b o t h these inequalities c a n n o t be false, since there is only one lattice point (and its image) in question. T h e n if we p u t
O = 4 { 1 - - 3 ( c - - b ) } { 1 - - 3 ( c - - d ) } ,
a necessary a n d sufficient condition for b o t h these inequalities to hold is 0 ~ 0 .
Using (7.10) a n d (7.13), we find
0 = 4{(1--3c)2@3(b@d)(1--3c)@9bd}
= 4 ( 1 - - 3 c ) 2 4 - 3 ( 1 - - 3 c ) ( 6 c + 1 ) + 9 ( 1 @ 3 c 2 - - I A 2)
= 9 c 2 - - 1 5 c + 1 6 - - 9 I A 2 .
F u r t h e r , we note t h a t O = 0 only if 3(c--b) I or 3 ( e - - d ) = 1, a n d t h e n t h e lattice L has eight points on (6'. Changing our n o t a t i o n , if need be, we m a y suppose these to be the points (~, y), (fl, ~), ( ~ + f l , y@5), (~--fl, y--~) a n d their images in O.
If m > m0, we m a y now p u t our problem, for lattices of t y p e I, in t h e following form. W e seek t h e lower bound, say A0*, of A corresponding to real lattices, with t h e conditions
(13.1) A > 0, c >_ --3, ~b(A, e) = 0, O > 0 .
We shall simply o m i t t h e provision concerning reality, a n d later verify t h a t the solution t h u s o b t a i n e d does in f a c t satisfy it.
Now the expression qS(A, c)--qS(A, --~) vanishes when c = _ 1 , a n d so has a factor 6c@1. B y e q u a t i n g coefficients, we f i n d t h a t t h e other f a c t o r is 770. Thus
282 C.S. Davis.
we h a v e
27qi(A, c)--27~b(A, --~-) ~- 4 ( 6 e + 1 ) O .
I t follows t h e n f r o m (13.1) t h a t A * is t h e lower b o u n d of A > 0 w i t h
+(A, --~) _< o.
On p u t t i n g c - - - { in (7.17), we f i n d
--27~b(A, - - ~ ) - - 4 3 2 J A a + 1 8 0 I A z - 1 2 5 . S u b s t i t u t i n g A - - - - , t h e e q u a t i o n ~ b ( A , - - ~ ) = 0 b e c o m e s 5
6z
z a - I z - 2 J = O ,
t h e r o o t s of w h i c h are z := m ~ l , - - 2 m . I t follows t h a t
w h e r e
qO(A, - - ~) = C ( A - - A 2 ) ( A - - A ; ) ( A - - , 4 ; ' ) ,
C ~ - - 1 6 J > 0 ,
F u r t h e r ,
5 r 5 I t 5
A2 - - > 0 , A ~ - - < 0 , A~ - - > 0 .
6(1 + m ) 6 ( l - - m ) 12m
A~' 5(1+3m)
I . , - - A . , - > 0 . lO.m( 1 + m )
I t t t
Since A~ > A 2 > 0 > ./12, we h a v e r - - ~ ) < 0 if, a n d o n l y if, A < A, < 0 or
I !
A~ > A > A2 > 0. T h u s A* = A z , a n d t h i s v a l u e is a t t a i n e d if, a n d o n l y if, c =- --~- or O ( A > c) - - 0; in e a c h of t h e s e cases t h e c o r r e s p o n d i n g l a t t i c e s h a v e e i g h t p o i n t s on C~;. W e r e m a r k here, f o r l a t e r reference, t h a t t h e r e l a t i o n m > m0 is n o t u s e d in d e r i v i n g t h e r e s u l t s t a t e d in t h e last s e n t e n c e , w h i c h is c o n s e q u e n t l y t r u e w i t h o u t this r e s t r i c t i o n .
14. W e p r o c e e d to s h o w t h a t t h e s e v a l u e s of A a n d c c o r r e s p o n d to real l a t t i c e s b y a c t u a l l y d e t e r m i n i n g t h e lattices.
L e m m a 7. I f the p o i n t s (~, y), (fl, 5), ( ~ + f i , y + 5 ) a n d (o~--fl, 7,--6) lie o n c~, t h e n fl = + ~, d = -T- ~.
M a k e t h e s u b s t i t u t i o n (7.1) a n d n o t e t h a t h e r e c = - - ~ , b y t h e r e m a r k a f t e r (7.19). H e n c e b = - - d b y (7.10), so we h a v e
(14.1) f ( x , y) = ~(~, ~) = ~ + 4 b ~ a , j - - ~ 2 ~ 2 - - 4 b ~ ? a + ~ ] 4 .
The Minimum of a Binary Quartic Form (I). 283 B u t ~0(~, ~) = F ( - - ~ , ~), a n d so
X !
f ( x , y) = f ( ~ 4 - f l V , y $ + 6 V ) = f ( f l ~ - - ~ V , 6~--~}~) = f ( , y') ,
say. T h e t r a n s f o r m a t i o n f r o m x, y to x', y ' is a n a u t o m o r p h i s m of ~1~, a n d so, b y L e m m a 6, m u s t be i n c l u d e d in x' = ~ x , y ' = + y a n d x ' = • y ' = + x , w i t h a n y choice of signs. W e easily find t h a t x ' = + x , y ' = + y ; x ' = + x , y ' = T - y a n d x ' = + y , y ' = + x e a c h i m p l y a = f l = y = 6 = 0, which is impossible.
F i n a l l y , t h e a u t o m o r p h i s m s x' = + y , y ' = ~ x give fl = + ~ , d = ~ c ~ .
L e m m a 8. I f - - ~ < m < --~-, there e x i s t u n i q u e n u m b e r s p a n d q, w i t h p ~ q > O, s u c h that
f ( p , q) = f ( p - - q , p + q ) = 1 . F u r t h e r , q = 0 o n l y i f m = - - ~ .
W e h a v e
(14.2)
a n d so
giving (14.3)
p 4 + 6 m p 2 q 2 + q 4 = 1 ,
( p - - q ) 4 + 6 m ( p 2 - - q 2 ) ~ + ( p + q ) 4 = 1 ,
0 = ( p - - q ) 4 + 6 m ( p 2 - - q 2 ) 2 + ( p + q ) 4 - - 1
= ( 2 + 6 m ) ( p 4 + 6 m p 2 q 2 + q 4 ) - 1 2 p 2 q 2 ( 3 m 2 + 2 m - - 1 ) - - 1 , 6 m + 1
p2q2 = = K ,
1 2 ( 3 m - - 1) (1-~m)
say. I f m---6,1 K = 0 a n d p---- 1, q = 0 . E x c l u d i n g t h i s case, we h a v e K > 0 a n d , s u b s t i t u t i n g f o r p2 or q2 f r o m (14.3) in (14.2), we find t h a t p4 a n d q4 a r e r o o t s of
~ z @ ( 6 m K - - 1 ) ~ @ K 2 = O . T h i s gives
= 12 {( 1 - - 6 m K ) + ~/[( 1 - - 6 m K ) 2 - - 4 K 2] } = 3(2 - - 3m) + y'[5(7 - - 3m) ( 1 - - 3m)]
12 (1 - - 3 m ) ( l @ m )
T h e v a l u e s of ~ are r e a l a n d distinct, since ( 7 - - 3 m ) ( 1 - - 3 m ) > O. Also, 2 - - 3 m > 0 a n d ( 1 - - 3 m ) ( l + m ) > 0, so t o s h o w t h a t ~ > 0 it suffices to p r o v e t h a t
9 ( 2 - - 3 m ) 2 > 5 ( 7 - - 3 m ) ( 1 - - 3 m ) , i.e. (6re@l) 2 > 0, which is true. This p r o v e s t h e l e m m a ,
284 C.S. Davis.
We thus have a lattice, say L2, given by
(14.4) x = p ~ - - q ~ , y = q~-?p~? ,
which, by Lemma 8, has eight points on ((~j, namely (p, q), ( - - q , p), ( p - - q , p - ~ q ) , (p~-q, p - - q ) and their images in O. The determinant of this lattice is d = p2~_q2 and, from (14.2) and (14.3), we find
(14.5) ~j2 = (p2_~q2)2 __ p4~_6mp2(t2_~q4~_2(l_3m)p2q2 _ 5 - - A z . 6(1 ~-m)
The lattice L2, then, satisfies the conditions we found above for a minimum, and all such lattices will be given by the transforms of L2 by the automorphisms of ~ . As before, we find in this way only two distinct lattices, L~ and its image in the y-axis, say L~.
15. I t remains to show t h a t there are no lattices of type II, other t h a n L 2 and
!
L~, with A < A 2. We have, from (8.5),
T ( A , c)--~[I(A, - - ~ ) ~- 6c - - c ~-6~-10 s
= 1(6c+
1)(Sc~--~c+~--2IA ~)
- 1 ( 6 c + 1 ) Z , say, and so
1 1
(15.1) T ( A , c) = T ( A , --~)~-~(6c~-l)Z
Now I and Az are both strictly decreasing functions of m for m < 0, and so, since
m > 1
4 5 2
2 I A ~ < 2(~)(z) = ~0.
Recalling that, by (8.2), 6c~-1 < 0 for lattices of type II, we have, if 0 < A < A2,
Again, we find
Z - _v~r2--3617 ( 6c ~- l )-~ ~4{ - 2 I A 2
- - 3 6 ~ 3 6 - - ~ - - -36 - - 2 ~ > 0 9
l08~(A, --88 ~ 432JA3~ - 180IA 2 - 1 2 5
= 4 3 2 J ( A - - A 2 ) ( A - - A ' ~ ) ( A - - A ' ~ ' )
< 0 ,
for 0 < A < A~. Then, from (15.l), we have ~(A, c) < 0 if 6c~-1 < 0, 0 < A < A2, with equality only if A =~ A2, c ~ --}, which is the required result.
The Minimum of a Binary Quartic Form (I). 285 16. I t follows n o w t h a t A* = /I 2 w h e n m 0 _< m _~ --s,' a n d so we h a v e t h e following result, w h i c h we p u t in t h e f o r m of a l e m m a , as we will l a t e r p r o v e it t r u e for t h e r a n g e -- 89 < m _< - - ~ a n d give t h e c o m p l e t e result as a t h e o r e m .
L e m m a 9. I f m o <_ m < --~, there is a point x, y, other than the origin, of every lattice L of determinant ~J, such that
]x4 ~-6mx2y2 ~-y41 <
56(l~-m)A] 2.This is the best possible result, the equality sign being required if, and only if, L is proportional to one of the lattices L2 or L,2. !
T h e critical f o r m , s a y ~v~(~, ~/), c o r r e s p o n d i n g to t h e t r a n s f o r m a t i o n (14.4) will be g i v e n b y (14.1) w i t h t h e a p p r o p r i a t e v a l u e of b. S u b s t i t u t i n g in (7.4), we f i n d
b : - - ( 1 - - 3 m ) p q ( p 2 - - q 2) . H e n c e
(16.1) w h e r e
(16.2) h = --4b = 4 ( 1 - - 3 m ) p q ( p 2 - - q 2) ~_ O.
W e d e t e r m i n e h b y p u t t i n g h = - - 4 b --~ 4d, A 2 = A 2 a n d c = --3- in (7.13), giving
(16.3) 3 h 2 : 12IA~-- 1 3 .
T h e n c e
25(1 ~-3m2)--39(1-}-m) 2 2 (6m-}- 1) ( 3 m - - 7)
3h 2 = _--
3(1~-m) 2 3 ( l ~ - m ) 2
a n d so
(16.4) h - - 1
3(1 ~-m) V{2(6m ~- l) ( 3 m - - 7)}.
W e m a y n o w i n t e r p r e t t h e r e s u l t g i v e n b y L e m m a 9 in t h e light of o u r i n t r o - d u c t o r y r e m a r k s , a n d we a g a i n s t a t e t h e conclusion as a l e m m a .
L e m m a 10. Let ~(~, ~) be any binary quartic f o r m of diseriminant ~]), which is transformable into the standard f o r m x4~-6mx2y2~-y 4 with m 0 < m <_ --~. Then there exist integers ~, ~], not both zero, such that
_ _ 2 1 1
I~(8, ~)l < w ) - ~ .
This is the best possible result, the equality sign being required if, and only if, ~0(~, V)
286 C.S. Davis.
is equivalent to a m u l t i p l e of the f o r m
Yq(~, ~l) ~- ~4--h~3~--$~'rt2 ~-h$rl~ ~-~ 4 , where h is given by (16.4).
17. T o c o m p l e t e t h e discussion of f o r m s w i t h f o u r d i s t i n c t c o m p l e x roots, we m u s t n o w consider t h e l a t t i c e - p o i n t p r o b l e m f o r -- 89 < m < m 0. i n t h i s case, t h e m e t h o d s u s e d b y )/[ORDELL
(6)
to deal w i t h a c e r t a i n t y p e of n o n - c o n v e x region m a y be e m p l o y e d to o b t a i n t h e b e s t possible result. H i s g e n e r a l t h e o r e m s c a n n o t b e a p p l i e d d i r e c t l y to this p r o b l e m , since his b o u n d a r y curves h a v e b y h y p o t h e s i s n o r e a l finite p o i n t s of inflection, b u t we shall find t h a t t h e s a m e ideas a r e successful here. W e shall follow closely t h e d e t a i l s of Mordell's p r e s e n t a t i o n .T h e r e s u l t we find is t h a t L e m m a s 9 a n d 10 still h o l d f o r t h e c o m p l e t e r a n g e
< m ~ - - ~ , a n d we p r o v e this b y s h o w i n g t h a t , f o r -- 89 < m < m0, e v e r y l a t t i c e of d e t e r m i n a n t A s has a p o i n t , o t h e r t h a n t h e origin, in t h e region . ~ , a n d , f u r t h e r , t h a t this is a n i n n e r p o i n t e x c e p t w h e n t h e l a t t i c e is L 2 or L 2. !
W e first p r o v e s o m e r e s u l t s r e q u i r e d later. W e s u p p o s e , unless t h e c o n t r a r y is s t a t e d , t h a t -- 89 < m < m 0 ~ --0-259.
L e m m a 11. 1 < p < ~ .
W r i t e r ~ - - q / p a n d s o p 4 f ( 1 , r) ~ 1. B y L e m m a 8, p > q > 0, i . e . 0 < r < 1, so f(1, r) = 1 - ~ 6 m r 2 - ~ r 4 - - 1-~-r2(r2+6m) < 1 - ~ r ~ ( l ~ - 6 m ) < 1 ,
a n d h e n c e p > 1. Also, p 4 < (p~+q~)2 _ 5 5 81 3
< < - - a n d so p < - .
6(1-~m) 4 1 6 ' 2
L e m m a 12. 0 < r < 41.
As a b o v e , r > 0. Also r 2 ~ K / p 4 < K , w h e r e K = p~q2 is g i v e n b y (14.3). B u t 1 4 ( 6 m + l ) - - 3 ( 3 m - - 1 ) ( 1 4 - m ) ( 7 - - 3 m ) (1-~3m)
K - - 16 - - 4 8 ( 3 m - - 1 ) ( 1 - ~ m ) = 4 8 ( 3 m - - 1 ) ( l ~ - m ) < 0 , a n d t h e result follows.
L e m m a 13. I f -- 89 < m ~ - - ~ , h i n (16.4) is a strictly decreasing f u n c t i o n o f m . F u r t h e r , 0 ~ h < 2, a n d these are the best possible bounds.
T h e M i n i m u m of a B i n a r y Q u a r t i c F o r m (I). 287 W e h a v e h > 0 b y (16.2), t h e v a l u e 0 being a t t a i n e d w h e n q = 0, i. e. w h e n m = - - ~ . B o t h I a n d A2 d e c r e a s e s t r i c t l y if m < 0, a n d (16.3) shows t h a t h does t h e s a m e . W h e n m - - ~, (16.4) gives h = 2 a n d so h < 2 for m > -- 89
L e m m a 14. I f - - 8 9 then h > l + r + r 2.
A f t e r L e m m a s 12 a n d 13, it is sufficient to p r o v e h(mo) > 1 - ~ +1~ = 1.3125. 1 1 P u t t i n g m = m 0 ~ - - 0 . 2 5 9 in (16.4), we find h ( m o ) ~ - 1 - 3 2 0 . . . > 1.3125; this p r o v e s t h e l e m n m .
18. W e m u s t n o w e x a m i n e o u r g e o m e t r i c a l c o n f i g u r a t i o n m o r e closely. T h e following discussion will be a i d e d b y reference to Fig. 1.
T h e l a t t i c e L 2 h a s t h e p o i n t s (p, q), (p--q, p+q), (--q, p), ( - - p - - q , p--q) a n d t h e i r i m a g e s in 0 l y i n g o n ~ . T h e s e p o i n t s define a s q u a r e .cS 1, of which t h e v e r t i c e s a n d t h e m i d - p o i n t s of t h e sides all lie on ~ . F u r t h e r , t h e s q u a r e ~ 2 which is t h e i m a g e of .of 1 in t h e y - a x i s h a s t h e s a m e p r o p e r t i e s . T h e a r e a of e a c h of t h e s e s q u a r e s is 4(p2~-q ~) = 4A 2.
W e d e n o t e b y ~ 1 t h e closed region b o u n d e d b y t h e line joining t h e p o i n t s
Fig. 1. The region Ix4--~x2y2~-y41 <_ 1.
