WITH CONSTANT COEFFICIENTS.

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WITH CONSTANT COEFFICIENTS.

BY L A R S G A R D I N G

of LUND.

Contents.

P a g e

I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

C h a p t e r I. P r o o f o f t h e o r e m I . . . . . . . . . . . . . . . . . . . 9

C h a p t e r 2. H y p e r b o l i c p o l y n o m i a l s . . . . . . . . . . . . . . . . . . . 1 4

R e d u c e d p o l y n o m i a l s . . . . . . . . . . . . . . . . . . . . . . . . 14

H y p e r b o l i c p o l y n o m i a l s . . . . . . . . . . . . . . . . . . . . . . . 16

T h r e e l e m m a s . . . . . . . . . . . . . . . . . . . . . . . . 2 6

T h e d u a l c o n e . . . . . . . . . . . . . . . . . . . . . . . . . . 28

C h a p t e r 3. T h e R i e s z k e r n e l . . . . . . . . . . . . . . . . . . . . . . 9 9

A l e m m a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

C o n s t r u c t i o n o f t h e k e r n e l . . . . . . . . . . . . . . . . . . . . . 31

T w o e x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . 3 4

T h e R i e s z k e r n e l a n d t h e e l e m e n t a r y s o l u t i o n . . . . . . . . . . . . 3 5

C h a p t e r 4. T h e R i e s z o p e r a t o r . . . . . . . . . . . . . . . . . . . . . 37

D e f i n i t i o n o f t h e R i e s z o p e r a t o r . T w o t h e o r e m s . . . . . . . . . . . . 3 7

P r o o f o f t h e o r e m 4. I . . . 9 . . . . . . . . . . . . . . . . 39

P r o o f o f t h e o r e m 4 . 2 . . . . . . . . . . . . . . . . . . . . . . . 4 5

A l e m m a . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7

P r o o f o f t h e o r e m I I . . . . . . . . . . . . . . . . . . . . . . . 4 8

C h a p t e r 5. T h e p r o b l e m o f C a u c h y . G e n e r a l i z a t i o n s . . . . . . . . . . . . 4 9

T h e p r o b l e m o f C a u c h y . . . . . . . . . . . . . . . . . . . . 4 9

T h e s u r f a c e S . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0

T h e R i e s z o p e r a t o r . . . . . . . . . . . . . . . . . . . . . . 5 3

C h a p t e r 6. T h e d o m a i n o f d e p e n d e n c e . . . . . . . . . . . . . . . . . 5 4

I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4

S t r u c t u r e o f t h e d o m a i n s o f d e p e n d e n c e o f I a n d J . . . . . . . . . . 56

L a c u n a s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0

B i b l i o g r a p h y . . . . . . . . . . . . . . . . . . 61

1 - 642127 A ~ m a t / ~ ' m a t ~ . 85

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2 Lars G~lrding.

Introduction.

Let C(oo) be the class of complex functions f ( x ) of n real variables xl, . . . , xn which are defined and infinitely differentiabte for all x. L e t q (~)-~q (~1 . . . . , Cn) be a polynomial in $1 . . . . , r with complex coefficients and let A (q) be the class of all functions f in C(oo) which satisfy

q ( O l a x ) , f ( x ) = o

for all x.

Every polynomial q can be written in the form p + r where p is homogeneous and, if q is not a constant, the degree of p is greater than the degree of r. I f q is a constant we p u t t ---- q. We call the p o l y n o m i a l p thus defined the principal part of q.

Let $-~ (~1,-.., $,,)~ o be a real vector. We say t h a t q is hyperbolic with respect to ~ if p(~) ~ 0 and if there exists a real number to such t h a t

q(t$+i~7) ~ 0

when t > to and ~ is any real vector. I f q is hyperbolic with respect to ~, it is clearly hyperbolic with respect to any positive multiple of ~ and we will show t h a t it is hyperbolic also with respect to any negative multiple of ~. W e say t h a t (i) is a hyperbolic differential equation if q is hyperbolic with respect to at l e a s t one ~.

L e t

f l , f 2 , . . . , f i , . . ,

be a sequence of elements in C(oo). I f fk and every derivative of fk tends to zero with

I / k

u n i f o r m l y on every compact 1 set in the plane ( y , ~ ) = y l ~ l + . . , + Y , ~ = o or in the entire space we say t h a t f ~ tends to zero in the plane (y, ~)-= 0 or in the entire space and write

(a)

and

(b)

respectively.

o

f k --> O

I t is clear t h a t (b) implies (a) but the converse is not true. I n Chapter I the following theorem is proved 2

i A set S w h o s e e l e m e n t s a r e real v e c t o r s x = ( x l . . . Xn) is c a l l e d b o u n d e d if I x l = m a x k l Xk [ i s b o u n d e d w h e n x is i n S, a n d closed, if i t t o g e t h e r w i t h t h e e l e m e n t s of a s e q u e n c e x(k) a l s o c o n t a i n s e v e r y x s u c h t h a t l i m Ix(k) - - x I = o. I t is c o m p a c t if i t i s b o t h b o u n d e d a n d closed.

i T h e t h e o r e m s I a n d ] I I w e r e a n n o u n c e d in G.~RDI.NG [3], a n o u t l i n e of t h e c o n s t r u c t i o n of t h e R i e s z k e r n e l a n d t h e s o l u t i o n of C a u c h y ' s p r o b l e m i n G~.RDING [2]. See, h o w e v e r , t h e first f o o t n o t e t o C h a p t e r 5-

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T h e o r e m I. Let f k E A (q), (k-~ I, 2 , . . . ) . I f there exists a point x such that (x, ~) ~ o and f i (X) tends to zero with I / k whenever fk tends stro~gly to zero in the plane (y, ~ ) = o, then q i s hyperbolic with respect to ~.

Put f(~, y ) = e (~,y-~) where .~ is a complex vector~such t h a t q(~)-~ o. T h e n f ( ~ , - ) is in A (q), it equals I at the p o i n t x, and the proof, whose origin was a r e m a r k by H a d a m a r d ([4] P- 4o), uses the fact t h a t if q is not hyperbolic with respect to ~, then we can always find a sequence of vectors ~(~) .. . . , ~(k), . . . such t h a t q(~(~))=o for all k and .f(~(k), . ) . § o (~).1

The main o b j e c t of the rest of the paper is the following t h e o r e m which is a strong converse of T h e o r e m I.

T h e o r e m II. Let f~ ~ A (q), (k = ~, 2 . . . . ) and let q be hyperbolic with respect to ~. Then i f f~ tends stro~gly to zero in the plane (y, ~ ) = 0 it tends strongly to zero in the entire space.

Combining t h e t w o t h e o r e m s we have the following concise theorem.

Theorem I I I . Let f ~ e A ( q ) , (k = i, 2 , . . . ) . Then a necessary and sufficient condition that (a) implies (b) is that q is hyperbolic with respect to ~.

The simplest not trivial h y p e r b o l i c equation is the wave equation in two variables which corresponds to the case n = 2 and q = r Then q is hyperbolic with respect to $ = ( I , o ) . I n fact, p ( ~ ) = q ( ~ ) = i r ~ o a n d q ( t ~ + i ~ ) =

= ( t + i ( ~ + ~ ] z ) ) ( t + i ( V l - - ~ ] 2 ) ) ~ o w h e n t > o (or t < o ) . Also i f f E A ( q ) one has the e l e m e n t a r y f o r m u l a

~ r ~ - x 1

(2) f ( x ) = ~ (f(o, x2 + xl) + f(o, x~ -- Xl)) + 89 f f ' (o, t) d t

Z 2 - - X 1

where f ' ( x ) = Of(x)/Oxl. H e n c e if A (q)~fk and j~. t e n d s strongly to zero in the plane (x, ~ ) = xl ~ o, i.e. on the x2-axis, it follows t h a t

fi(x)

tends to zero for all x and, more generally, t h a t fk t e n d s strongly to zero in the entire space.

This proves T h e o r e m I [ in our special case. The p r o o f in the general case is similar. I n fact, if q is not c o n s t a n t and hyperbolic with respect to ~', it is possible to c o n s t r u c t a linear f u n c t i o n a l K ( f ) = K(~, x, f), in the case j u s t considered given by the right side of (2), with the following properties.

1 T h e p r o o f r e s t s m a i n l y o n a l e m m a o n t h e r a t e of g r o w t h of c e r t a i n a l g e b r a i c f u n c t i o n s . T h i s l e m m a is p e r h a p s Of i n t e r e s t i n i t s e l f a n d i t p r o v e s a c o n j e c t u r e b y PETROWSKY ([9] foot- n o t e p. 24).

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4 Lars G~rding.

I t is a projection of C ( ~ ) upon A(q), i.e, it is defined for all f in C(c~) and is itself an element of A (q), and it reproduces the elements of A (q)so that

f(x) -~ K (~, .% f )

for all x if f belongs to A (q). If fk tends strongly to zero in the plane (y, ~ ) = o then K(~, ',fk) tends strongly to zero in the entire space. Moreover, if the derivatives of order < m of f vanish in a certain compact part B(x) of the plane (y, ~ ) = o , then

K ( ~ , x , f )

vanishes, m being the degree of q. Finally, the derivatives of order < m of

f - - K ( ~ , . , f )

vanish on the plane ( y , ~ ) = o . A t least when q is homogeneous one can write K explicitly in a form similar to (2) as a sum of certain integrals over

B(x).

The functional K also gives the solution of the problem of Cauchy to which we give the following seemingly sophisticated, b u t in fact simple and convenient form. Given an element g~ C(oo), find an element u in

A(q)

such t h a t the derivatives of u - - g of order < m vanish on the plane (y, ~ ) = o. In fact, one solution is simply

u (x) = K (~, x, g)

and because the difference v of any two solutions is an element in A(q) whose derivatives of order < m vanish on the plane (y, ~)-~ o it follows from the properties of K that v(x)----K(~, x, v)----o for all x and hence the solution is unique.

Conversely, assume t h a t for a given ~ # o and not constant q and an arbitrary g E C(oo) t h e problem of Cauchy has a unique solution

H(~,x, g)

w i t h the property that H ( ~ , x , gk) tends to zero with

I/k

for at least one x with (x, ~) # o whenever g~ -~ o (~). Then if A (q) contains every element of the sequence f l . . . . ,fk, . . . and fk -" o(~) we get that

fk(x) -~

H(~,

x,fk)

tends to zero with

I/k.

Hence the requirements of Theorem I are satisfied and it follows that q is hyperbolic with respect to ~. I t then follows t h a t H(~, x, g) --- K(~, x, g) for all x and all g e C (oo).

The continuity property of H used above is a variant of Hadamard's classical condition t h a t the problem of Cauchy should be correctly set ([5] PP. 4 o - - 4 I ) . Another variant was given by Petrowsky [9] who, however, restricts the be- haviour of the function g at infinity in the plane (y, ~)----o. The consequence is that in his case there are other than hyperbolic equations, e.g. the heat equation, for which one can find a suitable correctly set Cauchy problem.

Most equations (I) which so far have been classified as hyperbolic are hyperbolic in our sense, in particular the equations considered by Herglotz [6] and

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Petrowsky [8]. Slightly modified, Petrowsky's definition runs as follows. 1 A homogeneous polynomial p of positive degree is called hyperbolic with respect to r if p ( 2 ) # o and the zeros of the equation p ( t ~ + y ) = o are all real and different if ~ is real and not proportional s 2. I f m is the degree of p it then follows t h a t

p ( t ~ + i ~ ) - ~ i ' ~ p ( - - i t ~ + ~ ) # o

when t > o (or t < o ) and ~ is real, so t h a t p is hyperbolic in our sense. More generally, one can show t h a t if p is a homogeneous polynomial of degree m > o which in the sense of Petrowsky is hyperbolic with respect to ~, and

r'

is any polynomial of degree less than m, then

q' : p + r'

is (in our sense) hyperbolic with respect to ~. I f p is hyperbolic merely in our sense, this need not be true. A r a t h e r trivial example is given by

q ' :

~ + ~2, a less trivial one by

q'= ~ ( ~ - - ~ ) + ~.

I n both cases the principal parts are hyperbolic with respect to (I, o), but the polynomials are not.

Hence the hyperbolic character of a polynomial is in general not determined by i~s principal part alone. I t is, however, true t h a t if a polynomial is hyperbolic with respec t to a vector ~, then also its principal part is.

We study in the first section of Chapter 2 the effect of a linear transforma- tion,

x ' : x~I,

where 1~ is the transpose of a real, square and not singular matrix M, upon (I). I t is transformed into

q ' ( O / O x ' ) J ' ( x ' ) = o,

where

f (x') =f(x; ~i -~) ~-f(x)

and q' (~') = q (~). We call the polynomial q reduced if there is no M such t h a t q' is a polynomial in ~ , . . . , ~ alone where l < n.

The fact t h a t a polynomial is n o t always reduced introduces some complications in the proof of Theorem II. L e t $2(q) be the linear manifold of all real vectors such t h a t

q(ty + y')=q(~?')

for all real t and ~'. Then q is reduced if and only if $2 (q) contains only the element ~ = o.

Later in Chapter 2 we collect some facts concerning not constant hyperbolic polynomials. Let the polynomial q be hyperbolic with respect to ~. Then the same is true of its principal part p. L e t the common degree m of q and p

1 In t h e paper [8] Petrowsky considers only homogeneous equations w i t h constant coeffi cients, in [9] and [Io], however, he extends w h a t is s u b s t a n t i a l l y t h e definiiion given above t o very general s y s t e m s of differential equations which need n o t even be linear. For t h e m he solves t h e problem of Cauchy. - - The wellknown textbook Methoden der Math. Physik b y R. COURANT and D. HII.BERT (Berlin 1937) , has a terminology which differs from ours. There t h e equation (I) is called hyperbolic unless it is elliptic and i t is elliptic if t h e principal p a r t of q is a definite polynomial. An equation which is hyperbolic in the sense of Petrowsky is called t o t a l l y hyperbolic (1.c. I I p . 373--374).

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6 Lars G~rding.

be positive. Because ~ ( ~ ) ~ o , t h e degree of p ( t ~ + ~ ) with respect to t is m a n d we can write it in t h e form p (~) [ i ( t + u,.) where u,, -~ u~(~, ~) are certain

1

complex numbers. I t turns out t h a t if ~] is real t h e n also t h e n u m b e r s u~(~,~) are real. They need n o t all be different w h e n ~ is n o t p r o p o r t i o n a l to $, b u t if t h e y are, we have the case considered by Petrowsky. L e t

F(q, ~)

be the set of all real ~/ such t h a t min~ u~ (~, ~?)> o or briefly,

r ( q , = mi, , > o).

I t t u r n s o u t t h a t F(q, ~ ) =

F(p, ~)

is the i n t e r i o r of a convex cone c o n t a i n i n g ~:.

Also, if ~ ' e

F(q, ~)

t h e n q is hyperbolic with respect to b o t h ~' a n d - - ~ ' , a n d we have / ' ( q , ~ ' ) = F(q,~). W e also consider t h e dual cone

C-= C(q,~)

of

lz'= F(q, ~)

defined as t h e set of all real vectors x such t h a t ( x , ~ ) ~ o for all

~/e F, or briefly

C(q, ~) -~

^(x;

(x, ~) >-- o, ] e F(q, ~')).*

I t is convex and o r t h o g o n a l to Y2(q). I t s i n t e r i o r is n o t e m p t y if q is reduced, a n d the p a r t of C where (x, $')--~ b is closed a n d bounded if 8 ' e F.

The c e n t r a l question in the Chapters 3 a n d 4 is the effective d e t e r m i n a t i o n of t h e l i n e a r f u n c t i o n a l K(~, ~, f ) . W e use a m e t h o d of f r a c t i o n a l i n t e g r a t i o n developed by M. Riesz [ii] for the wave equation. Again, let t h e polynomial q be n o t c o n s t a n t a n d hyperbolic with respect to ~. L e t /'1 =

Fl(q, ~)

be t h e set of vectors ~' in / ' ~

F(q, ~)

for which there exists a to < I such t h a t q (t~' + i,/) ~ o when ~/ is real a n d t > to. I f ~ e F, t h e n a suitable positive multiple of ~ is i n / ' 1 . L e t ,] be real, let ~' be in F1, p u t ~ = ~' + i*] a n d define q(~)-~ as

e -all~

T h e n i t t u r n s o u t t h a t arg q a n d hence also q(~)-~ is if locally c o n t i n u o u s also singlevalued when ~ ' e / ' 1 a n d ~/ is real. D i f f e r e n t choices of a r g q at a p o i n t will affect q(~),~ only by a f a c t o r e - ~ " where k is a n integer. Assume for a m o m e n t t h a t q is reduced a n d t h a t ~ a > n. ~ T h e n q(~)-~ is the Fourier-Laplace transform" of a c o n t i n u o u s f u n c t i o n Q(a, x) which v a n i s h e s outside C, a n d we have the reciprocal f o r m u l a s

1 ~ m e a n s t h e real part o f a .

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w h e r e t h e i n t e g r a l s are t a k e n over t h e whole space. W h e n q = ~ - - $~ . . . ~ , in w h i c h case (i) becomes t h e wave e q u a t i o n , t h e n q is h y p e r b o l i c with r e s p e c t to any 8 such t h a t q ( 8 ) > o . One finds t h a t F = ( ~ ; 8 1 ~ 1 > o , q ( * / ) > o ) , t h a t

C - ~ ( x ; x151 >-o, q(x)>-o)

a n d t h a t w i t h a suitable choice of a r g q,

Q x) = q r r ( . - ( , , - 2))

w h e n x E C a n d zero elsewhere. This, w i t h a c h a n g e d to l a , is t h e k e r n e l of M. Riesz.

R e t u r n i n g to the g e n e r a l case, we p r o c e e d as follows. L e t S : S(8) be t h e plane (y, 8 ) : o a n d T : T ( 8 ) t h e r e g i o n (y, 8 ) > o . W h e n h E C ( o o ) , x E T a n d

a ~ n we define t h e Riesz o p e r a t o r I ~ by t h e formula.

I ~ h (x) = f q (a, x - - y) h(y) d y .

T

All y such t h a t x - - y e C a n d y e T + S, i.e. such t h a t

( x - - y , 8)<--(x, 8),

con- s t i t u t e a c o m p a c t set

C(x),

a n d t h e i n t e g r a n d v a n i s h e s outside

C(x).

H e n c e t h e i n t e g r a l always exists. L e t

a = a , e C(~),

let

az(y)=

I w h e n

y e C ( x ) a n d

let a~ (y) = o when ] y ] = maxk ]yk ] is large e n o u g h . L e t 8' fi/'1 (q, 8), p u t ~ = 8' + i ~/and

H~ (~) = f h (y) az (y) e - (;, :t) d y.

.r

T h e n by v i r t u e of P a r s e v a l ' s t h e o r e m , a n o t h e r f o r m of

I~h(x)

is

;7~ - - n

loh(x)=(z ) f

W h e n q is n o t n e c e s s a r i l y r e d u c e d , we define

I ~h(x)

by this f o r m u l a . T h e n it

8'

is i n d e p e n d e n t of 8' a n d az as l o n g as e F 1 and a, equals one on

C(x);

a n d t h e f o r m u l a is valid as long as t h e i n t e g r a l is absolutely c o n v e r g e n t , i.e. w h e n a > o . I t is s h o w n in C h a p t e r 4 t h a t w h e n

x e T , I~h(x)

is an e n t i r e func- t i o n of a, t h a t f o r all values of a

t i n u o u s in T a n d a t the same t i m e

= F ' h ( x )

and t h a t

I - k h ( x ) = q ( O /

on

C(x)

t h e n

I ~ h ( x ) = 0

f o r all a.

are c o n t i n u o u s in T + S and those

all its d e r i v a t i v e s w i t h r e s p e c t to x are con- e n t i r e f u n c t i o n s of a, t h a t

q (0/0 x ) I ~§ h(x) -~

0x) kh(x) when k ~ o , 1 , 2 , . . . I f h vanishes F u r t h e r , all t h e d e r i v a t i v e s of

l h (x) ---- 11 h(x)

of o r d e r ~ m v a n i s h on S.

L e t I ~_ be t h e Riesz o p e r a t o r c o n s t r u c t e d as above b u t with 8 c h a n g e d to

-- 8. T h e n I ~ - h (x) is defined w h e n x e T - : T ( - - 8 ) a n d it vanishes if h vanishes on t h e c o m p a c t set C - (x) : (y;

x -- y E C(q,

- - _~) = - - C, (y, 8) >--- (x, 8)). I t t u r n s out t h a t all t h e d e r i v a t i v e s of

11__ h(x) -- I h (x)

v a n i s h w h e n x e S. W e p u t

Ih(x) : I [ h (x)

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8 Lars GArding.

w h e n x E T - . T h e n

I h E C(oo)

a n d one can prove t h a t i f h k -~ O, t h e n also

Ihk ~ o.

Also, if all the derivatives of h (x) of order < m vanish when x E S, or briefly, if h (x) (m) o, (x E S), t h e n

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h (x) -~ I q (O/Ox) h(x).

I n t e r m s of t h e operator / , the linear f u n c t i o n a l K(2, x , f ) is given by the f o r m u l a

K (2, x,.f) : f(x) -- I q (0 / 0 x)f(x).

t t follows f r o m (3) t h a t if f a n d g are in C(oo) and

f ( x ) - - g ( x ) ('')o, (xES),

the r i g h t s i d e of this f o r m u l a does n o t c h a n g e if we change f to g. P u t 1

m - - 1

P i f ( x ) = ~.j (a, 2)-': (x, 2)k f (k) (x --

(a,

~)-1 (x, 2) a)/k[

o

where a is a vector such t h a t (a, 2 ) # o a n d

f(k)(x)=(a, O/Ox)kf(x).

T h e n

P~fEC(oo) and f(x)--P~f(x)~m)o, (xES),

a n d consequently a n o t h e r f o r m of

K($, x , f )

is

K (2, x , f ) -= P~f(x) -- I q (O / O x) P~f(x).

Now P ~ f depends only on t h e values of f a n d its derivatives of order < m in t h e plane (y, 2) = o a n d it is easy to see t h a t if fk -+ o (2) t h e n P ~ f i ~ o. H e n c e if fk ~ o($) it follows t h a t K(~, ",fk)--> o. Moreover, if

f E A ( q )

t h e n

f ( x ) = K(2, x, f).

This proves Theorem I I w h e n q is n o t a constant. I f q is a constant, it is n o t zero so t h a t A (q) c o n t a i n s only t h e element f---- o a n d t h e t h e o r e m is trivially true.

I t is clear t h a t

K(2, x , f )

depends only on t h e values of f a n d its derivatives i n t h e pointset

B(x) -~

(y; x - - y E C1, (•, 2) = o), where Cx = + C according as x E T or x E T , , a n d B (x) = x when x E S. Moreover, B (x) is b o u n d e d a n d closed, i.e. compact, a n d

K ( ~ , x , f )

vanishes if t h e derivatives of f of o r d e r < m vanish on B (x).

I n C h a p t e r 5 we consider the

problem

of C a u c h y w h e n a suitable surface plays t h e part of the plane S---S(~), b u t only for the case t h a t q is homogeneous a n d reduced. 2

1 p ~ f i s t h e b e g i n n i n g of a Taylor series for f w i t h respect t o t h e variable (x, ~).

s In GARDING [2] t h e results of t h i s c h a p t e r were announced for arbitrary hyperbolic and reduced q, n o t necessarily homogeneous. See t h e first footnote to Chapter 5. The third footnote to t h e same chapter contains a correction to G),~DINO [4],

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In Chapter 6, finally, we give some remarks concerning the d o m a i n of de- pendence of the o p e r a t o r I a n d t h e operator J defined by J f ( x ) - ~ f ( x ) - -

- - I q ( O / O x ) f ( x ) . I t summarizes t h e i m p o r t a n t progress in the t h e o r y of phe- n o m e n a connected with H u y g e n s ' principle t h a t h a s been m a d e recently in a paper by P e t r o w s k y [8] a n d also in a paper by t h e a u t h o r [ 4 ] . - I want to t h a n k here C. Hyltdn-Cavallius, who proved L e m m a 2.2, a n d H . Jacobinski f o r a critical reading of parts of t h e m a n u s c r i p t .

C h a p t e r x.

P r o o f o f T h e o r e m I.

L e t q be an a r b i t r a r y p o l y n o m i a l in n variables w i t h complex coefficients, let ~ = (~1 . . . . , ~ n ) # 0 be a n a r b i t r a r y real vector a n d define A (q) as in the introduction. W h a t is m e a n t by f k -~ 0 a n d 9r ~ 0(~) when f k , (k = I, 2 , . . . ) , is a sequence of elements .in C(oo) is explained in the i n t r o d u c t i o n . I t is assumed t h a t t h e r e is a real point x ---- (Xl, 9 9 x,~) such t h a t (x, ~) ---- Xl ~1 + ' " + xn ~,, # 0 a n d f ~ ( x ) - + 0 w i t h I / k whenever A ( q ) ~ f k ~ 0(~) a n d we have to show t h a t in this case q is hyperbolic with respect to ~. I t is shown in L e m m a 2.2 in the next c h a p t e r t h a t if q is hyperbolic w i t h respect to ~ it is also hyperbolic w i t h respect to - - ~ . H e n c e c h a n g i n g if necessary ~ to - - ~ we m a y suppose w i t h o u t loss of generality t h a t (x, ~ ) > o.

L e t ~ be a complex vector and t a complex n u m b e r (if any) such t h a t

q(tt

Then A (q) c o n t a i n s t h e f u n c t i o n

f (t, ~, y) : e ('-x,t~+~).

I t is clear t h a ~ f ( t , ~ , x ) = I a n d when ( y , ~ ) - ~ o one has

(2) f ( t , ~, y) = e -t(~'~) e Iv-x,~-)

Clearly o u r a s s u m p t i o n implies t h a t we c a n n o t find a sequence t (k), ~(~') satisfying (I) such t h a t f ( t (~), $(k), .)_> o(~). L e t us first assume t h a t there exists a vector -~ ~' such t h a t (I) is satisfied for all t. L e t D f be a fixed derivative of f with respect to y a n d B a c o m p a c t set in the plane (y, ~) = o. T h e n if t is real, per- f o r m i n g t h e differentiation and p u t t i n g (y, ~)---~ o a f t e r w a r d s we get as in (2)

D f t -~ D r ( t , ~', V) = 0 (t M) e -t (x, ~), ( M >-- o),

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10 Lars G&rding.

u n i f o r m l y in B. L e t t i n g t ~ co it follows t h a t j ~ - § 0(2). H e n c e t h e r e can be no v e c t o r s s u c h t h a t (I) is satisfied f o r all t.

L e t s be a c o m p l e x n u m b e r a n d p u t ~ = s t ' w i t h a r b i t r a r y b u t fixed ~'.

T h e n t h e p o l y n o m i a l q(z, a) = q (r ~ + a~') in t h e i n d e t e r m i n a t e s ~ a n d a is n o t zero f o r a n y c o m p l e x value s of a. 1 L e t t h e d e g r e e of q ( v , a ) w i t h r e s p e c t to b o t h i n d e t e r m i n a t e s a n d t h e i n d e t e r m i n a t e v be m' a n d m r e s p e c t i v e l y . W e a r e g o i n g to s h o w t h a t m ~ m'. W r i t e q(z, a) a c c o r d i n g to d e s c e n d i n g p o w e r s of ~,

= + . .

I f q~(a) is n o t a c o n s t a n t t h e n m ' > m a n d t h e r e exists a c o m p l e x n u m b e r so s u c h t h a t q ~ I s 0 ) ~ - o . I n a c e r t a i n n e i g h b o r h o o d of s = s o, e v e r y zero t = t (s) of q ( t , s ) - ~ o is of t h e f o r m t(s) = o or

(3) t(s) = a ( s - - so) ~ (I + o(I)),

w h e r e a r o, b is r a t i o n a l a n d o(I) -~ o as s -~ so. N o t all t(s) are b o u n d e d w h e n s -~ So, b e c a u s e t h e n q (t', So) = lim q,~ (s) II (t' - - t(s)) = o f o r e v e r y c o m p l e x n u m b e r t'.

8 ~ 8 o

H e n c e we m a y a s s u m e t h a t b < o in (3). W e also c h o o s e a r g ( s - - s o ) so t h a t a ( s - - s o ) b is r e a l a n d positive. T h e n ~ t ( s ) = la] I s - - s 0 ] b ( I + o(I)) a n d i t is e a s y to see t h a t

D f ~ = D f ( t ( s ) , s$', y) = O(]s --s01 -M) e -(~, ~)~t('s), ( M > o),

u n i f o r m l y in B so t h a t f, ~ o(2) as s--, so. N e x t a s s u m e t h a t qm is a c o n s t a n t b u t t h a t r n ' > m, in w h i c h case m is n e c e s s a r i l y positive. I n a c e r t a i n neigh- b o r h o o d of s = c o , e v e r y zero t = t ( s ) of q ( t , s ) = o is of t h e f o r m t ( s ) = o or

(4) t (,~) = a s b ( I + o (I)),

w h e r e a # o , b is r a t i o n a l a n d o ( I ) - + o as s - ~ o o . N o t e v e r y b is --< I b e c a u s e o t h e r w i s e q(t's, s) = q,~ II (t's -- t(s)) = O(s '~) f o r e v e r y c o m p l e x n u m b e r t' w h i c h c o n t r a d i c t s t h e a s s u m p t i o n m ' > m. L e t b > I in (4) a n d choose a r t s so t h a t a s b is r e a l a n d p o s i t i v e a n d c o n s e q u e n t l y ~Rt(s)-= lal Is lb(I + o(I)). T h e n one g e t s

Df~ =- D f ( t (s), s r y) ~- 0 (1 s I M,) e M, I, I e-(~, ~) ~t (.~), (3/i, M2 > o), u n i f o r m l y in B so t h a t f~ --, o (~:) as s -~ co.

N o w l e t p be t h e p r i n c i p a l p a r t of q, so t h a t q = p + r w h e r e p is h o m o -

T o s a y t h a t q(v, s) i s z e r o m e a n s b e c a u s e v i s a n i n d e t e r m i n a t e , t h a t i t i s i d e n t i c a l l y z e r o , c o n s i d e r e d a s a p o l y n o m i a l i n z .

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g e n e o u s and t h e d e g r e e of p is g r e a t e r t h a n t h e d e g r e e of r, or if q is a con- s t a n t , r = o: Because q is n o t i d e n t i c a l l y zero, p is n o t i d e n t i c a l l y zero. H e n c e we can choose $' so t h a t p(~') ~ o. I f p(~) = o t h e n f o r q(3~ + a~') one would h a v e m ' > m which is impossible. H e n c e p ( ~ ) ~ o, a n d m is t h e c o m m o n d e g r e e of p a n d q. I f m = o t h e n q ( t ~ + i ~ ) = p ( ~ ) ~ o . C o n s i d e r t h e case r e > o , let

~7 be real and c o n s i d e r t h e zeros t = t ( i ~ ) of t h e e q u a t i o n q(t~ + i ~ ) = o . L e t

~t(i~l) a t t a i n its m a x i m u m t ' ( 8 ) i n t h e d o m a i n max~]vk } ~ s w h e n ~ = ~ ( s ) a n d t ( i ~ ) = t(s). By v i r t u e of t h e l e m m a p r o v e d n e x t in this c h a p t e r , f o r sufficiently l a r g e s one has t ' ( s ) = o or

(5) t'(s) = + 0(,))

w h e r e a is real a n d n o t zero, b is r a t i o n a l and 0 ( I ) - + o as s - ~ co. I t is clear t h a t t ( s ) = O(s v) f o r some b ' > o , (actually b ' - ~ I). I f t ' ( s ) w e r e n o t b o u n d e d f r o m a b o v e w h e n s -> c~, one would h a v e a > o and b > o in (5) a n d t h e n

Df~ : D f ( t (s), V (s), y) = 0 (s ~r) e -(x, ~)t' (~), ( M > o), u n i f o r m l y in B so t h a t f~-~ o (8) as s - + co.

I f t ' ( s ) ~ t o one has q ( t ~ + i ~ ? ) ~ o when t > t 0 a n d ~ is real. T h i s r e d u c e s t h e p r o o f of T h e o r e m I to t h e p r o o f of t h e f o l l o w i n g lemma.

L e m m a . L e t q(T, e l , . . . , a~) be a complex polynomial in the i~determi, ates 3, al . . . a, such that when sl . . . 8, are real, the degree with respect to 3 of the polynomial q ( T ) = q(3, s~ . . . . , s~) is positive a~d independe~t of s ~ , . . . , sn. ~ L e t M(s) be the maximum of the real parts of the zeros of the equation7 q (3) = o when maxl~ Isk ] ~ s. Then for sufficiently large s, either 51 (s) = o or

+

where a is real and not zero, b is ratio,el a~d o(I) -~ o as s-+ c>o.

P r o o f . L e t ~ r = M ~ ( S l , . . . , s~) be the m a x i m u m of t h e real p a r t s of t h e zeros of the e q u a t i o n q ( 3 ) - ~ o. I t is clearly a c o n t i n u o u s f u n c t i o n of s ~ , . . . , s , . L e t I S l , . . . , s ~ I be t h e g r e a t e s t of t h e n u m b e r s I S l [ , . . . , Is,,I. L e t M ~ =

= ]Ie(s~ . . . s~, s) be t h e m a x i m u m of M~ w h e n s~, . .., se are fixed a n d se+~ . . . . , s~

vary so t h a t Is~+~ . . . . , s,,] ~ s. I t is also a c o n t i n u o u s f u n c t i o n of s ~ , . . . , s~, s.

This is a l m o s t evident, b u t we give h e r e a f o r m a l proof. P u t M~. ~ Me (s~,..., s~, s')

1 I f w e w r i t e q i n t h e f o r m X' qk (el . . . . ' an) vk w h e r e qm (61 . . . . , am) i s n o t ( i d e n t i c a l l y ) z e r o

0

t h i s m e a n s t h a t m > o a n d t h a t q m ( S l . . . . , sn) i s n e v e r z e r o .

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12 Lars G~rding.

a n d s u p p o s e t h a t I S l - - S ' l , . . . , & - - s ' k , s - - s ' [ <--J. Choose s1:+~ . . . . , sn s u c h t h a t M n -~ M n ( s l . . . . , Sn) = M k a n d p u t sj = csj, (j > k), w h e r e c ~- I w h e n s' --~ s a n d c : s ' / s w h e n s ' < s . T h e n [ s - - s ~ , . . . , s n - - s ~ [ - - ~ l t ~ , ( I - - c ) sl ~ [ ( ~ , s - s ' l ~ a n d also [ s ~ . + l , . . . , s~,[ ~ c s ~--s'. H e n c e by t h e definition of M~. we g e t M ~ - - ~ M ; ~ = !

= i n (81,' . . . , S~)' so t h a t ~T/i --> ~/~ - I1]/, - M g l . N o w w h e n It - - t l , . . . , tn--t,,' I -~

a n d ]tl . . . . , t~, t~, . . . , t~] is less t h a n s o m e c o n s t a n t g r e a t e r t h a n Is1, . . . , s n , sl . . . . , snl t h e n by u n i f o r m c o n t i n u i t y , ] M n ( t i , . . . , in) - - . ~ l n ( t ' , . . . , t')] ~-- S(~) w h e r e e(6) -~ o as ~ -~ o. H e n c e Ms ~ M k - - e ( J ~ a n d by s y m m e t r y , M k ~ - - M ' k - - e ( 5 ) so t h a t I M k - - ~ / ; I -< e (6).

L e t C -~ C [ u 1 . . . . , ud be t h e r i n g of all r e a l p o l y n o m i a l s in t h e i n d e t e r m i n a t e s u l , . . . , u t . A n d e m e n t q E C is called a p r o p e r f a c t o r of p E C if p ~ q q ' w h e r e q' E C a n d q a n d q ' are n o t r e a l n u m b e r s . A n d e m e n t p is called p r i m i t i v e w i t h r e s p e c t to ul if it c o n t a i n s no p r o p e r f a c t o r i n d e p e n d e n t of Ul.

L e t Ak be t h e class of all real p o l y n o m i a l s / 0 = - - P ( v , al . . . . , a~, a ) ~ o 1 s a t i s f y i n g

(6) P (Mk, Sl . . . . , Sk, S) = 0

f o r all r e a l sl . . . . , sk, s s u c h t h a t s ~ o a n d h a v i n g no p r o p e r f a c t o r w i t h t h e s a m e p r o p e r t y . I t t h e n f o l l o w s t r i v i a l l y t h a t P h a s no p r o p e r m u l t i p l e f a c t o r s b u t also t h a t it is p r i m i t i v e with r e s p e c t to T. I n f a c t , l e t P = P 1 P 2 w h e r e /)2 is p r i m i t i v e a n d Px is i n d e p e n d e n t of T. T h e f o r m u l a (6) s h o w s t h a t P 2 ( M k , sl . . . . , & , s ) : o a t e v e r y p o i n t w h e r e /01(sl . . . . ,sk, s ) ~ o . B u t t h e s e p o i n t s a r e d e n s e in t h e r e g i o n s---o a n d M , is c o n t i n u o u s t h e r e so t h a t (6) f o l l o w s f o r P2 a n d c o n s e q u e n t l y /)1 is a c o n s t a n t so t h a t P is p r i m i t i v e .

T h a t An h a s a t l e a s t one e l e m e n t c a n be seen as follows. T h e r e is c e r t a i n l y a r e a l p o l y n o m i a l Q' (T, a l , . . . , an) ~ o w h i c h v a n i s h e s w h e n aj = sj, ( j = I, . . . , n), a n d T ~ 89 (tj + t~), (j, k = i, . . . , m), w h e r e ti, 9 9 tm a r e t h e m > o zeros of t h e e q u a t i o n q(T, Sl, 9 9 s,) ~ o. C o n s e q u e n t l y it h a s a t l e a s t o n e f a c t o r Q in An.

A s s u m e n o w t h a t k > o a n d t h a t P E A k . W e a r e g o i n g to c o n s t r u c t a n e l e m e n t P ' E A k - 1 . I f P ~ = O P / O a k : o , t h e n /0 is i n d e p e n d e n t of ak, SO t h a t b e c a u s e Mk is c o n t i n u o u s i t is i n d e p e n d e n t of sk f o r s u c h sl . . . sk, s t h a t P ( T , sl . . . . , sk, s) ~ o. B u t t h e s e a r e d e n s e in t h e r e g i o n w h e r e Mk is defined so t h a t b e c a u s e Mk is c o n t i n u o u s it is i n d e p e n d e n t of sk f o r all values of t h e o t h e r a r g u m e n t s . H e n c e P ' = P E Ak-~. A s s u m e t h a t /ok ~ o a n d let

1 F r o m n o w o n i n t h i s c h a p t e r , s m a l l G r e e k l e t t e r s i n d i c a t e i n d e t e r m i n a t e s . T h a t _ P = o t h e n m e a n s t h a t a l l t h e c o e f f i c i e n t s o f t> v a n i s h .

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M k - 1 : M k ( 8 1 , . . . , 8 k - l , Jk, 8)

f o r fixed s l , . . . , sk-1 a n d s. I f

Iss

we h a v e one of t h e equalities (7) P ( M k - 1 , sl, . . . , Sk-1, +_ S, s) = O.

Assume n e x t t h a t Is~[ ~ s. L e t t h e values of Ot)/OT a n d Pk be cl a n d c~ rr speetively when T----M~-I, a l : S l , . . . , a k : S ~ a n d a : s . I f n o t c l = c 2 - ~ o , t h e plane c u r v e whose p o i n t s are (sk, Mk), (]S~ I < S), has a t a n g e n t at t h e p o i n t (s~, Mk-1) a n d because Mk--~ M k - i it follows f r o m e l e m e n t a r y c o n s i d e r a t i o n s t h a t this t a n g e n t m u s t be parallell to t h e sk-axis a n d t h i s a g a i n implies t h a t c2----o.

H e n c e we g e t

s 81 . . . . , 8 k - 1 , 8k, 8) -~- 0 p

(S)

P k (Mk-l, 81 . . . . , 8k--1, 81, 8) ~- O, a n d t h e s e e q u a t i o n s are also t r u e if cl = c2-= o.

C o n s i d e r t h e discriminaDt /~ of P w i t h r e s p e c t to ak. I t b e l o n g s to C = C[T, a l , . . . , ak-1, a] a n d we w a n t to p r o v e t h a t it does n o t vanish. P u t C1 = C [T, al, ., ak, a], let C' be t h e q u o t i e n t field o f C a n d let C' [ak] be t h e r i n g of all real p o l y n o m i a l s in ak with coefficients in C'. I t is clear t h a t an e l e m e n t in C' [a~.] whose d e r i v a t i v e with r e s p e c t to ak vanishes is i n d e p e n d e n t of ak. H e n c e because /~ d e p e n d s on ak, it follows I t h a t if B = o t h e n P is of t h e f o r m P 2 P 2 w h e r e /)1 a n d /)2 are in C' [ak] a n d /)1 d e p e n d s on ak. B u t t h e n 2 we can also w r i t e P as P [ P ~ where P1 ---~ Pl/)1 a n d t)2 : P 2 P~ are in C1 a n d Pl a n d P2 are suitable e l e m e n t s of C'. H e n c e P has t h e p r o p e r m u l t i p l e f a c t o r 151 so t h a t P ~ Ak a g a i n s t t h e a s s u m p t i o n . C o n s e q u e n t l y R ~ o.

I t follows f r o m (8) t h a t

(9) /~ (Mk-1, 81, . . . , 8k-1, 8) ⁼ ^{O .}

M o r e o v e r , / ) + ~ P (T, al, 9 9 _+ a, g) ~ o because o t h e r w i s e JP has t h e f a c t o r ak T a which implies t h a t P is n o t p r i m i t i v e w i t h r e s p e c t to v a g a i n s t the a s s u m p t i o n t h a t P E A k . H e n c e if / ' I = P + P - R we h a v e P x ~ O a n d by v i r t u e of (7) a n d (9)

P I ( M k - 1 , .el, . . . , 8k-1, 8)-= 0

w h e n s ~ o. H e n c e P has a t least one f a c t o r P ' in Ak-1. S t a r t i n g f r o m Q in A , we can t h u s c o n s t r u c t a n e l e m e n t Qn-IEAn-1 and, c o n t i n u i n g , finally a n

1 VAN DER W A E R D E N [ I 2 ] I p. 93"

2 1.C.p. 75--77.

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14 Lars G~rding.

e l e m e n t G in Ao, Because

M ( s ) ~ Mo(s)we

g e t

G(M(s),

8 ) : o, (s ~ o). N o w in a n e i g h b o r h o o d N of s = o % every s o l u t i o n t of G ( t , s ) - ~ - o is e q u a l t o o n e o f a finite n u m b e r of different c o n v e r g e n t series of c e r t a i n real f r a c t i o n a l d e s c e n d i n g powers of s, one of which m a y vanish i d e n t i c a l l y while t h e o t h e r s have t h e f o r m

(1o) a s b + . . . . a s b ( I + o(I)),

w h e r e a ~ o, b is r a t i o n a l and s b is t h e h i g h e s t p o w e r of s t h a t occurs in t h e series, so t h a t o(I)--> o as s - ~ co. All t h e s e series assume d i f f e r e n t values in a suitable N a n d because

M(s)

is c o n t i n u o u s it is i d e n t i c a l with one of t h e m t h e r e a n d we assume t h a t it is (Io). T h e n a is real because it is t h e limit of

M(s)s -b

as s-+ 0% a n d this proves t h e lemma, w h i c h of course also is t r u e if we by

M(s)

m e a n t h e m i n i m u m of the real p a r t s of the zeros of q ( T ) ~ o in t h e r e g i o n maxk l skl -< 8.

C h a p t e r 2.

H y p e r b o l i c P o l y n o m i a l s .

R e d u c e d p o l y n o m i a l s . L e t q ( ~ ) ~ q ( ~ l , . . . , C~) be a p o l y n o m i a l in ~1 . . . . , ~n w i t h complex coefficients a n d consider t h e d i f f e r e n t i a l e q u a t i o n

(I) q (0/0 x)f(x) =

o,

w h e r e

f ( x ) : f ( x l , . . . , x,)

is a c o m p l e x a n d infinitely differentiable f u n c t i o n of n r e a l variables x l , . . . , x , . W r i t e x----(Xx . . . . , x~) a n d c o n s i d e r a real l i n e a r t r a n s f o r m a t i o n

X' ~ X ]~V

w h e r e _~r is t h e transpose of a r e a l q u a d r a t i c n o n - s i n g u l a r m a t r i x M. ] t t h e n follows t h a t

O/Ox-=(O/Ox')_~[

so t h a t (I) becomes

(2) q (o/o~' M) f(x' 5t-~) = o.

L e t us p u t

q'($')=q(~'M)

a n d

f ' ( x ' ) = f ( x ' ] ~ - l ) .

I t is clear f r o m (2) t h a t

f ~ - f '

is a l i n e a r one-to-one m a p p i n g of t h e solutions of (I) u p o n t h e solutions of

(3)

q' (O/Ox') f ' (x') = o.

Because t h e a r g u m e n t

O/Ox

of q in (1) a n d t h e a r g u m e n t x of f t r a n s f o r m d i f f e r e n t l y a n d we h a v e to c o n s i d e r a r g u m e n t s of q which like x are vectors w i t h n u m e r i c a l c o m p o n e n t s , it is c o n v e n i e n t to do as follows. C o n s i d e r t w o v e c t o r spaces E a n d E* w h e r e E consists of all vectors w i t h n real c o m p o n e n t s a n d

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E* consists of all vectors w i t h n complex components. W e denote t h e elements of E by L a t i n letters x, y , . . . a n d t h o s e of E* by Greek letters ~, ~, ~ , . . . I f E* is subjected to t h e linear t r a n s f o r m a t i o n ~-~ ~ ' M , the elements of E should be t r a n s f o r m e d according to the f o r m u l a x - ~ x ' ~ - l . I n such a way the scalar p r o d u c t

( x , = x : + . - . +

r e m a i n s i n v a r i a n t if we s u b s t i t u t e x' f o r x and $' f o r ~. W h e n it is a complex or real vector, the a r g u m e n t of q should always be t h o u g h t of as a n element of E*, while the a r g u m e n t of a solution of (I) o u g h t to be considered an element of E. W e have tacitly stuck t o this convention in the preceding chapter.

A suitable choice of M m a y m a k e (3) easier to h a n d l e t h a n (I). L e t l be the least of all integers l' for which there exists a m a t r i x M" such t h a t q(~'M') is a polynomial in ~ { , . . . , $'t, only w h e n ~ { , . . . , ~n are considered as indeter- minaLes. I f M is a m a t r i x corresponding to l it is clear t h a t x'z+l . . . . , xn p e n t e r into (3) only as parameters. A polynomial for which l ---- n will be called reduced.

L e t q be a n a r b i t r a r y polynomial in n variables with complex coefficients.

The following concept is useful.

Definition. L e t ~9(q) be the set of all real vectors ~' in E* such t h a t q (7 + t = q (7)

for all real n u m b e r s t and real vectors ~ in E*.

L e m m a 2.1. The set ~ (ql is linear over the real numbers. A polynomial q is reduced , f and only ~:f ~ ( q ) - ~ o . I f the

independent and 0 (t+:) . . . . , 0 (') constitute a

= q (~i 0 (1) + "'" + ~n 0 (~)) is reduced.

real vectors 0(1), . . . , ()(") are linearly basis of Q(q) then

q'(~'~,..., ~;)-~

Proof. I f ~/' a n d ~/" are in f2 (q) t h e n

q(~? + t'~ I' + t"~?")----q(~ + t'~l')=q(~l)

for all real ~/ a n d real t' a n d t". H e n c e ~Q(q) is linear. Assume t h a t q is n o t reduced a n d let /,(1), . . . , #(,) be the columns of such a m a t r i x M t h a t q ( ~ ' M ) =

_ _ ' ( 1 ) '

- - q ( ~ l / , + "'" + ~n/, (~)) is i n d e p e n d e n t of ~n. T h e n q(~ + t / , ( n ) ) = q ( ~ ) f o r all re~l ~ a n d t. H e n c e ~(q) contains t h e e l e m e n t /,(n)~ o. Conversely, suppose t h a t o # y'GS2(q) a n d let

/,(1),.,., p(n--1)

a n d /,(n)__ 7' be a basis for all real vectors ~]. T h e n

q (~{ #(z) + ... + ~;, #(,)) _ q (~ #(:) + .,. + 5 - : #(n-l))

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16 Lars Gtirding.

for all real ~ , . . . , $~. B u t t h e n the same equality holds for i n d e t e r m i n a t e $~, . . . , ~,~

a n d consequently q is n o t reduced. As to the last assertion of the lemma, the same a r g u m e n t shows t h a t q ( ~ 0/1) + ... + ~ 0(~1) is a p o l y n o m i a l in ~ , . . . , ~ alone, say q' (~, . . . , $~). I f q' were n o t reduced, t h e n one could find real n u m b e r s 71,-.., T~

n o t all zero such t h a t

q'(vl + t v l , . . . , 7, + t T ; ) = q ( w , . - . , W)

for all real t a n d T 1 , . . . , 7 ~ - B u t t h e n q ( 7 ) : q ( v + tT") for all real t a n d T i f T" = T~ t~l) + "" + T~O(~). H e n c e o ~ V" E t? (q) a n d 0 (l+l), . . . , O(') is no~ a basis of Q(q) against assumption.

H y p e r b o l i c p o l y n o m i a l s . L e t E consist of all real elements in E*, i.e. of all vectors with n real components. L e t q be a . p o l y n o m i a l in n variables w i t h complex coefficients, let it be hyperbolic 1 w i t h respect to ~ E E a n d let t h e degree m of q be positive. I f p is the principal p a r t of q a n d T e E , t h e n because p(~) # o, t h e degree of q(t~ + i T ) = p ( ~ ) t ~ + ... w i t h respect to t is m. Hence

there are complex n u m b e r s v~ (~, iT) , (v ---- I, . . . , m), such t h a t

~n

(4) q (t + i 7) = p 1-[ (t + v, i T))

1

f o r a n y complex t, L e t ~ t be the real and ~ t t h e i m a g i n a r y p a r t of t. Because q is hyperbolic w i t h respect to ~ we g e t

q(t~ + i v ) : q ( ~ t ~ +

i(3t~ +T)) ~

o

if ~ t > to. Now q(t ~ + iT) vanishes when t = -- v, (~, iv). H e n c e m a x , -- ~ v, (~, i T) --~ to

so t h a t

(5) min,. ~ v , (~, iv) --~ - - to

for all real T. Conversely, if p (~) # o a n d (5) is satisfied, it follows from (4) t h a t q(t~ + i T ) # o w h e n ~ t > to so t h a t q is hyperbolic w i t h respect to ~.

I t follows directly f r o m the definition t h a t if q is hyperbolic w i t h respect to ~, i~ is also hyperbolic w i t h respect to a n y positive m u l t i p l e o f ~. The same conclusion is, however, also t r u e for the negative multiples of ~. I n order to prove this it is sufficient to prove t h e following lemma.

L e m m a 2.9.. I f a polynomial q is hyperbolic with respect to ~, it is also hyp~'- bolic with respect to --~.

1 See the definition given in the beginning of the introduction.

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P r o o f 3 I f q is a c o n s t a n t it is n o t zero, a n d i t follows t h a t q is h y p e r b o l i c with r e s p e c t to all real vectors, in p a r t i c u l a r - - ~ . I f q is n o t a c o n s t a n t we c a n use (4). T h e sum /~ = ~ v l ( ~ , i ~ ] ) + .-. + ~ v ~ ( ~ , iv) is a p o l y n o m i a l in

~ , . . . , Un of d e g r e e ~ I a n d by v i r t u e of (5) b o u n d e d f r o m below a n d h e n c e it m u s t be c o n s t a n t . B u t t h e n w i t h r , - ~

,~v,(~,i~)

we g e t

r , = R - - r~ . . . r , _ ~ - - r,+~ r~ ~ R -F ( m - - I) t o

f o r all ~. B u t t h e n it follows f r o m (4) t h a t

q(-- t~+i~) ~ o

w h e n t > R + ( m - - I ) to.

Also p ( - - ~ ) ~ ( - - 1)~p(~) ~ o. H e n c e t h e l e m m a is proved.

T h e d e g r e e of

p(t~ + 7 ) : P ( ~ ) W + " "

w i t h r e s p e c t to t is m. H e n c e t h e r e are m c o m p l e x n u m b e r s u,($, ~), (v = I . . . . , m), such t h a t f o r a n y c o m p l e x t,

(6)

p(t~ + ~)=p(~) f i (t + u,C~,V))

a n d in p a r t i c u l a r w h e n t-~--o,

(7) p (7) = p ] I

1

T h e f o l l o w i n g i d e n t i t i e s in w h i c h a ~ o is a c o m p l e x n u m b e r , ~' a v e c t o r in E*

such t h a t p ($') ~ o, U an a r b i t r a r y e l e m e n t in E* a n d a suitable l a b e l l i n g of t h e n u m b e r s u,(~, U) is u n d e r s t o o d are i m m e d i a t e c o n s e q u e n c e s of (6) a n d t h e homo- g e n e i t y of p,

u, (~, ~) = I, u, (~, a 7) = a u, (~, U),

(8)

u , ( a ~ , u ) : a - Z u , ( ~ , ~ ) , u,(~,$ + a T ) = ~ + au,($,~),

I t is clear t h a t (6), (7) a n d (8) are valid when p is a n y h o m o g e n e o u s poly- nomi~l of d e g r e e m aud p ( $ ) ~ o.

Lemma

2.3. A necessary and sufficient condition that a homogeneous polynomial p of positive degree is hyperbolic with respect to ~ is that p (~)~ o and that the

numbers u, (~, 7) defined by

(6)

are all real when ~ is real.

P r o o f . L e t p be h y p e r b o l i c with r e s p e c t to ~. A p p l y i n g (8) we get if a a n d

~/ are real

I o w e t h e p r o o f t o C. I - I Y L T ~ N - C A V A L L I U S . 2 - 642127 Acts mathematiea. 85

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18 Lars Ghrding.

By virtue of (5), tile left side is bounded from below for all v a n d real a . B u t this clearly implies t h a t ~ u , (~, ~/)---o for all v.

Conversely, if 19 (~) ~ o a n d t h e numbers u, ($, ~7) are real when ~/is, a p p l y i n g (8) we get

p(t~ + i~)=p(~)ii(t

+ i u , ( ~ , r ~ o

1

w h e n t > o (or t < o), so t h a t p is hyperbolic with respect to ~.

R e m a r k . M u l t i p l y i n g both sides of (7) by p(~)-I we g e t p (~)-~/9 (~) = I [ u,

(~, ~).

1

H e r e the r i g h t side is real so t h a t p ( ~ ) - l p ( r l ) is a real p o l y n o m i a l in r/.

Our n e x t 1emma is classical.

L e m m a 2.4. L e t

a n d

t m + a I t m-1 -{- " ' " -~- a,, : fi (t -- t,)

1

m

t m + bl tin-1 + . . + b , ~ = l I ( t - - s ~ )

1

be two p o l y n o m i a l s w i t h complex coeffieients. T h e n there e x i s t ; a labelling o f the n u m b e r s s x . . . sm such that m a x , It, - - s, I tends to zero w h e n ax, 9 9 am are f i x e d a n d m a x ,

la,--b,I

tends to zero.

Ostrowski x proved the more precise result t h a t if

e(a, b ) = 4 m m a x , ( I , I a, li/", Ib, I 1/')

(Y,

[a~ - - btz]2) 1.2~n ,

/z

t h e n there exists a labelling of the n u m b e r s s x , . . . , sm such t h a t max, It, ^- s,I -<

e(a, b).

L e m m a 2.5. I f a p o l y n o m i a l q is hyperbolic w i t h res/geet to ~, then also its p r i n c i p a l p a r t is.

Proof. I f q is a constant, t h e n /9 = q a n d the lemma is trivial. H e n c e assume t h a t t h e degree m of q is positive, let (6), (7) a n d (8) refer to t h e prin-

I [7] 1 'j. 2 0 9 - - 2 1 2 .

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cipal p a r t p of q, let 7 a n d s ~ o be real a n d p u t q~(t) = s - ' n p ( ~ ) -~ q ( s t ~ + i s n ) . T h e n as s ~ co, qx(t) = t '~ + ... c o n s i d e r e d as a p o l y n o m i a l in t t e n d s t o p ~ (t) =

= p ( ~ ) - l p ( t ~ + i ~ ) . N o w by v i r t u e of (4) a n d ( 6 ) t h e zeros of q x ( t ) = o a n d pl(t) = o are t -~ - - s -1 v, (~, i s 7) a n d t = - - u, (~, i7) r e s p e c t i v e l y , (v = I . . . . , m).

H e n c e t h e p r e c e d i n g l e m m a c o m b i n e d w i t h (8) shows ~hat m i n , s - X ~ v ~ ( ~ , i s T ) -+ m i n , ~ t ' u , ( ~ , 7)

as s - + c o . H e r e by v i r t u e of (5), the l i m i t of the l e f t side is > o so t h a t min~ ~ i u , ( ~ , 7 ) > - - o . H e n c e c h a n g i n g z/ to - - ~ a n d u s i n g (8) we g e t

o --< min ~ t u ~ (~ e, - - 7 ) = min~ - - ~ i u ~ ( ~ , 7) ~--- - - m a x , ~ i u , ( ~ , 7),

so t h a t m a x , ' ~ i u , ( 8 , 7)--<o. H e n c e all t h e n u m b e r s u , ( ~ , n ) are real w h e n ~/ is.

H e n c e L e m m a 2.3 shows tha6 ~o is h y p e r b o l i c with r e s p e c t to ~.

T h e converse of this l e m m a is n o t true. I n f a c t , p = ~ is h y p e r b o l i c w i t h respee~ to ~ = (I,O) b u t p u t t i n g q = C~ + ~ we have

q (t, + i n ) = (t + i7 ) + = (t + i n l + V )(t + i7 - Vi-7 ) so t h a t

w h i c h is n o t b o u n d e d f r o m below. A less t r i v i a l e x a m p l e is given by

Now t h e r e is one i m p o r t a n t case when t h e converse of o u r l a s t l e m m a is t r u e , ~ n a m e l y w h e n q is n o t d e g e n e r a t e . L e t (6) r e f e r to t h e p r i n c i p a l p a r t p of q. W e say t h a t q is n o t d e g e n e r a t e if u , ( ~ , 7 ) # u v ( ~ , n ) w h e n v # / z a n d 7 is real a n d n o t p r o p o r t i o n a l to ~. To prove o u r assertion write q ( s ) = q ( s ~ + i~), p ( s ) = p ( s ~ + i7) a n d r ( s ) = r ( s ~ + i7) w h e r e r - ~ q - - p a n d resolve qi0 -x i n t o

p a r t i a l f r a c t i o n s as follows

~-~ r ( - - i u , ) ,

(9) q ( 8 ) p ( 8 ) - l m I + ,'(8)~9(.~) -1 = t + ~=l~V~..-i: ~ ) [ 8 + i,,~) -1

w h e r e p ' ( s ) = d p / d s and u, = u,(~, 7) a n d w h e r e we have used (8). L e t E~ be a l i n e a r subspace of t h e space E of all v e c t o r s with n r e a l c o m p o n e n t s such t h a t E~ does n o t c o n t a i n ~ a n d ~ a n d E~ t o g e t h e r s p a n E. By a s s u m p t i o n , rain, ]p'(--iu,,)]---min, [ P ( ~ ) [ I I I - - u , + ut, [ has a positive m i n i m u m M1 w h e n

tt=l

WITH CONSTANT COEFFICIENTS.