The Ritt theorem in several variables
C. A. BEREI~STEII# a n d M. A. DOSTAL
University of Maryland and Stevens Institute of Technology*)
w l. Formulation of the problem
T h e m o t i v a t i o n for t h e p r o b l e m t r e a t e d in this n o t e arises in t h e t h e o r y of con- v o l u t i o n e q u a t i o n s . L e t U be a locally c o n v e x space o f f u n c t i o n s or d i s t r i b u t i o n s . I t is a s s u m e d t h a t b y m e a n s of a t r a n s f o r m a t i o n ~ o f F o u r i e r typeS), t h e s p a c e
U is i s o m o r p h i c to a s u b a l g e b r a s of t h e a l g e b r a ,vg = ~-~(C n) of all e n t i r e f u n c t i o n s in {3% U s i n g t h e i n v e r s e t r a n s f o r m a t i o n ~ 1 one can t r a n s f e r t h e o p e r a t i o n o f m u l t i p l i c a t i o n f r o m ~7 to U. T h e r e s u l t i n g ring m u l t i p l i c a t i o n in U is called t h e c o n v o l u t i o n a n d is d e n o t e d r * ~0 (r ~v C U). More generally, let G E ~A b e such t h a t t h e m u l t i p l i c a t i o n b y t h e f u n c t i o n G is a c o n t i n u o u s e n d o m o r p h i s m c}l'la o f U. I f %~ is t h e c o n t i n u o u s e n d o m o r p h i s m o f U c o r r e s p o n d i n g to c ~ a u n d e r t h e i s o m o r p h i s m ~, t h e n ~(fG is called a c o n v o l u t o r of t h e s p a c e U. S o m e t i m e s - - b u t n o t a l w a y s - - t h e r e exists a d i s t r i b u t i o n S (or a n o t h e r " g e n e r a l i z e d f u n c t i o n " ) s u c h t h a t t h e F o u r i e r t r a n s f o r m of S is t h e f u n c t i o n G, a n d for e a c h r C U, ~r162 c a n be i n t e r p r e t e d as t h e c o n v o l u t i o n S * r in s o m e g e n e r a l i z e d sense.
G i v e n f C U, consider t h e e q u a t i o n
S * u = f (1.1)
w h i c h is equivalent to the equation S(~)~;(~) = f(~). It is natural to ask for necessary a n d sufficient conditions for the solvability of (1.1) in U. A n obvious necessary condition is that
f/S
be entire. In order to conclude that ~ S is in U one usually*) The authors wish to thank Instituto de Matematica Pura e Aplicada, l~io de Janeiro, for its support and hospitality.
1) cj can be either the classical Fourier-Laplace transformation or the Fourier-Borel trans- formation or another similar transformation depending upon the nature of the space U. The inverse transformation will be denoted by cj-1; instead of cj(r we shall write r Similarly,
stands for cj(U), etc.
268 C . A . B E I ~ E N S T E I N A N D ]VL A . D O S T A L
has to use additional properties of S and U, and this is often done b y combining a theorem of Paley-Wiener t y p e 2) with the following intermediary step: There is another subMgebra ~ of cA containing S and f and such that
(S) whenever F, G C ~ are such that F/G E ~ , then F/G E ~?~.
Obviously, if ~3 = (/ has property (S), then the above necessary condition for the solvability of (1.1) is also sufficient. However, this is rarely the case, and one usually has to consider a different subMgebra '%)6. In the sequel, subalgebras
~/~ with the property (r will be called stable. Denoting b y F~N the quotient field of ~?d, stability of ~N means F ~ N ~4 = 9:0.
As an illustration of the previous general argument, consider the following two examples:
Example 1. Let U = c~A ' be the dual of the space ~>4 endowed with the compact-open topology. B y means of the Fourier-Borel transform (eft w 2), U is isomorphic to the algebra U = E x p = the set of all entire functions of exponential type in {Y. The stability of E x p is a classical result of LindelSf [18] (eft also below).
Hence a necessary and sufficient condition for the solvability of equation (1.1) in the space of analytic functionals in 13" is that 0~S C :~q. Non-triviM examples of the use of the stability of Exp in the s t u d y of convolution equations can be found in H6rmander [13] and MMgrange [19, 20].
Example 2. Let U =- cE' be the space of distributions with compact support in R ~. Consider equation (1.1) with S , f C ~'. Then, although the function f/S is of exponential t y p e (provided it is entire), in generM j~S ~ c~,, unless S is in- vertible (cf. [12]). Hence c~, is not stable, and it is the non-stability of c?~, which causes serious difficulties in the s t u d y of such convolution equations. In order to overcome these difficulties, one often has to resort to intricate methods (cf. [13, 20]).
I t therefore seems interesting to examine the stability of those subalgebras of d which arise in the study of convolution equations. Below we list some of them:
~P: polynomials in (In;
E: finite exponential sums, i.e. entire functions of the form
2) For example, the Paley-Wiener-Schwartz theorem for the case of distributions, or the PSlya-Ehrenpreis-Martineau theorem (cf. w 2) when analytic functionMs are involved, etc.
T H E I~ITT T H E 0 1 ~ E ~ 117 S E V E I ~ A L V A R I A B L E S 269
H(~) = ~ hje<o~ "~>, (~ e C") (1.2)
j = l
where hj are complex numbers, and 0~ C (Y, also called frequencies of H, are given points in lY 3);
E2: exponential polynomials in C ~, i.e. functions of the form (1.2) with hj E 7 ; F,7: entire functions of the form HIP for some H C E2, P C ~?;
~ o : Fourier transforms of distributions q ) q ~g' such that, for some constants t > _ 0 , r > 0 , ~ > 0 and A real (all depending on q)),
^
m a x ]~b(~')l >_ c e x p [Aco(~) @ hi~l(~)], (V ~ C C"), (1.3) where A(~; r) denotes the p o l y d i s k of center ~ and radius r; ~ = ~ § i~; h[~ 1 is the supporting function of the s u p p o r t of q~; a n d ~o(~) = In (2 -t- 1~]). ~) (For the properties of such classes, see [3, 4, 5]);
~ : Fourier transforms of Schwartz distributions with compact support; or, more generally,
(r the same for Beurling distributions (cf3));
Exp: entire functions in C ~ of exponential type.
The stability of ~ is simple (ef. e.g. [9]). T h a t E is stable for n = 1 constitutes the so-called R i t t t h e o r e m [23]. F o r n > i this was p r o v e d b y Avanissian and Martineau [1]. E 2 is clearly non-stable. Indeed, for n = 1,
sin z
- - e E~ \ E~ (z e C). (1.4)
Z
I t can be easily established t h a t c~ 0) is stable [5]. ~ ' ' and more generally c~, 6 0 ) ,
are not stable (ef. E x a m p l e 2 above). The stability of E x p was p r o v e d b y E. Lindel6f [18] for n = 1. The extension to a r b i t r a r y n >_ 1 is due to Ehrenpreis [10] and Malgrange [19].
I t remains to be found, first how " u n s t a b l e " E 2 really is a), and secondly,
3) I n w h a t follows it will a l w a y s be a s s u m e d t h a t t h e frequencies Oj are pairwise d i s t i n c t a n d t h e coefficients hj are all non-zero. Besides, < , > denotes t h e bilinear p r o d u c t in On: <0, ~> = 01~1 + 9 . . + O n ~ .
4) A c t u a l l y , for c0 one m a y t a k e a n y c o n t i n u o u s s u b a d d i t i v e f u n c t i o n in R" satisfying c e r t a i n g r o w t h conditions. T h e n , i n s t e a d of ~ ' , one has to t a k e t h e space cd,~o of B e u r l i n g d i s t r i b u t i o n s (of. [2]).
~) i.e., to describe t h e s t r u c t u r e of e n t i r e f u n c t i o n s of t h e f o r m Y/G where F , G E E 2 (or, e q u i v a l e n t l y , F , G C E 9 .
270 C . A . B E I ~ E I ~ S T E I N A I ~ D M . A . D O S T A L
w h e t h e r E 2 is s t a b l e or not. O u r m a i n o b j e c t i v e in this n o t e is to s h o w t h a t b o t h questions h a v e a c o m m o n answer:
MAIN T~EOREM. The subalgebra E2 is stable. Hence the most general form of an entire function of the form FIG, F, G E E 2, (cf.5)) is
F H
G P (H ~ E2, P c ~). (1.5)
G i v e n a n e x p o n e n t i a l p o l y n o m i a l H , t h e g r e a t e s t c o m m o n d i v i s o r o f its coeffi- cients,
dH = (hi, . . . , hs), (1.6)
will be called the content of H. F u r t h e r m o r e , l e t f = HIP be a n y e l e m e n t o f E2"
T h e n we shall s a y t h a t HIP is a reduced form o f f, p r o v i d e d (P: dH) = 1. I f t h i s is so a n d f = H*/P*, for s o m e H* C E 2 a n d P* C ~ , t h e n it is e a s y to see t h a t PIP* a n d dHIdH.. H e n c e t h e following l e m m a holds:
LnMMA. Every function in E2 has a unique reduced form 6). Furthermore, let H, F E E~ be such that Q = H / F E ~ . Then Q[dH.
T h e n t h e m a i n t h e o r e m c a n be r e f o r m u l a t e d as follows:
T~EOI~EM 1. Let F, G E E 2 be such that FIG C ~i. Then there exist unique 6) H E E 2 and P E ~ such that F / G = H / P and (P, dH)= 1.
As a n a p p l i c a t i o n o f T h e o r e m 1 one obtains:
T ~ E O ~ E ~ 2. Let F, G E E 2 and F/G E d . Let HIP FIG. Then PIdG.
COROLLARY 1. Let F, G be exponential polynomials in C" such that FIG is entire and dc = 1. Then FIG is also an exponential polynomial.
COROLLARY 2. E is stable (n >_ 1).
As w a s m e n t i o n e d a b o v e , Corollary 2 for n - - 1 is t h e R i t t t h e o r e m [23]. O t h e r proofs o f R i t t ' s t h e o r e m were g i v e n b y H . Selberg [24], 1 ). D. L a x [16] a n d A. Shields [25]. Shields p r o v e s t h a t , for n ~- 1, t h e h y p o t h e s e s F E E 2, G E E a n d FIG E d i m p l y FIG E E,~. H e also m e n t i o n s t h a t , a c c o r d i n g to a n u n p u b l i s h e d r e s u l t o f
6) Unique up to a constant multiple of H and P.
be the reduced form of
THE I ~ I T T TttEORE)/s IN SEVEI~A~, VAll, IABLES 271 W. D. B o u w s m a , Corollary 1 holds w h e n n = 1. F i n a l l y , Corollary 2 is due t o V. A v a n i s s i a n a n d A. M a r t i n e a u (unpublished [1]). 7)
T h e m a i n t h e o r e m is e s t a b l i s h e d in Section 2. T h e o r e m 2, as well as a n o t h e r a p p l i c a t i o n o f T h e o r e m 1 are discussed in t h e concluding Section 3.
T h e o r e m s 1 a n d 2 were a n n o u n c e d in our n o t e [7].
F o r applications o f e x p o n e n t i a l p o l y n o m i a l s in one v a r i a b l e see [15].
w 2. Proof of the main theorem
F o r ~ E C ", ~ d e n o t e s t h e c o m p l e x c o n j u g a t e o f ~, i.e. ~ = ( ~ 1 , - . . , ~n).
W h e n ~ is considered as a p o i n t in R 2'~, t h e coordinates o f ~ are (Re gl, I m ~1 . . . . , R e ~n, I m ~ ) . T h e E u c l i d e a n i n n e r p r o d u c t in R m will be d e n o t e d < , >~. W e recall f r o m w 1 t h a t t h e bilinear p r o d u c t in C n, d e n o t e d s i m p l y b y < , >, is
< z , ~ > = Z l ~ l + . . . + z n ~ , z , ~ C ~.
I f F is an e x p o n e n t i a l p o l y n o m i a l w i t h frequencies a ~ , . . , %, i.e.
P
F(~) ~- Z Pj(~) e<"j'r (2.1)
j = l
w h e r e Pj are polynomials, we will d e n o t e b y [F] t h e c o n v e x hull o f t h e p o i n t s 51 . . . . , @ in R 2". L e t h[f ] be t h e s u p p o r t i n g f u n c t i o n of t h e set [F], i.e. for each ~ E C " = It 2"
h [ F ] ( ~ ) = m a x < x , C>2n = m a x <(~j, ~>2n ~ - m a x ] ~ e < a j , ~ > . ( 2 . 2 )
xE [F] 1 < j < p l < j <_p
W e shall n e e d a few simple facts a b o u t a n a l y t i c functionals, i.e. e l e m e n t s o f d ' (cf. [14, 17, 21, 26]). T h e F o u r i e r - B o r e l t r a n s f o r m /~(~) = #(e <'' :>) establishes a n i s o m o r p h i s m of t h e spaces ~4' a n d E x p . G i v e n # E ~ ' , t h e indicator p , of
# is d e f i n e d b y
- - log I~(t~)l p~(~) = lim
t-+co t
a n d its u p p e r - s e m i c o n t i n u o u s r e g u l a r i z a t i o n 15, 15,(~) = lim p,(~'), is p l u r i s u b h a r m o n i c .
7) Professor H. S. Shapiro has kindly informed us that several years ago he proved Corollary 2 by means of an inductive argument. His proof has not been published. Added in proof: In the meantime K. Kitagawa [27] published a proof of Corollary 1. His proof is rather sketchy.
A complete proof of Corollary 1 was recently announced by V. Avanissian and 1~. Gay in [28, 29].
272 C. A . t 3 E I % E N S T E I I ~ A:ND ~I. A . D O S T A L
A carrier of a n a n a l y t i c f u n c t i o n a l # is a c o m p a c t subset K of 6 ~ such t h a t for e v e r y n e i g h b o r h o o d U o f K t h e r e is a c o n s t a n t C such t h a t
1~(r _< c sup Ir l,
z E U
for all r E Q/. A c o m p a c t c o n v e x set K is called a convex support o f # if K is a m i n i m a l c o m p a c t c o n v e x carrier o f #, i.e. K is a carrier o f # such t h a t i f L is a n o t h e r carrier o f # a n d L c _ K , t h e n ch. L = K , w h e r e ch. L d e n o t e s t h e c o n v e x hull of L.
I n t h e sequel t h e P d l y a - E h r e n p r e i s - M a r t i n e a u t h e o r e m will be u s e d in t h e following f o r m u l a t i o n (cf. [14], Th. 5.2, Cor. 5.3):
T~nORE~ I. Let # E ~ ' , then
~,(~) ~- inf{hK(~): K carries #}.
Hence # has a unique convex support if and only i f ~5, is a convex function.
(2.3)
I n [4, 6] we established t h e following lower e s t i m a t e for e x p o n e n t i a l polynomials:
T~nORE~ I I . Let P E E2, i.e. P is an exponential polynomial. Then, for each
> O, there exists a constant C ~-- C(s, P) such that if f is an analytic function in the polydisk zl(~, s) = A,
A = {~' E C": m a x I~j - - $/] ~ e}, (2.4) J
then
If(~)[ chip](0 ~ C m a x [f(z)P(z)]. (2.5)
z E A
To e v e r y e x p o n e n t i a l p o l y n o m i a l t h e r e corresponds, via t h e F o u r i e r - B o r e l t r a n s - form, a u n i q u e #p E c9/' such t h a t /~p(~) = P(~). H e n c e we h a v e
COROLLARY 1. l~or every P E E2, [P] is the unique convex support of #p and
~.~(~) --p.~(~) ~ h~(~). (2.6)
(Indeed, it suffices to set
f
~ 1 in T h e o r e m I I a n d a p p l y T h e o r e m I).L e t A be a c o m p a c t c o n v e x subset o f R m. F o r a n a r b i t r a r y 0 E R m, set
A ~ I f A ~ consists o f one p o i n t only, 0 is called a
regular direction o f A: T h e set o f all r e g u l a r directions will be d e n o t e d reg A.
T h e set o f all e x t r e m a l points will be d e n o t e d e x t A.
LnM~A 1 (cf. [8]). Let A and B be compact convex sets in R m. Then for each O,
T H E R I T T T H E O R E M I N S E V E R A L V A R I A B L E S 273
Moreover, every z in
A ~ gl e x t A = e x t (A~ (2.7)
(A -~ B ) ~ = A ~ ~- B ~, (2.8)
e x t (A § B) c__ e x t A -~ e x t B. (2.9) e x t (A + B) has a unique decomposition z = z 1 ~- z 2, z 1 E e x t A, z 2 E e x t B. Although the inclusion in (2.9) cannot be replaced by equality, one has for every 0 C reg A [J reg B,
(ext (A @ B)) ~ = (ext A ) ~ @ (ext B ) ~ (2.10)
( R e l a t i o n s (2.7)--(2.9) a r e o b v i o u s . T h e u n i q u e n e s s o f t h e d e c o m p o s i t i o n z = z 1 -}- z 2 w a s p r o v e d in [11]. E q u a t i o n (2.10) follows f r o m (2.7) a n d (2.8)).
F i n a l l y , a s i m p l e l e m m a on pieeewise linear f u n c t i o n s in R = will be n e c e s s a r y . G i v e n x 0C R '~ a n d 9 > 0, set
Bin(x0; ~) = {x e R~: Hx - -
x011
= m a x Ix i - Xo, i[ < ~}.l < i < m
W e shall w r i t e B~(~) for B~(0; 9). L e t ~s be t h e class o f all c o n t i n u o u s f u n c t i o n s on B,,(1) w i t h t h e following p r o p e r t y : for e a c h r E ~ t h e r e e x i s t -N distinct v e c t o r s O j e R m ( j = 1 , . . . , - N ) s u c h t h a t for e a c h x e B m ( 1 ) , r 1 6 2
( x , 0i} ~ for s o m e j . G i v e n a f u n c t i o n f on a n o p e n c o n v e x s e t G _c R ", f will b e called piecewise linear on G if f o r e a c h x 0 E G t h e r e e x i s t a 0 > 0 a n d a n a f f i n e m a p p i n g Z of 1t m (i.e. Z(z) = A z @ x o for s o m e n o n - s i n g u l a r m • m a t r i x A) s u c h t h a t Z(0) = x0, z(B~(1)) --~ Bm(xo; O) c__G, a n d if r d e n o t e s t h e r e s t r i c t i o n o f f ~ t o B~(1), t h e n r is in c~.
LEMMA 2. Let f be a piecewise linear function defined on an open convex set G c t l m. T h e n f is convex i f (and only if) f is subharmonic.
Proof. I n v i e w o f t h e local c h a r a c t e r of c o n v e x i t y t h e l e m m a will follow if we show t h a t a n y s u b h a r m o n i c f u n c t i o n r r E ~ , is c o n v e x in B,,(~) for s o m e
~o _< t. W e c a n a s s u m e r = 0. Leg N ( r be t h e n u m b e r N c o r r e s p o n d i n g to b y definition. O u r claim is t r i v i a l w h e n e i t h e r N ( r = 1 or m = 1. A s s u m e t h a t it h a s b e e n p r o v e d for all i n t e g e r s 1, 2 . . . . , N - - 1 a n d a r b i t r a r y m (-N > 2).
F i x r to be a n y f u n c t i o n in c22 for w h i c h N ( r L e t 01 . . . . ,0N be t h e c o r r e s p o n d i n g v e c t o r s . Set V = { x E R m : ( x , 0 i - - 0 y ) = 0 , V i , j, i % j } a n d d = d i m V. Since N > 2 a n d 0~ are d i s t i n c t v e c t o r s , d < m, i.e. 0 < d < m - - 1.
Consider f i r s t t h e case d = 0, i.e. V = {0}. I f x 0 EBm(1) is a r b i t r a r y , x0 ~ 0, let -N0 be t h e t o t a l n u m b e r o f 0y'S for w h i c h r = (x0, 0y},,. T h e n _N o < 57, b e c a u s e d = 0. B y c o n t i n u i t y , in s o m e Bin(z0; d)c__Bm(1), one needs o n l y N 0 linear f u n c t i o n s to d e f i n e r B y t h e i n d u c t i o n h y p o t h e s i s r is c o n v e x in s o m e B,,(x0; do), 0 < do _< 8. H e n c e , it r e m a i n s to be s h o w n t h a t r is c o n v e x a t t h e
2 7 4 C~. A . B E R E ~ T S T E I k ' f f A N D M . A . I ) O S T A L
origin. L e t x~, x~ be a n y two distinct points in B~(1) such t h a t x 1 = 5X 2 for some cr ~ 0 . One has to show
r 1~- ( 1 - - ~)x2) < 2 r Jr ( 1 - - 2)r 0 < ~ < 1. (2.11) F o r m--~ 1, this is trivial because s u b h a r m o n i c i t y coincides w i t h convexity.
I f m > 1, there will be a vector y E Bin(1 ) linearly i n d e p e n d e n t of x 1 such t h a t x s - ~ y E B ~ ( 1 ) (i---- 1,21. Then for a n y k = 1 , 2 , . . . t h e s e g m e n t w i t h end- points X~, k = x~ d- k-~y does n o t contain t h e origin a n d b y t h e local c o n v e x i t y of r in B i n ( l ) ~ {0}, t h e i n e q u a l i t y (2.11) is satisfied for Xl, k, X2, k i n s t e a d of x 1, x 2. L e t t i n g k---> ~ , (2.11) follows b y continuity. Finally, for t h e case d > 1, one can assume t h a t
V = {x E Rm: xs = O for i > d}.
F o r x E R ~, set x = ( X s + l , . . . , x z ) a n d r r I t suffices to prove t h e c o n v e x i t y of r in B m s(0; ~) for some ~ < 1. However, since
dim {~ C Rm-S: <~, O~ -- Oj> m d = 0 V j, i i :/: j } = O, we are in t h e preceding case.
COROLLARY 2. Let f be a plurisubharmonic function in (~ which is piecewise linear in R 2'~- G ~. Then f is convex in R 2".
Proof of Theorem 1. L e t
be such t h a t
P
F(~) = ~ Pj(~)e<"J ' ~>, (P :e ~ ) (2.12)
j 1 q
G(~) ---- ~ Oj($le<bJ '~>, (Qj e ~1 (2.13 /
/ = 1
K = ~ c ~;d. F (2.14)
T h e n K C Exp. L e t ~o, #, #0 C d ' be such t h a t K = ~0, G = ~, F = ~o. Obviously, p~o _ < p , ~ p~o. Since P~0 =/5,0 = h[F], P , = /5, = big ] (cf. (2.6)), we o b t a i n from Theorems I, I I t h a t for e v e r y s > 0, there are constants C1, C 2 depending o n l y on s , F a n d G such t h a t for every ~ C ( F ,
eP."(~)]K(~)l _< C 1 m a x [F(z)l < C2ep.-0 (r162 (2.15)
z e ~ (~; ~)
This shows t h a t p~o < p."o --p.", hence b y (2.6)
P~0 --~ /5~0 = hM -- h[c]" (2.16)
THE RITT THEOREM IN SEVERAL VARIABLES 275 B y T h e o r e m 2, p:0 is a c o n v e x function. Since p~0 is also p o s i t i v e l y h o m o g e n e o u s o f o r d e r t, t h e r e exists a c o m p a c t c o n v e x set [K] _c R 2" such t h a t p:0 :-- h[g], hence b y (2.16),
IF] = [G] + [K]. (2.17)
( B y T h e o r e m I of this section, t h e set [K] is o b v i o u s l y t h e u n i q u e c o n v e x s u p p o r t of t h e f u n c t i o n a l ~0).
L e t ~/ be t h e f a m i l y of all linear varieties in R :~ of dimension 2n - - 1, each o f which contains a t least t w o d i f f e r e n t p o i n t s of t h e f o r m
P q
i = l j = l
where all t h e coefficients l~, mj are integers. T h e n t h e set 9~ = {0 C S 2~ ~: 0 /- A, for some A E CY} has m e a s u r e zero in S 2n-~, t h e u n i t sphere in R 2=. (Indeed, f i x a r b i t r a r y Zx, 9 9 zT (T _> 2) of t h e f o r m (2.18); t h e n t h e n o r m a l v e c t o r s to all A E ~ such t h a t z~ C A (i = 1 , . . . , T), d e f i n e a n algebraic s u b v a r i e t y of S 2~ 1 o f dimension < 2n - - 2.)
Obviously, one can assume t h a t t h e c o m p a c t sets IF], [G], [K] lie in R ~ , t h e positive o r t h a n t in R 2~. T h e set ~ l being o f m e a s u r e zero in S 2 n - l , o n e can f i n d v E (S 2~-~ ~ ~}?) [3 R+ ~. Since ~ ~ ~ / , v is a regular d i r e c t i o n for b o t h [F]
a n d [G]. H e n c e h~F~(v)----<c7j,~>2~ for e x a c t l y one ~j.. l ~ e n u m b e r i n g t h e a~'s, if necessary, a n d using t h e f a c t t h a t ~ q <]7, one can assume t h a t
h[F](V) = <a,, ~>2, > <a2, u>:, > . . . > <dr, r>z= > 0 (2.19) Similarly,
h[q(v) --~ <b~, v>:~ > . . . > <b~, v>2~ > 0. (2.20) Set k ~ - - - - a x - - b r B y L e m m a 1, k ~ E x t [ K ] , a n d k~ is t h e o n l y p o i n t o f [K]
for which h[Kl@) = <$~, v>2=.
Set
d ~ - < b 1 - - b2, ~)>2n, g A : ~__. {X e R2n: <X, ~'>2n > <~1, ~)>2n - - d}, H - : R 2n ~ H :b.
L e t r be such t h a t 5 ~ H + for i = 1 , . . . , r a n d 5 ~ r for i - - - - r - b l . . . . , p . Using t h e n o t a t i o n (2.12)--(2.14), set
[
fl(~) = F ( ; ) - p1(~)~<ol, ~>, g1($) G($) - - Ql(~)e <b'' ~>, K I ( g ) QI(g)K(r - - p i ( $ ) e <k,, ~>, F I ( $ ) Qx($)fl(g) - Pl(~)g~(g) e<k1" ~>T h e n fl, El are e x p o n e n t i a l polynomials, K 1 is entire, a n d
(2.21)
F 1 = K1G. (2.22)
276 C. A . B E I ~ E N S T E I N A N D ~r A. D O S T A L
N e x t we claim t h a t
~ @ 5 2 C H - ( j = 2 . . . . ,q). (2.23)
Indeed, b y (2.19) a n d (2.20), <kl -5 bj, v}2~ = <51, u}2, -5 ( - - 51 -5 bj, v}2n <_
(5~, ~'>2~ + ( - b~ .5 b2, ~'>~ = (~1, ~>2,~ - - d.
Since 52 . . . . ,5~ E H +, it follows f r o m (2.23) t h a t none of the t e r m s w i t h fre- quencies aj (2 ~ j ~ r) can be cancelled in F 1 b y a t e r m coming from Pl(~)gl(~)e <k''~>. Moreover, it also shows t h a t a 1 c a n n o t be a f r e q u e n c y of F 1.
Hence, if x C [ F 1 ] , x @52 , a n d
h[F~](r) = (52, v}2, > (x, @2,. (2.24)
r _f _/
Thus, the frequencies of F1 are a2, . . . , a , a;+ 1 .. . . , %~, where { a t + l , . . . , a p l } i 8
a subset of (a~+l . . . . , @ , 5 1 - - b l @ t ) 2 , . . . , 5 1 - - E ) l " s b ~ } ~ H - and [F1]c__[F].
Indeed, [F~] E eh. ([f~] O { ~ "5 bj}j>_2) c [F] O ([K] "5 [G]) = [F].
N e x t we proceed w i t h ~1, G, K 1 in t h e same fashion as above w i t h F , G, K . H e n c e there is a ~1 E ~v4' such t h a t ;1 = / ( 1 a n d ~1 has a unique convex support [K1], a n d [F~] = [K~] "5 [G]. I n particular, b y (2.24) a n d (2.20),
(a~, ~)~o = hi~,](~ ) .5 hiG](~) = hi~,~(~) + (b,, ~}~.
}tenee h[K~](r) = (/c2, f}2,~ for a unique /c 2 C [K~]. On t h e other h a n d , b y L e m m a 1, /c 2 = a 2 - b 1. Set
I f~(~) = F~($) -- Q~(~)P~($)e <"~' :>
g~(~) = g~(~) (2.2~)
K~(~) = K~(~) - - P~(~)e <~-'' :>
[ F~($) = f~(~) -- P~($)g~(~)e <~o' ~>
Then f~ , F~ ~ E2, K~ ~ :~i a n d
Fe = K~G, (2.26)
a: is n o t a f r e q u e n c y of F~, b u t each a~, i = 3 , . . . , r is. The r e m a i n i n g frequencies a ~ + l , . . . , %~ form a subset of
{at+l, . .., %, a 1 -- 51 @ 5 2 .. . . , Ct 1 -- b~ ~- bq, a~ -- b I -~ b 2 .. . . . a 2 -- b~ "5 b~}.
Hence {5"+1, . .., ~ } ~ H - . Moreover, [Fe] ~ [F~], because
IF:] _~ ch ([fi] U {~: +
b~}j~:)~
[F1] U ([K1] -5 [G]) = [F~].Continuing in t h e same fashion, one f i n a l l y constructs F~ ~ E}, a n d K~ ~ such t h a t (i) F~ = K~G, (ii) [F~] ~ _ H - [ 7 [F]. Since ~ e oH, t h e frequencies a} ~) of F~ can be n u m b e r e d so t h a t 5~ ~) is the o n l y point in [F~] for which (5~ ~), ~ ) ~ = h~,.~(~) a n d (~), ~)~ > (~), ~)~o > . . . . ~ 5 (~)~, ~)~o > 0. Set H I + =
{ x ~ R : ~ : ( x , r } : ~ > (c7~ ~ ) , v } ~ - d } , H i = R 2 ~ H +, a n d let r ~ > 1 be such
T H E : R I T T T I t E O I ~ E 1 V s I N S E V E R A L V A R I A B L E S 277 t h a t 5! ~) C H + for i = 1 , . . . , r~ a n d 5~ ~) C H~- for j > r~. I t is n o w clear t h a t we can r e p e a t t h e same p r o c e d u r e indefinitely. I f a t some p o i n t we o b t a i n
~Y = F~+~,~ .... +~N = 0, t h e t h e o r e m follows. H o w e v e r , this m u s t a c t u a l l y h a p p e n w h e n N is s u f f i c i e n t l y large. For, let N be so large t h a t H ~ [3 R+ ~ = O, h e n c e [ ~ ] c _ [ F J 1 3 H ~ = 0 a n d ~ 0 .
w 3. Applications
B y T h e o r e m 1, if F a n d G are e x p o n e n t i a l p o l y n o m i a l s such t h a t t h e q u o t i e n t K = FIG is entire, we can w r i t e K in t h e r e d u c e d form, K = H / P , w h i c h is u n i q u e l y d e t e r m i n e d (cf. w 1 and6)). N o w t h e question arises w h e n P ~ 1. T h e n e x t t h e o r e m gives a simple sufficient condition.
THEOREM 2. Let F, G e E 2 be such that FIG E ~ . Let H I P be the reduced form of .FIG. Then P divides d~. I n particular, P ~ 1 whenever de = 1.
Proof. Set
P
~(~) = ~ aj(=)e<~ '~
j - - 1 q
H(~) = ~ bj(~)e<~J ,~>.
j - - 1
F i r s t we shall p r o v e t h e following special case b y i n d u c t i o n on p.
(3.1)
(A) (i) P is irreducible (ii) de = 1. Then P ~ 1.
I f p = 1, t h e n b y (ii), al(z ) is a c o n s t a n t , a 1 ~ 0. H e n c e H / P is t h e r e d u c e d f o r m o f t h e e x p o n e n t i a l p o l y n o m i a l (1/al)e -< .... >F(z). I n view o f t h e u n i q u e n e s s o f t h e r e d u c e d form, P m u s t be c o n s t a n t .
Suppose n o w t h a t (A) holds w h e n e v e r G has a t m o s t p - 1 frequencies, p ~ 1 . T h e r e are t w o possible cases: e i t h e r P]bj for all j = 1 , . . . , q or P ~ b / for some j. I n t h e f i r s t case, P ~ 1 b y d e f i n i t i o n o f r e d u c e d form. H e n c e it suffices t o consider t h e second case when, a f t e r r e a r r a n g i n g t h e flj's if necessary, t h e r e is a q0 ~ 1 such t h a t P § bj, j = l . . . . , qo a n d bj = b ' P , bj* e ~ , for j = q0 + 1 , . . . , q. W e claim t h a t it suffices to consider t h e case q0 = q. I n d e e d , if qo < q, set
q0
F*(~) = F(~) - G(z) ~ b*(~)e<~i , % H*(z) = ~ bj(~)e<~J ' z>
J ~> qo j - - I
T h e n F*/G is e n t i r e a n d H * / P is its r e d u c e d form. T h e r e f o r e we shall a s s u m e
p + bj (V j). (3.2)
2 7 8 C . A . B E I t E N S T E I N A N D l'r A . ] : ) O S T A L
I t will be shown t h a t (3.2) leads to contradiction if P r constant, a n d this will prove (A). I t follows from w 2 t h a t [PF] = [H] + [G], a n d
[ P F ] = ch {s, + ~: i = 1 , . . . , p, j = 1 , . . . , q}.
L e t y be a f i x e d e x t r e m e point of the polyhedron [PF]. B y L e m m a 1, y = c~0 @ flL for e x a c t l y one i o a n d J0. R e n u m b e r i n g the frequencies one can assume t h a t i o = p , i.e.
@ 4- fiJo J= ~ + f i j ( i < p , V j ) . (3.3) Consider all jo's for which (3.3) holds. R e n u m b e r i n g t h e b/s one can assume t h a t there is some J, 1 < J < _ q , such t h a t (3.3) holds for all J0>--J, b u t does n o t hold for j0 < J . H e n c e each of the frequencies ~p + fij, j >_ J, appears in t h e p r o d u c t HG = P F e x a c t l y o~ce. B y the l e m m a in w l, this means ~hat Plaebj for j > J , t h u s b y (3.2) a n d (i),
Set
Then
a e ~-- ~tpP for some gp E ~ . (3.4)
G*(z) = a ( z ) _ aAz)e<~p , z >
0(~) = r
F ( z ) = F ( z ) - - ~,(z)e<~ '~>H(z), [I(z) = H(z)dc.(z).
(3.5)
t t / P is t h e reduced form of F / d . (3.6) Indeed, b y (3.5), _F/G = [I/P, a n d since (C/H, P) = 1, (da, P) = 1 means b y (i) t h a t P § da., where d~. = (al . . . . , % 1). However this follows from (ii) a n d (3.4). Since G has p -- 1 terms, the i n d u c t i o n hypothesis shows t h a t P is constant, which contradicts (3.2).
(B) N e x t assume t h a t P is irreducible a n d d G arbitrary. Assume t h a t P ~ d G.
I n particular P ~ constant. W r i t i n g G = dGG1, one can a p p l y (A) to FIG 1 = HdG/P. Hence P is a constant, a contradiction.
(C) Finally, let P a n d dG be arbitrary. I f P = P1 " " " Pr is t h e factorization of P into irreducible factors, t h e n t h e t h e o r e m follows b y applying (B) to each of the equations
( F - ~ P , ) l a = H I P j ( j - - 1 . . . . . ~).
i c j
A n o t h e r application of T h e o r e m 1 is t h e following s t a t e m e n t (Theorem 3), which gives a simple necessary condition for a quotient of two exponential poly- nomials to be entire.
T H E t ~ I T T T H E 0 ~ E M I N S E V E R A L V A R I A B L E S 279 Given a r b i t r a r y f i n i t e sets B = {/~1 . . . . , fiq}, C = { y ~ , . . . , y,} of points in R ~ we shall s a y t h a t t h e fii's are rational affine combinations of t h e yCs, if for some j0, ko a n d all j
fiJ -- fiJ. = i Wjk(yk -- yko), Wjk e Q. (3.7) k--1
I t is clear t h a t if (3.7) holds for some j0, k0, it holds w i t h suitable r a t i o n a l s wj,~ for a n y o t h e r pair j0, ko. T h e n e x t s t a t e m e n t is an e a s y consequence o f T h e o r e m 1.
THEOREM 3. Let F, G be exponential polynomials such that FIG is entire. Then the frequencies of G are rational affine combinations of the frequencies of F.
T h e p r o o f follows along similar lines as t h e p r o o f o f t h e t h e o r e m in Section 1 of [22].
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280 C. A. : B E R E N S T E I N A N D M. A. D O S T A L
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Added in proof (ef. footnote 7):
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2 9 . - - ~ - - Sur les fonetions enti~res arithm6tiques de type exponentiel et le quotient d'exponentielle-polyn6mes de plusieures variables (ibidem).
17eceived February d, 1974 C. A. Berenstein
University of Maryland I ) e p a r t m e n t of Mathematics
College Park, Maryland 20742, U.S.A.
M. A. Dostal
Stevens I n s t i t u t e of Technology D e p a r t m e n t of Mathemaflics
Hoboken, New Jersey 07030, U.S.A.
