2 and any even n a t u r a l number d &gt

TRANSVERSAL ZEROS AND POSITIVE SEMIDEFINITE FORMS by

* ) * * ) Man-Duen C h o i , M a n f r e d Knebusch ,

afc a|c afc \ j|r $ $ \

T s i t - Y u e n Lam , B r u c e R e z n i c k E r s c h i e n e n i n :

G e o m e t r i e - A l g e b r i q u e

. . • • •. P r o c e e d i n g s , Rennes19 81

I n t r o d u c t i o n - L e c t u r e N o t e s Math. 959 S p r i n g e r 1982

Fc * any n a t u r a l number n > 2 and any even n a t u r a l number d > 2 we c o n s i d e r t h ^ convey cone P(n,d) c o n s i s t i n g o f t h e p o s i t i v e s e m i d e f i n i t e

( = psd) forms o v e r JR i n n v a r i a b l e s x-,...,x o f degree' d, and t h e 1 n

convex subcone I ( n , d ) c o n s i s t i n g o f t h e f i n i t e sums o f s q u a r e s o f forms o f degree" d/2 i n t h e v a r i a b l e s k-|/.../X^N. As i s w e l l known

I ( n , d ) f P(n,d) e x c e p t f o r v e r y s p e c i a l p a i r s ( n , d ) , namely t h e p a i r s wxth n = 2 o r d = 2 o r (n. d) • •= .(3,4) ( H i l b e r t r c f .'^J [CI/] f o r an elemen- t a r y p r o o f ) .

.In t h i s p a p e r we a s k f o r r e l a t i o n s between t h e s e t s EP(n,d) and E l ( n ^ d ) o f e x t r e m a l e l e m e n t s o f t h e cones P.(n,d) and I ( n , d ) . N o t i c e

t h a t , s i n c e o u r cones a r e c l o s e d ( a f t e r a d d i n g t h e o r i g i n ) , e v e r y e l e - ment i n P(n,d) r e s p . X(n,d) i s a f i n i t e sum o f e l e m e n t s i n EP(n,d) r e s p . E I t n , d ) . Thus t h e s e t s EP(n,d) and EX(n,d) d e s e r v e s p e c i a l a t t e n t i o n .

Our main r e s u l t , Theorem 6.1 i n §6, i s t h e d e t e r m i n a t i o n o f a l l p a i r s (n,ü) s u c h t h a t E I ( n , d ) i s c o n t a i n e d i n E P ( n , d ) , w h i c h means E I ( n , d ) = EP(n,d) fl I ( n , d ) . T h i s answers P r o b l e m B i n t h e s u r v e y a r t i - c l e [ C L11 .

In o r d e r t o o b t a i n t h e r e s u l t a g e n e r a l o b s e r v a t i o n t u r n s o u t t o be h e l p f u l :

a) L e t H be an i r r e d u c i b l e i n d e f i n i t e f o r m i n I R [ x . , . • , ,x ] o f degree i n

r . Then f o r any F £ P(n,d)

F 6 EP(n,d) <=> F H² £ EP(n,d+2r) , F € EX(n,d) <^ F H² £ E I ( n , d + 2 r ) ,

*) s u p p o r t e d by NSERC o f Canada

*"*). s u p p o r t e d by DFG d u r i n g a s t a y a t B e r k e l e y 1980

***) s u p p o r t e d by NSF

w

c f . ^;Theorem 5.1. We a l s o f e e l t h a t the f o l l o w i n g o b s e r v a t i o n sheds l i g h t on the p r o b l e m :

b) I f F 6 EP(n,d) t h e n F² € E I ( n , 2 d ) , c f . Theorem 5.2.

Our " c o u n t e r e x a m p l e s " G £ E I ( n , d ) , G £ EP(n,d) a r e o f t h e f o r m 2 2

G = H F w i t h H a p r o d u c t o f i r r e d u c i b l e i n d e f i n i t e forms and F an i r r e d u c i b l e p s d f o r m o f some d e g r e e e w h i c h i s n o t e x t r e m a l i n P ( n , e ) . B a s i c c o u n t e r e x a m p l e s w i l l be e x p l i c i t l y c o n s t r u c t e d i n §6 f o r

(n,d) = (3,12) and (n,d) = ( 4 , 8 ) .

The o b s e r v a t i o n s a) and b) r e l y on t h e p r e s e n c e o f " t r a n s v e r s a l z e r o s " f o r some forms coming up. i n t h e p r o o f s . A t r a n s v e r s a l z e r o o f a p o l y n o m i a l F ( x1 , xn) o v e r IR " i s a p o i n t c € IRn such t h a t F changes s i g n i n e v e r y n e i g h b o u r h o o d o f c. I f F has no m u l t i p l e i r r e d u c i b l e f a c t o r s t h e n a p o i n t c o f t h e z e r o s e t Z(F) c 3Rⁿ t u r n s o u t t o be a t r a n s v e r s a l z e r o i f and o n l y i f Z(F) has l o c a l d i m e n s i o n n-1 a t c, c f . Theorem 3.4.

The f i r s t h a l f o f o u r p a p e r i s d e v o t e d t o a g e o m e t r i c s t u d y o f t r a n s v e r s a l z e r o s and t o t h e q u e s t i o n how f a r a p o l y n o m i a l i s d e t e r - mined by i t s t r a n s v e r s a l z e r o s . We t r y t o do a l l t h i s on a n a t u r a l l e v e l o f g e n e r a l i t y . T h i s l e a d s us t o s t u d y the s e t ]D] o f r e a l p o i n t s o f an e f f e c t i v e W e i l d i v i s o r D on a n o r m a l a l g e b r a i c v a r i e t y X o v e r 3R . But f o r t h e a p p l i c a t i o n s o f the t h e o r y o f t r a n s v e r s a l z e r o s made i n §5 and §6 i t s u f f i c e s t o c o n s i d e r t h e case when X i s a p r o j e c -

n— 1

t i v e space IP , o r - i f one wants t o s t u d y a l s o m u l t i f o r m s - a d i r e c t JR

p r o d u c t o f p r o j e c t i v e s p a c e s .

We s u s p e c t t h a t many o f o u r c o n s i d e r a t i o n s on t r a n s v e r s a l z e r o s a r e more o r l e s s ¹¹ f o l k l o r e " , w e l l known t o t h e e x p e r t s . However, t o our knowledge, no c o h e r e n t a c c o u n t o f t h i s u s e f u l t h e o r y seems t o e x i s t i n t h e l i t e r a t u r e . Thus we f e e l t h a t t h e s e P r o c e e d i n g s a r e a good p l a c e t o e x p l i c a t e the b a s i c f a c t s .

In t h e whole p a p e r we a d m i t any r e a l c l o s e d f i e l d R as g r o u n d f i e l d i n s t e a d o f t h e f i e l d IR o f r e a l numbers. U s i n g some s t a n d a r d r e s u l t s f r o m s e m i a l g e b r a i c t o p o l o g y , a l l c o n t a i n e d i n [DK] and §1 o f t h e p r e - s e n t p a p e r , t h i s does n o t c a u s e a d d i t i o n a l d i f f i c u l t i e s . Thus we n e v e r need T a r s k i ' s p r i n c i p l e t o t r a n s f e r e l e m e n t a r y s t a t e m e n t s f r o m K t o o t h e r r e a l c l o s e d f i e l d s .

§ 1 The p u r e d i m e n s i o n a l p a r t s o f a s e m i a l g e b r a i c s e t

We s t a r t w i t h a v a r i e t y X o v e r a r e a l c l o s e d f i e l d R, i . e . a r e - duced a l g e b r a i c scheme o v e r R. The s e t X(R) o f r a t i o n a l p o i n t s o f X i s a s e m i a l g e b r a i c space i n t h e sense o f [DK], and we use f r e e l y the l a n g u a g e o f " s e m i a l g e b r a i c t o p o l o g y " d e v e l o p e d i n t h a t p a p e r . I n p a r - t i c u l a r we make use o f t h e d i m e n s i o n t h e o r y i n [DK, §8].

L e t N be a s e m i a l g e b r a i c s u b s e t o f X ( R ) . F o r any p o i n t x o f N the l o c a l d i m e n s i o n dim N o f N a t x i s d e f i n e d as t h e i n f i m u m o f the d i -

_ _ ^x

mensions o f a l l s e m i a l g e b r a i c n e i g h b o u r h o o d s o f x i n N [DK, §13]. We i n t r o d u c e t h e s e t s (k =0,1,2,...)

XV( N ) := {x £ N | dim N > k} .

Jc x

Of c o u r s e (N). i s empty x f k e x c e e d s t h e d i m e n s i o n d o f N. I t i s c l e a r from [DK, §8] t h a t e v e r y I ^ f N ) i s a c l o s e d s u b s e t o f N ( i n the s t r o n g t o p o l o g y , as a l w a y s ) . We s h a l l need some e l e m e n t a r y f a c t s a b o u t the s e t s E^CN) ( a c t u a l l y o n l y a b o u t Z ^ ( N ) ) , n o t c o v e r e d by the p a p e r [DK] *

P r o p o s i t i o n 1.1. ^ ( N ) i s s e m i a l g e b r a i c f o r e v e r y k > 0.

I t i s t r i v i a l t o v e r i f y t h i s lemma u s i n g t h e theorem t h a t e v e r y a f f i n e s e m i a l g e b r a i c space can be t r i a n g u l a t e d [DK^]. A more elemen- t a r y p r o o f , w h i c h a l s o g i v e s a d d i t i o n a l i n s i g h t , r u n s as f o l l o w s . L e t d = d i m ( N ) . F o r k > d t h e r e i s n o t h i n g t o p r o v e . We now d e a l w i t h t h e c a s e k = d. We may assume t h a t X i s a f f i n e . L e t Y denote the Z a r i s k i c l o s u r e o f N i n X, and l e t S d e n o t e t h e s i n g u l a r l o c u s o f Y.^Then

N¹ := (Y(R) \ S ( R ) ) 0 N

i s an open s e m i a l g e b r a i c s u b s e t o f N and t h e complement i n N, i . e . N f> S ( R } , has d i m e n s i o n a t /nost d-1 . Suppose we know a l r e a d y t h a t

t ^ ( N ' ) i s s e m i a l g e b r a i c . Let L bo t h e c l o s u r e o f ^ ( N¹) i n N. T h i s i s c»gain a s e m i a l g e b r a i c s e t . N ^ L i s open i n N and has d i m e n s i o n a t most d-1. Thus N v L i s d i s j o i n t f r o m Z ^ ( N ) . On t h e o t h e r hand L i s c o n -

t a i n e d i n Xß(N), s i n c e ^ ( N ) i s c l o s e d and c o n t a i n s X^dSJ¹). Thus Z^(N) c o i n c i d e s w i t h t h e s e m i a l g e b r a i c s e t L.

R e p l a c i n g N by N1 and X by X ^ S we assume now t h a t Y i s smooth.

L e t Y,j, ... ,Y^t d e n o t e t h e c o n n e c t e d components o f Y. The s e t I^(N) i s t h e u n i o n o f t h e s e t s Id( N n Y ^ ( R ) ) , and i t s u f f i c e s t o p r o v e t h a t t h e s e s e t s a r e s e m i a l g e b r a i c . N n Y^(R) i s Z a r i s k i dense i n Yⁱ- Re-

p l a c i n g N by anyone o f the s e t s N n (R) we assume t h a t i n a d d i t i o n Y i s c o n n e c t e d , hence i r r e d u c i b l e .

We^; have N •= U ... U N^r w i t h non empty s e t s

. N^± =" f x € Y(R) J g^±( x ) = 0^f f ^ . (x) > 0, j = 1, . . . , s±},

o

e where a^ e f u n c t i o n s i n t h e a f f i n e r i n g R [Y] . I f gi i s n o t z e r t h e n dim" 5 n-1. B u t i f g^ i s z e r o t h e n 1NL i s open i n Y ( R ) , henc

c: I ^ ( N ) , s i n c e Y i s smooth and t h u s Y(R) has l o c a l d i m e n s i o n d a t e v e r y / p o i n t [DK, §8]. I t i s now c l e a r t h a t £^d(N) i s t h e c l o s u r e o f t h e u n i o n o f a l l Nⁱ w i t h gⁱ = 0 i n t h e s e t N. Thus I^d( N ) i s i n d e e d s e m i a l - g e b r a i c .

" C o n s i d e r now t h e open s e m i a l g e b r a i c s u b s e t N¹ := N > I^d( N ) o f N.

C l e a r l y

Zd - 1( N ) = z d ( N ) u zd - T(V -

We know from t h e p r o o f a l r e a d y g i v e n t h a t Id_J ]( N1) and ^d( N ) a r e s e m i - a l g e b r a i c . Thus £^d_-| (N) i s s e m i a l g e b r a i c . R e p e a t i n g t h i s argument we see tha.t a l l I^.(N) a r e s e m i a l g e b r a i c , and o u r lemma i s p r o v e d .

P r o p o s i t i o n 1.2. F o r e v e r y k > 0 t h e s e m i a l g e b r a i c s e t . I°(N). := X^k( N ) - I^{k + 1}( N ) ,

c o n s i s t i n g o f a l l p o i n t s x 6 N w i t h dim N = k, i s p u r e o f d i m e n s i o n k, o o ^

i . e . dim I , (N) =k f o r e v e r y x £ I, (N) .

• • x • J C • x.

P r o o f . L e t x be a p o i n t o f ^{a n (}3- l^{e t U Q} ^^{e a n} open s e m i a l g e b r a i c n e i g h b o u r h o o d o f x i n N w i t h dim U = k. F o r any open s e m i a l g e b r a i c n e i g h b o u r h o o d U c U^Q o f x i n N we t h e n a l s o have dim U = k. M o r e o v e r f o r e v e r y such U t h e r e e x i s t s an open s e m i a l g e b r a i c s u b s e t V o f U w h i c h i s s - e m i a l g e b r a i c a l l y i s o m o r p h i c t o an open non empty s u b s e t o f R^k[ D K , §8], C l e a r l y V i s c o n t a i n e d i n t£(N) fl U. Thus

dim(l£(N) n U) = k. Q.E.D.

We c a l l I (N) t h e purr* k-d i mons i o n a l p a r t o f N. More s p e e i f i oa 11 y , i f (1 I It) N d wri t<« I I I |j ( N) $ ^d (N ) I lui j»nr n jw* i I »>l N

Example 2.3. I f X i s i r r e d u c i b l e o f d i m e n s i o n n, and i f t h e s e t X ( R )^{r e g} o f r e g u l a r p o i n t s o f X i n X(R) i s n o t empty, t h e n t h e p u r e p a r t In( X ( R ) ) o f X(R) i s t h e c l o s u r e o f X(R) . i n X ( R ) .

Indeed, X ( R )^{r e}^ i s .pure o f d i m e n s i o n n, and X(R) has l o c a l dimen- s i o n at.most n-1 a t e v e r y s i n g u l a r p o i n t w h i c h i s n o t c o n t a i n e d i n the c l o s u r e o:. X(R)

r e g

§ 2 . T r a n s v e r s a l z e r o s o f a l g e b r a i c f u n c t i o n s

We assume i n t h i s s e c t i o n t h a t t h e v a r i e t y X o v e r R i s i r r e d u c i b l e , t h a t t h e s e t X(R) o f r e a l p o i n t s i s n o t empty, and t h a t X i s r e g u l a r a t e v e r y p o i n t o f X ( R ) . Then X(R) i s an n - d i m e n s i o n a l s e m i a l g e b r a i c m a n i f o l d [DK,- §13] w i t h n = dim X. We a l s o assume t h a t X i s a f f i n e , and we denote t h e r i n g R[X] o f r e g u l a r f u n c t i o n s on X by A. On t h e space X(R) e v e r y f € A t a k e s v a l u e s i n R. We a r e i n t e r e s t e d i n t h e z e r o s and t h e s i g n b e h a v i o u r o f t h e f u n c t i o n s f : X(R) -> R.

D e f i n i t i o n 2.1. L e t L be a s u b s e t o f X(R) on w h i c h f does n o t v a n i s h e v e r y w h e r e . We s a y t h a t f i s p o s i t i v e s e m i d e f i n i t e ( r e s p . p o s i t i v e d e f i n i t e ) on L i f f (x) > 0 ( r e s p . f ( x ) > 0) f o r a l l x £ L. I n t h e same way we use t h e words " n e g a t i v e s e m i d e f i n i t eM and " n e g a t i v e d e f i n i t e " .

I f t h e r e e x i s t p o i n t s x £ L and y £ L w i t h f ( x ) > 0 and f (y) < 0, t h e n we c a l l f i n d e f i n i t e on L.

D e f i n i t i o n 2.2, ; L e t f be a non z e r o e l e m e n t o f A. A t r a n s v e r s a l z e r o o f f i s a p o i n t x £ X(R) such t h a t f i s i n d e f i n i t e on e v e r y s e m i a l g e - b r a i c n e i g h b o u r h o o d V o f x i n X ( R ) . N o t i c e t h a t f c a n n o t v a n i s h e v e r y - where on V s i n c e d i m V = n.

We d e n o t e by Z ( f ) t h e s e t o f z e r o s o f f on X(R) and by Z^t( f ) t h e s e t o f t r a n s v e r s a l z e r o s o f f . We f i n a l l y d e n o t e • b y N ( f ) t h e c l o s e d r e d u c e d subscheme o f a l l z e r o s o f f on X. Thus Z ( f ) i s t h e s e t o f r e a l p o i n t s o f N ( f ) and Z^t( f ) i s a s u b s e t o f Z ( f ) . The s e t Z ( f ) i s c l o s e d and s e m i a l g e b r a i c i n X ( R ) . The s e t Z ^ f ) i s t h e i n t e r s e c t i o n o f the c l o s u r e o f t h e s e t o f p o i n t s o f X(R) where f i s p o s i t i v e w i t h t h e c l o s u r e o f t h e s e t where f i s n e g a t i v e . Thus Z^t( f ) i s a l s o c l o s e d and s e m i a l g e b r a i c i n X(R)".

P r o p o s i t i o n 2.3. F o r e v e r y non z e r o r e g u l a r f u n c t i o n f on X t h e s e t Z^t( f ) o f t r a n s v e r s a l z e r o s i s e i t h e r empty o r p u r e o f d i m e n s i o n n-1.

P r o o f . L e t a be a g i v e n p o i n t o f Zt( f ) . We choose an open n e i g h b o u r - hood V o f a i n X(R) w i t h a s e m i a l g e b r a i c i s o m o r p h i s m cp : V V¹ o n t o an open s e m i a l g e b r a i c convex s u b s e t V o f Rⁿ. ( R e c a l l t h a t X(R) i s a s e m i a l g e b r a i c m a n i f o l d . ) We t h e n choose a p o i n t Xq £ V w i t h f ( xQ) > 0 and an open s e m i a l g e b r a i c s u b s e t U c V such t h a t f ( y ) > 0 f o r e v e r y y £ U and such t h a t Ü' := tp(U) i s c o n v e x i n Rⁿ. We f i n a l l y choose a h y p e r p l a n e H o f Rn w i t h H fl Uf ^ 0 and n o t c o n t a i n i n g t h e p o i n t

7 - ; ' := <P(*^C). Now c o n s i d e r t h e c e n t r a l p r o j e c t i o n

it : Rⁿ v {x» } H

."O ,

o n t o H w i t h c e n t e r x'. We c l a i m t h a t o

(*) TT ° ( p(z^t( f) n v) 3 H h u^f.

I n d e e d , l e t y' 6 H n U¹ be g i v e n and l e t y¹ : [ 0 , 1 ] be t h e s t r a i g h t p a t h f r o m x ^ t o y',

Y • ( t ) = ( 1 - t ) x ^ + t y¹ .

Then y := <P ^ *.Y¹ i s a s e m i a l g e b r a i c p a t h i n V r u n n i n g from t h e p o i n t x^Q t o t h e preimage y o f y¹ . S i n c e f ( x^Q) > 0 and f ( y ) < 0 t h e r e e x i s t s some p o i n t x 6 ] 0 , 1 [ where t h e s e m i a l g e b r a i c f u n c t i o n f o Y on [ 0 , 1 ] changes s i g n . Y(T) i s c l e a r l y a t r a n s v e r s a l z e r o o f f . The p o i n t Y^F ( x ) ^ l i e s i n t p ( Z^t( f ) fl V) and maps under n t o t h e p o i n t y¹ . Thus t h e

i n c l u s i o n (*) h o l d s t r u e . T h i s i m p l i e s t h a t dim Z ( f ) fl V > n-1,

s i n c e d i m (H n U^f ) = n-1. B u t Z ( f ) fl V has d i m e n s i o n a t most .n-1 s i n c e t h i s s e t i s c o n t a i n e d i n N ( f ) . Thus Z^t( f ) n V has d i m e n s i o n n-1 f o r e v e r y open s e m i a l g e b r a i c n e i g h b o u r h o o d V o f a.

Q.E.D.

C o r o l l a r y 2.4. L e t f and g be non z e r o r e g u l a r f u n c t i o n s on X. L e t a c X(R) be a t r a n s v e r s a l z e r o o f f and assume t h a t Zt( f ) n ü i s c o n - t a i n e d i n Z (g) f o r some n e i g h b o u r h o o d U o f a. Then f and g have a non t r i v i a l common f a c t o r i n t h e r e g u l a r l o c a l r i n g 0 . {Recall t h a t Ö^{x a} i s a u n i q u e f a c t o r i z a t i o n domain.J

P r o o f . F o r e v e r y a f f i n e Z a r i s k i n e i g h b o u r h o o d W o f a i n X t h e s e m i - a l g e b r a i c s e t W n ü fl Z^t( f ) h a s d i m e ^ ^ i o n n-1 by P r o p o s i t i o n 2.3 above. Our h y p o t h e s i s i m p l i e s t h a t t h i s s e t i s c o n t a i n e d i n t h e i n t e r - s e c t i o n N ( f ) fl N(g) fl W o f t h e h y p e r s u r f a c e s f = 0 and g = 0 on W.

Thus t h e ( a l g e b r a i c ! ) d i m e n s i o n o f N ( f ) fl N(g) n W c a n n o t be s m a l l e r , t h a n n-1 f o r any Z a r i s k i n e i g h b o u r h o o d W o f a. T h i s i m p l i e s t h a t

* t h e r e e x i s t s some h 6 A w h i c h i s a p r i m e e l e m e n t i n 0V and has t h e

A , a p r o p e r t y t h a t

N(h) n W c N(f) n N(g) n w

f o r s m a l l Z a r i s k i n e i g h b o u r h o o d s W o f a. By t h e l o c a l N u l l s t e l l e n s a t z , h d i v i d e s b o t h f and g i n C>

^ x,a I n t h e same v e i n we o b t a i n

C o r o l l a r y 2.5. L e t a g a i n f and g be non z e r o f u n c t i o n s on X. Suppose t h a t f o r some open s e m i a l g e b r a i c s u b s e t U o f X(R) t h e s e t 2t( f ) fl U i s n o t empty and c o n t a i n e d i n Z .(g) . Then t h e complex h y p e r s u r f a c e s N ( f ) and N(g) have a common i r r e d u c i b l e component. I n p a r t i c u l a r , i f A i s f a c t o r i a l t h e n f and g have a non t r i v i a l common f a c t o r i n A.

P r o p o s i t i o n 2.6. L e t f and g be non z e r o r e g u l a r f u n c t i o n s on X, and assume t h a t t h e h y p e r s u r f a c e s N ( f ) and N(g) have no i r r e d u c i b l e compo- n e n t i n common.' Then

2t( f g ) = Zt( f ) U Zt( g ) .

P r o o f . a) L e t a be a p o i n t o f X(R) w h i c h i s n o t c o n t a i n e d i n

Z ^ ( f ) U Z^t( g )^# Then t h e r e e x i s t s a n e i g h b o u r h o o d U o f a i n X(R) such t h a t b o t h f and g a r e s e m i d e f i n i t e on U ( p o s i t i v e o r n e g a t i v e ) . Then a l s o t h e p r o d u c t f g i s s e m i d e f i n i t e on U, and a i s n o t a t r a n s v e r s a l z e r o o f f g . T h i s p r o v e s t h a t Z^t( f g ) i s c o n t a i n e d i n Z^t( f ) U Z^t( g ) •

(Our h y p o t h e s i s , t h a t N ( f ) and N(g) have no common component, i s n o t y e t needed f o r t h a t . )

b) We show t h a t t h e s e t M :•= Z ^ ( f ) i s c o n t a i n e d i n Z ^ ( f g ) , w h i c h w i l l f i n i s h t h e p r o o f . We may assume t h a t M i s n o t empty. By P r o p o s i t i o n 2.3 M i s p u r e o f d i m e n s i o n n-1. On t h e o t h e r hand t h e s e t

N ;= Z^t( f ) fl Z^t( g ) has d i m e n s i o n a t most n-2, s i n c e ' N i s c o n t a i n e d i n t h e i n t e r s e c t i o n o f t h e h y p e r s u r f a c e s N ( f ) and N(g) w h i c h have no common i r r e d u c i b l e component. Thus t h e s e t M ^ N i s dense i n M (a t r i v i a l argument, c f . [DK, §13]). S i n c e Zt( f g ) i s c l o s e d i t s u f f i c e s t o v e r i f y t h a t M v N i s c o n t a i n e d i n Zt( f g ) .

L e t x be a p o i n t o f M^N, w h i c h means t h a t x € Z.^t(f)# ^ f. Z^t( g ) • We choose a n e i g h b o u r h o o d UQ o f x on w h i c h g i s s e m i d e f i n i t e . Now f i s

i n d e f i n i t e on e v e r y n e i g h b o u r h o o d U c u o f x . Thus a l s o f g i s i n d e f i - n i t e on e v e r y s u c h U. T h i s i m p l i e s t h a t x 6 Z^t( f g ) .

Q.e.d.

C o r o l l a r y 2.7. Assume t h a t A i s f a c t o r i a l . L e t f be a non z e r o e l e - ment o f A and l e t

be t h e d e c o m p o s i t i o n o f f i n t o powers o f p a i r w i s e non a s s o c i a t e d prime e l e m e n t s p¹#.-..#P^t- / w i t h u a u n i t o f A Then Z'^t(f) i s t h e u n i o n o f t h * s e t s Z^t(p'ⁱ) w i t h eⁱ odd.

e.

P r o o f . A p p l y P r o p o s i t i o n 2.6 and o b s e r v e t h a t Z (p. 1) i s empty i f

~ e i t i ei i s even and Zt( p . ) = z t( Pj L) i f e, i s odd.

I n t h e same v e i n we o b t a i n f o r t h e s e m i a l g e b r a i c s e t germ Z . ( f )

u. a o f a non z e r o f u n c t i o n f € A a t any p o i n t a € X ( R ) :

C o r o l l a r y 2.8. L e t

e e

• f = u p¹ ... p^t

be t h e d e c o m p o s i t i o n o f f i n t o p r i m e e l e m e n t s i n the l a c t o i i a l i \ mj 0 ^. Then 2 (f)- i s t h e u n i o n o f t h e s e t germs Z⁴_ ( p . )^{r a} w i t h e. o d d .

A , a "C a t l a I

§ 3 P u r e l y i n d e f i n i t e d i v i s o r s

We s t i l l assume t h a t X i s an i r r e d u c i b l e n - d i m e n s i o n a l v a r i e t y o v e r R and t h a t t h e s e t X(R) i s n o t empty and c o n t a i n s no s i n g u l a r p o i n t s o f X. B u t we no l o n g e r assume t h a t X i s a f f i n e . Our

t e r m i n o l o g y from §2 t h e n t a k e s o v e r from f u n c t i o n s t o e f f e c t i v e d i v i - s o r s D > 0 on X, by w h i c h we a l w a y s mean e f f e c t i v e W e i l d i v i s o r s . D e f i n i t i o n 3.1. L e t D be an e f f e c t i v e d i v i s o r on X and l e t a be a p o i n t o f X ( R ) . L e t f be t h e l o c a l e q u a t i o n o f D on some a f f i n e Z a r i s k i open n e i g h b o u r h o o d V o f a. We c a l l D i n d e f i n i t e a t a, i f f i s i n d e f i - n i t e on e v e r y n e i g h b o u r h o o d o f a i n V ( R ) . S i m i l a r l y we c a l l D semide- f i n i t e ( r e s p . d e f i n i t e ) a t a, i f f i s p o s i t i v e o r n e g a t i v e s e m i d e f i - n i t e ( r e s p . d e f i n i t e ) on some n e i g h b o u r h o o d o f a i n V ( R ) . The p o i n t s o f X(R) where D i s i n d e f i n i t e a r e c a l l e d t h e t r a n s v e r s a l p o i n t s o f D, and t h e s e t o f t h e s e p o i n t s i s d e n o t e d by |D|^. T h i s s e t i s a c l o s e d s e m i a l g e b r a i c s u b s e t o f t h e s e t o f r e a l p o i n t s |D|^R :~ | D| fl X(R) o f t h e s u p p o r t | D| o f D.

L e t D = e^D¹ + . .. + e^tD^t be t h e d e c o m p o s i t i o n o f D i n t o i r r e d u - c i b l e components.

P r o p o s i t i o n 3.2. l^D l t ^{i s t h e u n i o n o f} a l l s e t s | Dⁱ 1¹ w i t h e.^ odd.

T h i s i s c l e a r f r o m P r o p o s i t i o n 2.6 i n §2, o r i t s c o r o l l a r y 2.8.

D e f i n i t i o n '3.3. We c a l l an e f f e c t i v e d i v i s o r D i n d e f i n i t e , i f | D lt i s n o t empty, i . e . i f D i s i n d e f i n i t e a t some p o i n t o f X ( R ) . We c a l l D s e m i d e f i n i t e , i f |D.|^t i s empty, and we c a l l D d e f i n i t e i f |D|^R i s empty. F i n a l l y , we c a l l p p u r e l y i n d e f i n i t e , i f D ^ 0 and t h e r e does n o t e x i s t a s e m i d e f i n i t e e f f e c t i v e d i v i s o r E ^ 0 w i t h E <s D. T h i s means t h a t D i s non z e r o , h a s no m u l t i p l e components, and t h a t a l l

i r r e d u c i b l e components o f D a r e i n d e f i n i t e .

I t i s c l e a r f r o m P r o p o s i t i o n 2.3 i n §2 t h a t f o r e v e r y e f f e c t i v e d i v i s o r D on X t h e s e t J D | ^T i s e i t h e r empty o r p u r e o f d i m e n s i o n n-1.

T h i s r e s u l t c a n be i m p r o v e d .

Theorem 3.4. Assume t h a t D has no m u l t i p l e components. Then t h e s e m i - a l g e b r a i c s e t | D Jt o f t r a n s v e r s a l p o i n t s o f D c o i n c i d e s w i t h t h e p u r e

( n - 1 ) - d i m e n s i o n a l p a r t I v ( |Dl r > ) o f t h e s e t |-D | p o f r e a l p o i n t s on n— I R -t\

P r o o f . I t r e m a i n s t o v e r i f y t h a t D i s i n d e f i n i t e a t any g i v e n p o i n t a o f j p ]^R w i t h dim ^a | D j^R = n- 1 . We choose a l o c a l e q u a t i o n f o f D on some a t f i n * Z a r i s k i open n e i g h b o u r h o o d W o f a i n X. L e t U be any semi- a l g e b r a i c open n e i g h b o u r h o o d o f a i n W(R). The s e t U IT J D |^R has d i - m e n s i o n n- 1 , b u t t h e s e t o f p o i n t s i n ID J _ w h i c h a r e s i n g u l a r on JDJ has d i m e n s i o n a t most n-2. Thus U n } D-J. c o n t a i n s some r e g u l a r p o i n t b o f | D | . There e x i s t s a r e g u l a r s y s t e m o f p a r a m e t e r s f ^ f ^ . . . , ^ o f t h e r e g u l a r l o c a l * r i n g ö , such t h a t f 1 d e f i n e s t h e germ o f t h e

v a r i e t y J D | a t b. The f u n c t i o n s and f d i f f e r i n 0^{X b} o n l y by a v j n i t , hence we may assume t h a t f = f^. By t h e i m p l i c i t f u n c t i o n t h e o - rem t h e s y s t e m ( f ^ # . . . , fn) y i e l d s a s e m i a l g e b r a i c i s o m o r p h i s m o f some open s e m i a l g e b r a i c n e i g h b o u r h o o d U¹ c U o f b i n X ( R ) o n t o some open s e m i a l g e b r a i c s u b s e t o f R^N. S i n c e f ¹ (b) = 0 c e r t a i n l y f = changes s i g n on U1. A f o r t i o r i f i s i n d e f i n i t e on U.

• " Q.e.d.

We mention" tha't *the theorem "now p r o v e d i m p l i e s a g e n e r a l i z a t i o n o f t h e " S i g n - C h a n g i n g C r i t e r i o n " o f D u b o i s and E f r o y m s o n f o r e x t e n d i n g an o r d e r i n g P o f a f i e l d k t o a g i v e n f u n c t i o n f i e l d o v e r k ([DE, Th.2.7J, c f . a l s o [ELW, §4 b i s ]

C o r o l l a r y 3.5. (Dubois - Efroymson f o r V = A £ ). L e t k be an o r d e r e d f i e l d and R be a r e a l c l o s u r e o f k w i t h r e s p e c t t o t h e g i v e n o r d e r i n g . L e t V be an a b s o l u t e l y i r r e d u c i b l e v a r i e t y w i t h o u t s i n g u l a r p o i n t s o v e r k and D a p r i m e d i v i s o r on V . L e t V d e n o t e t h e v a r i e t y o v e r R o b t a i n e d from V by b a s e e x t e n s i o n and l e t D d e n o t e t h e e f f e c t i v e d i v i - s o r on V - o b t a i n e d f r o m D by base e x t e n s i o n . Then t h e o r d e r i n g o f k can be e x t e n d e d t o t h e f u n c t i o n f i e l d k ( D ) o f D i f and o n l y i f D i s i n d e f i n i t e .

P r o o f . The o r d e r i n g o f k exce: ^.s t o k ( D ) i f and o n l y i f t h e r e e x i s t s a f i e l d c o m p o s i t e o f k ( D ) and R o v e r k, w h i c h i s f o r m a l l y r e a l . These f i e l d c o m p o s i t e s a r e t h e f u n c t i o n f i e l d s R ( D1) , . . . , R ( D ) o f t h e i r r e - d u c i b l e components ^{D ^ , . . . , D G} o f t h e d i v i s o r ^D. The p r i m e d i v i s o r s ^{D^} a l l o c c u r w i t h m u l t i p l i c i t y one i n D . Thus D i s i n d e f i n i t e i f a t l e a s t one i s i n d e f i n i t e . By Theorem 3.4 a g i v e n D . ^ i s i n d e f i n i t e i f and o n l y i f t h e s e t o f r e a l p o i n t s D ^ ( R ) o f D ^I h a s d i m e n s i o n n-1 w i t h n := d i m V = d i m VD. B u t dim D . ( R ) = n-1 means t h a t t h e v a r i e t y D. has n o n s i n g u l a r r e a l p o i n t s , c f . § 1 . Now i t i s a w e l l known f a c t , due t o A r t i n , t h a t D ^I h a s n o n s i n g u l a r r e a l p o i n t s i f and o n l y i f t h e f i e l d R ( DI) i s f o r m a l l y r e a l ([A, § 4 ] , c f . a l s o [ E ] ) .

We r e t u r n t o o u r i r r e d u c i b l e v a r i e t y X o v e r R.

P r o p o s i t i o n 3 . 6 . L e t D be an e f f e c t i v e d i v i s o r f 0 w i t h o u t m u l t i p l e components. Then | D |^r i s Z a r i s k i dense i n j D j i f and o n l y i f D i s p u r e l y i n d e f i n i t e . I n t h i s c a s e even |D| i s Z a r i s k i dense i n |D|.

P r o o f . L e t D^{L F}. . . / D^R d e n o t e t h e i r r e d u c i b l e components o f • D . C l e a r l y

| D ]^R = D1( R ) U ... U DR( R )

i s Z a r i s k i dense i n D i f a n d o n l y i f e v e r y (R) i s Z a r i s k i dense i n Dⁱ. T h i s means t h a t (R) has t h e s e m i a l g e b r a i c d i m e n s i o n n - 1 , i . e . t h a t ^ⁿ - i ( ^Di ( ^R ) ) ^{i s n o t} empty/ and i n t h a t c a s e o f c o u r s e a l r e a d y

Iⁿ_«j(D^(R)) i s Z a r i s k i dense i n D. . The p r o p o s i t i o n now f o l l o w s f r o m t h e p r e c e d i n g Theorem 3 . 4 .

T h i s i s p e r h a p s t h e a p p r o p r i a t e p l a c e t o i n d i c a t e a r e l a t i o n b e - tween o u r i n v e s t i g a t i o n s and t h e r e a l N u l l s t e l l e n s a t z o f D u b o i s - R i s l e r -

S t e n g l e C s^f Theorem 2 ] . Assume t h a t X i s an a f f i n e v a r i e t y o v e r R and t h a t W.is a c l o s e d s u b v a r i e t y o f X. L e t A d e n o t e t h e a f f i n e r i n g o f X and -Oi-the i d e a l o f f u n c t i o n s i n A. v a n i s h i n g on W. Then t h e r e a l N u l l - s t e l l e n s a t z s a y s i n p a r t i c u l a r t h a t W(R) i s Z a r i s k i dense i n W i f and o n l y i f t h e i d e a l s i s " r e a l " , i . e .

h^ + ... -h h ^ eM- h¹ e^cv,... , h^r e ^

f o r a r b i t r a r y e l e m e n t s h ¹ , h ^r o f A. ( T h i s i s e s s e n t i a l l y R i s l e r ' s v e r s i o n o f t h e r e a l N u l l s t e l l e n s a t z [ R i ] , [ R i ^ ] . ) Thus i f X i s i r r e - d u c i b l e and has no s i n g u l a r r e a l p o i n t s t h e n t h e p r o p o s i t i o n we j u s t p r o v e d s a y s t h e f o l l o w i n g * :

C o r o l l a r y 3. 7^ L e t X be a f f i n e and 1 ( D ) denote t h e i d e a l o f f u n c t i o n s i n R [ X ] v a n i s h i n g on JD| f o r D an e f f e c t i v e d i v i b o i / o w i t h o u t j t i u l t i - p i e components. Then 1 ( D ) i s r e a l i f and o n l y i f D i s p u r e l y i n d e f i - n i t e .

I f D i s a p r i m e d i v i s o r t h e n c l e a r l y 1 ( D ) i s r e a l i f and o n l y i f t h e f u n c t i o n f i e l d R(D) i s f o r m a l l y r e a l , and we a r e back t o t h e a r g u - ments w h i c h l e d t o t h e S i g n - C h a n g i n g C r i t e r i o n above ( C o r o l l a r y 3 . 5 ) . D e f i n i t i o n 3.8» We c a l l a s e m i a l g e b r a i c s u b s e t M o f X(R) p u r e and

f u l l o f d i m e n s i o n k i n X, i f d i m M = k (hence t h e Z a r i s k i c l o s u r e Z o f M i n X has d i m e n s i o n k ) and M i s t h e p u r e p a r t X , ( Z ( R ) ) o f Z ( R ) .

'In t h i s t e r m i n o l o g y we can say a c c o r d i n g t o Theorem 3 . 4 and Propo- s i t i o n 3 . 6 t h a t f o r e v e r y non z e r o p u r e l y i n d e f i n i t e d i v i s o r on X the

D | ^T i-3 p u r e and f u l l o f d i m e n s i o n n- 1 i n X. We now p r o v e a con- v e r s e o f t h i s s t a t e m e n t .

Theorem 3 . 9 . L e t M be a pure' and f u l l ( n- 1) - d i m e n s i o n a l s e m i a l g e b r a i c s u b s e t o f X ( R ) . Then t h e r e e x i s t s a u n i q u e p u r e l y i n d e f i n i t e d i v i s o r D on X such t h a t M c o i n c i d e s w i t h t h e s e t |D|t o f t r a n s v e r s a l p o i n t s o f D . The v a r i e t y | D | i s t h e Z a r i s k i c l o s u r e o f M i n X.'

P r o o f . L e t Z d e n o t e t h e Z a r i s k i c l o s u r e o f M i n X and l e t Z'1/...,Z d e n o t e t h e i r r e d u c i b l e components o f Z. The s e t M i s t h e u n i o n o f t h e c l o s e d s e m i a l g e b r a i c s u b s e t s M. := M fl Z . ( R ) , i = 1, . . . , r . D e n o t i n g by Z!^ t h e Z a r i s k i c l o s u r e o f Mi i n .X we have Z| c= z ^ and

Z i U U Z¹ = Z, U . . . U Z' ,

1 r I r

and we c o n c l u d e t h a t . Z^ = Zi f o r i = 1, . . . , r . T h i s means t h a t e v e r y 2YL i s Z a r i s k i dense i n Z. . S i n c e Z. i s n o t c o n t a i n e d i n t h e u n i o n o f t h e Z j . w i t h j £ i , a l s o Mⁱ i s n o t c o n t a i n e d i n .the u n i o n o f t h e NL w i t h j f i . Thus

M! := M N U M.

i s a non empty open s u b s e t o f M, w h i c h i s t h e r e f o r e p u r e o f d i m e n s i o n n- 1. T h i s i m p l i e s d i m = n- 1 and dim Z^ = n- 1 f o r e v e r y i =1 ,. . . , n . The s e t Z.(R) c o n t a i n s M^, hence has a g a i n d i m e n s i o n n- 1 . We now con- c l u d e from Theorem 3 . 4 t h a t f o r e v e r y i =1, . . . , r t h e p r i m e d i v i s o r Z^

i s i n d e f i n i t e . W e ' i n t r o d u c e t h e p u r e l y i n d e f i n i t e d i v i s o r D := Z1 + ... + Zr.

By c o n s t r u c t i o n |D| i s t h e Z a r i s k i c l o s u r e Z o f M. S i n c e M i s p u r e and f u l l , ^M c o i n c i d e s w i t h ^ D _ I ( I D I R ) - By Theorem ^3.4 t h i s l a s t s e t i s

J D |^T. I t i s now a l s o c l e a r t h a t D i s t h e o n l y p u r e l y i n d e f i n i t e d i v i - s o r w i t h |D|t = M, s i n c e by P r o p o s i t i o n 3 . 6 f o r any s u c h d i v i s o r Df

t h e v a r i e t y |D' j i s t h e Z a r i s k i c l o s u r e o f M i n X.

Q.e.d.

A m i l d g e n e r a l i z a t i o n o f t h e s e r e s u l t s i s p o s s i b l e . Assume o n l y t h a t X i s an i r r e d u c i b l e n - d i m e n s i o n a l v a r i e t y w h i c h i s normal a t e v e r y r e a l p o i n t , and t h a t X(R) has d i m e n s i o n n. L e t X1 denote t h e open s u b v a r i e t y o f a l l r e g u l a r p o i n t s o f X. Then X(R) ^ X1( R ) has d i - mension a t most n-2. I n p a r t i c u l a r X¹( R ) i s n o t empty. L e t D be an

e f f e c t i v e d i v i s o r on X and l e t D¹ d e n o t e t h e r e s t r i c t i o n o f D t o X'.

D e f i n i t i o n 3.10. We c a l l D i n d e f i n i t e ( r e s p . s e m i d e f i n i t e , r e s p . p u r e l y i n d e f i n i t e ) i f D' i s i n d e f i n i t e ( r e s p . s e m i d e f i n i t e , r e s p .

p u r e l y i n d e f i n i t e ) . We d e n o t e by | D J^t t h e c l o s u r e o f t h e s e m i a l g e b r a i c s e t |D^f| i n X ( R ) .

I t i s e v i d e n t t h a t a l l t h e t h e o r e m s , p r o p o s i t i o n s and c o r o l l a r i e s i n t h i s * s e c t i o n , e x c e p t C o r o l l a r y 3.5, r e m a i n t r u e word by word i n t h e p r e s e n t more g e n e r a l s i t u a t i o n . C o r o l l a r y 3.5 r e m a i n s t r u e f o r a n o r - mal i r r e d u c i b l e v a r i e t y V o v e r k i n s t e a d o f a r e g u l a r v a r i e t y .

§ 4 A remark on s e m i d e f i n i t e p r i m e d i v i s o r s

As b e f o r e l e t X be an i r r e d u c i b l e n - d i m e n s i o n a l v a r i e t y o v e r R s u c h t h a t X(R) i s a l s o n - d i m e n s i o n a l and c o n t a i n s o n l y n o r m a l p o i n t s . We r e g a r d on X(R) b e s i d e t h e s t r o n g t o p o l o g y a l s o t h e c o a r s e r Z a r i s k i t o p o l o g y . T h i s i s t h e t o p o l o g y on X(R) i n d u c e d by t h e Z a r i s k i t o p o l o g y o f X. E v e r y Z a r i s k i c l o s e d s u b s e t . M o f X(R) i s a f i n i t e u n i o n o f i r r e - d u c i b l e c l o s e d s u b s e t s M^1f-...,M^r w i t h Mⁱ <fi M_. f o r i ^ j . We c a l l t h e s e s u b s e t s t h e i r r e d u c i b l e components o f M. They a r e u n i q u e l y d e t e r -

mined by - M.

E v e r y i r r e d u c i b l e Z a r i s k i c l o s e d s u b s e t M o f X(R) w h i c h has dimen- s i o n n-1 i s c l e a r l y t h e s e t o f r e a l - p o i n t s o f an i n d e f i n i t e p r i m e d i - v i s o r D on X u n i q u e l y d e t e r m i n e d by M ( c f . Theorem 3.9, w h i c h s a y s much more t h a n t h i s . ) We now p r o v e a weak a n a l o g u e o f t h i s s t a t e m e n t

f o r l o w e r d i m e n s i o n a l i r r e d u c i b l e Z a r i s k i c l o s e d s u b s e t s ' o f X(R) . U n i q u e n e s s o f t h e p r i m e d i v i s o r D "can no l o n g e r be e x p e c t e d . Thus t h e f o l l o w i n g theorem i s l e s s v a l u a b l e t h a n Theorem 3.9.

Theorem 4.1. Suppose t h a t X i s a l s o q u a s i p r o j e c t i v e , i . e . a l o c a l l y c l o s e d subscheme o f some p r o j e c t i v e space IP L e t -M be an i r r e d u c i b l e

Z a r i s k i c l o s e d s u b s e t o f X(R) o f d i m e n s i o n a t most n-2. Then t h e r e e x i s t s some s e m i d e f i n i t e p r i m e d i v i s o r D on X such t h a t M = D ( R ) .

F o r t h e p r o o f we r e p l a c e X by i t s n o r m a l i z a t i o n , w h i c h does n o t change a n y t h i n g f o r t h e s p a c e X ( R ) . Now t h e z e r o d i v i s o r d i v ( f )+ and t h e p o l e d i v i s o r d i v ( f ) _ o f any non z e r o r a t i o n a l f u n c t i o n f on X a r e h o n e s t l y d e f i n e d as W e i l * d i v i s o r s .

N The s e t X(R) i s c o n t a i n e d i n t h e a f f i n e open subscheme V o f P ^D

2 2 • w h i c h i s t h e complement o f t h e h y p e r s u r f a c e x^Q + ... ^{+ X N =} °- We

i n t r o d u c e t h e Z a r i s k i c l o s u r e X¹ o f X fl V i n V. Then X(R) = X¹ (R) and X\j i s an a f f i n e v a r i e t y . L e t W d e n o t e t h e Z a r i s k i c l o s u r e o f M i n

We choose r e g u l a r f u n c t i o n s g¹, . . . , g^r on X^ such t h a t W i s t h e r e d u c e d subscheme N^x ( g¹) fl ... fl N^x ( g^r) o f a l l common z e r o s o f g¹, . .. , g^r on

. F o r t h e r e g u l a r f u n c t i o n 1 1

on X-j we have

q^A + ... + g 2 2

^1 r r

M = {x € X¹ (R) | g ( x ) =0}.

We now e x t e n d t h e r e g u l a r f u n c t i o n g | X fl V. t o a r a t i o n a l f u n c t i o n f

on X i n t h e u n i q u e p o s s i b l e way. The domain o f d e f i n i t i o n o f f c o n t a i n s X fl V, hence X ( R ) . Thus the p o l e d i v i s o r E : = d i v ( f ) _ has i n i t s sup- \ .port no r e a l p o i n t s , i . e . E i s d e f i n i t e . On t h e o t h e r hand we have f o r

t h e z e r o d i v i s o r D := d i v ( f )+

|D|R = {x £ X(R) | f ( x ) = 0} = {x e x1( R ) ] g ( x ) = 0} = M.

L e t D r ^eiD| + ... + e s D s be the d e c o m p o s i t i o n o f D i n t o p r i m e d i v i - s o r s . M i s t h e u n i o n o f the Z a r i s k i c l o s e d s u b s e t s D-j (R) , ...rD (R).

S i n c e M i s I r r e d u c i b l e , M c o i n c i d e s w i t h one o f t h e s e s e t s , say

M = ( R ) . The p r i m e d i v i s o r i s s e m i d e f i n i t e a c c o r d i n g t o Theorem 3.4, o r a l r e a d y P r o p o s i t i o n 2.3, and o u r t h e o r e m i s p r o v e d .

§ 5 E x t r e m a l p o s i t i v e s e m i d e f i n i t e forms and e x t r e m a l s q u a r e s

L e t X be t h e ( n - 1 ) - d i m e n s i o n a l p r o j e c t i v e space TP¹!' (n > 2 ) . E v e r y e f f e c t i v e d i v i s o r D on X i s t h e d i v i s o r d i v ( F ) o f a form F ( x . j , . . . , x ) w i t h c o e f f i c i e n t s i n R u n i q u e l y d e t e r m i n e d by D up t o a m u l t i p l i c a t i v e c o n s t a n t . I n t h i s way t h e p r i m e d i v i s o r s c o r r e s p o n d w i t h t h e i r r e d u c i - b l e forms,, the i n d e f i n i t e d i v i s o r s c o r r e s p o n d w i t h t h e i n d e f i n i t e forms i n t h e u s u a l s e n s e - n o t i c e t h a t X(R) i s c o n n e c t e d -, and t h e s e m i d e f i - n i t e ( r e s p . d e f i n i t e ) d i v i s o r s c o r r e s p o n d w i t h t h e p o s i t i v e s e m i d e f i - - n i t e ( r e s p . d e f i n i t e ) f o r m s , o f c o u r s e a l s o w i t h t h e n e g a t i v e s e m i d e f i - n i t e ( r e s p . d e f i n i t e ) f o r m s .

We c a l l a f o r m F € R[x,j , . . . ,x^ft3 p u r e l y i n d e f i n i t e , i f t h e d i v i s o r d i v ( F ) i s p u r e l y i n d e f i n i t e . T h i s means t h a t F i s n o t c o n s t a n t , a l l i r r e d u c i b l e f a c t o r s o f F a r e i n d e f i n i t e , and no i r r e d u c i b l e f a c t o r s o c c u r w i t h m u l t i p l i c i t y > 1.

F o r any i n t e g r a l number r > 0 we d e n o t e by F ( r ) t h e s e t o f a l l non z e r o forms o f degree r i n R[ x jj , . . . , x ] and by F t h e u n i o n o f a l l F ( r ) . F o r any even number d > 0 we d e n o t e by P(d) t h e convex cone i n F ( d )

• c o n s i s t i n g o f a l l p s d (= p o s t i v e s e m i d e f i n i t e ) forms o f d e g r e e d i n R.[x,j, . . . ,x ] , and by P t h e u n i o n o f a l l P ( d ) . S i m i l a r l y we d e n o t e by 1(d) t h e convex subcone o f P ( d ) c o n s i s t i n g o f a l l f i n i t e . s u m s o f

d

s q u a r e s o f non z e r o forms i n R [ x ^ , . . . , xⁿ3 o f degree ^ and by I t h e u n i o n o f t h e s e t s 1 ( d ) .

The cones P ( d ) U { 0 } and 1 ( d ) U { 0 } a r e w e l l known t o be c l o s e d s e m i a l g e b r a i c s u b s e t s o f t h e v e c t o r space F ( d ) U { 0 } . Our t h e o r y i n §2 has some a p p l i c a t i o n s t o t h e t h e o r y o f t h e s e t s E (P (d)) and EX I.'(d) ) o f e x t r e m a l p o i n t s o f t h e cones P (d) and 1 ( d ) . We r e f e r t h e r e a d e r t o t h e p a p e r [CL.] f o r t h e b a c k g r o u n d , some r e s u l t s , and c o n c r e t e e x a m p l e s i n t h i s t h e o r y . L e t a g a i n E ( P ) d e n o t e t h e u n i o n o f s e t s E ( P ( d ) ) and E ( I ) t h e u n i o n o f t h e s e t s E ( I ( d ) ) .

I f n o t h i n g e l s e i s s a i d a l l forms i n t h e s e q u e l a r e u n d e r s t o o d t o be forms i n x - j , . . . , x o v e r R. F o r any two s u c h forms we mean by

"F > G" t h a t F - G l i e s i n P U { 0 } . I n p a r t i c u l a r t h e n F and G must have, t h e same d e g r e e . S i m i l a r l y we mean by " F » Gⁿ t h a t F - G l i e s i n

I U { 0 } . C l e a r l y an e l e m e n t F o f P l i e s i n E (p) i f and o n l y i f

F > G > 0 i m p l i e s G = XF w i t h some c o n s t a n t X. S i m i l a r l y an e l e m e n t F o f I l i e s i n E ( I ) i f and o n l y i f F » G » 0 i m p l i e s G = XF w i t h some c o n s t a n t X. Of c o u r s e i n b o t h c a s e s t h e c o n s t a n t X l i e s i n t h e i n t e r - v a l [ 0 , 1 ] .

Theorem 5.1. i ) L e t F and G be psd forms. Assume t h a t F £ E.( P) and G d i v i d e s F. Then G G E ( P ) .

i i ) Assume t h a t F € E ( I ) and F = G-H² w i t h some forms G and". H. Then G £ E ( X ) .

i i i ) L e t G be a p s d form and H a p u r e l y i n d e f i n i t e form. Then G l i e s i n E ( P ) i f and o n l y i f G H² l i e s i n ' E ( P ) .

i v ) L e t a g a i n G be a p s d form and H a p u r e l y i n d e f i n i t e form. Then G • l i e s i n E ( I ) i f 'and o n l y i f G H² l i e s i n E ( I ) .

P r o o f . i ) We have F = G H w i t h some p s d form H. Suppose t h a t

G > G^v > 0. We have t o v e r i f y t h a t G¹ = XG w i t h .some c o n s t a n t X. S i n c e H > O w e have GH > G'H > 0. S i n c e F i s e x t r e m a l t h i s i m p l i e s

G'H = XGH w i t h some c o n s t a n t X and then G* = XG.

i i ) We may i n d u c t on t h e number o f i r r e d u c i b l e f a c t o r s o f H and t h u s assume t h a t H i s i r r e d u c i b l e . S i n c e F i s an e x t r e m a l sum o f / s q u a r e s F

" 2 " "

i s a c t u a l l y a s q u a r e L . Now H d i v i d e s L. We have L = H S w i t h some

2 ? ?

form S and t h e n F = H S . From t h i s we o b t a i n G = S . I n p a r t i c u l a r G € I . We see now by t h e same argument as i n i ) t h a t G i s e x t r e m a l i n

^*

2

i i i ) I f GH. i s e x t r e m a l t h e n a l s o G i s e x t r e m a l as has been p r o v e d above. Assume now t h a t G i s e x t r e m a l . I t s u f f i c e s t o c o n s i d e r t h e c a s e t h a t H i s i n d e f i n i t e and i r r e d u c i b l e , s i n c e we t h e n o b t a i n t h e f u l l r e -

2

s u i t by i t e r a t i o n . L e t L be a non z e r o form w i t h GH > L > 0. The s e t o f r e a l z e r o s Z(H) i s c o n t a i n e d i n Z ( L ) . By a m i l d a p p l i c a t i o n o f

C o r o l . 2.5 we see t h a t H d i v i d e s L. ( R e s t r i c t H and L t o t h e n - s t a n d - n*" 1

a r d open a f f i n e s u b v a r i e t i e s o f IP_ .) S i n c e H i s i n d e f i n i t e t h e n a l s o

2 * 2 *

H d i v i d e s L, c f . P r o p o s i t i o n 3. 2 . We have L = H L¹ w i t h some p s d form

2 2

L V and o b t a i n f r o m GH > L1H > 0 t h a t G > L1 > 0. S i n c e G i s e x t r e m a l

2

t h i s i m p l i e s L1 = XG w i t h some c o n s t a n t X and t h e n L = XGH .

i v ) We a g a i n r e t r e a t t o t h e c a s e t h a t H i s i r r e d u c i b l e and i n d e f i n i t e .

2

I f GH l i e s i n E ( I ) , t h e n ' b y i i ) a l s o G l i e s i n E ( I ) . Assume now t h a t G € E ( I ) . Suppose t h a t G H2 » L » 0. We have

L = M² + ... + M²

1 r

w i t h some forms ,...^fM o f same d e g r e e . The s e t Z(H) i s c o n t a i n e d i n e v e r y z e r o s e t Z d Y L ) . Thus by C o r o l l a r y 2.5 we have IVL = HN. w i t h some forms N. and L = H2L1 / where

We can a p p l y t h e same argument t o t h e sum o f s q u a r e s GH - L and have GH - L = H S- w i t h some £ I . We o b t a i n G = L. + S . S i n c e G i s ex-2 2

I I l i

t r e m a l i n I t h i s i m p l i e s L1 = XG w i t h some c o n s t a n t X £ [ 0 , 1 ] and then 2 2

L = XG H . Thus GH i s i n d e e d e x t r e m a l i n Z. Theorem 5.1 i s now com- p l e t e l y p r o v e d .

2

We may ask f o r w h i c h forms F t h e s q u a r e F i s e x t r e m a l i n I o r even i n P. By p a r t i i i ) o f Theorem 5.1 t h e l a t t e r i s t r u e f o r any p r o - d u c t F o f i r r e d u c i b l e i n d e f i n i t e f o r m s . We a l s o know from p a r t s i ) and i i ) o f t h e theorem t h a t

( F¹F²)² E E ( I ) > F|. £ E(Z) , F² € E ( Z ) ; ( F1F2)2 € E ( P ) F2 € E(P) , Y\ £ E ( P ) .

To p u r s u e t h i s q u e s t i o n f u r t h e r we may o m i t i n a g i v e n f o r m F a l l i r r e d u c i b l e i n d e f i n i t e f a c t o r s , a c c o r d i n g t o Theorem 5.1, and assume t h a t F i s p s d . We have t h e f o l l o w i n g p a r t i a l r e s u l t .

2

Theorem 5.2. L e t F be a f o r m i n E ( P ) . Then F has t h e f o l l o w i n g 2 2

p r o p e r t y : I f F = G + H w i t h some p s d form H and some form G t h e n 2 2

G = eF w i t h some c o n s t a n t e. (Of c o u r s e e l i e s i n t h e i n t e r v a l [ 0 , 1 ] . ) I n p a r t i c u l a r F2 e E ( Z ) .

P r o o f , We may assume t h a t F £ ±G. We d i s t i n g u i s h two c a s e s .

Case 1: F - G i s s e m i d e f i n i t e . I f F - G w o u l d be n e g a t i v e s e m i d e f i n i t e 2 2

t h e n a l s o F + G w o u l d be n e g a t i v e s e m i d e f i n i t e , s i n c e F - G > 0. Thus t h e sum 2F o f F - G and F + G w o u l d be n e g a t i v e s e m i d e f i n i t e , w h i c h i s

2 2

n o t t r u e . Thus F - G > 0. S i n c e F -G = (F - G) (F + G) i s p s d , a l s o F + G > 0. From t h e r e l a t i o n

F = (F 4- G)/2 + (F - G)/2 we o b t a i n , s i n c e F i s e x t r e m a l ,

(F - G)/2 = XF, (F 4- G)/2 = yF

w i t h c o n s t a n t s X > 0, y > 0 s u c h t h a t X + y = 1. T h i s i m p l i e s G » ( y - X ) F and t h e n G² = (y - X )²F², as d e s i r e d .

Case 2, F - G i s i n d e f i n i t e . A c c o r d i n g t o P r o p o s i t i o n 3.2 t h e r e e x i s t s an i r r e d u c i b l e i n d e f i n i t e form P w h i c h d i v i d e s F - G w i t h an odd m u l t i - p l i c i t y . S i n c e F2 - G2 > 0 t h e form P o c c u r s i n F2 - G2 w i t h even m u l t i - p l i c i t y , a g a i n by P r o p o s t i o n 3.2. Thus P d i v i d e s a l s o F + G, hence P d i v i d e s b o t h F and G. S i n c e F i s p s d even P d i v i d e s F. We have

2

F = P P1 w i t h a f o r m € E ( P ) by Theorem 5 . 1 . i . We a l s o have G = P G ' w i t h some form G1 and t h e e q u a t i o n

4 2 2 2

;P = P G¹ + H.

Thus H = P2Hf w i t h a form H' £ P, and 2 2 2 P F^ = G + H1 .

The z e r o s e t Z ( p ) i s c o n t a i n e d i n Z(G') and a l s o i n Z(H') . Thus by §2 t h e i r r e d u c i b l e i n d e f i n i t e from P d i v i d e s b o t h G¹ and H^f, t h e l a t t e r

2

one w i t h an even m u l t i p l i c i t y . We o b t a i n G1 = P G1 , H1 = P w i t h H¹ € P, and

F².= G^{2 +} H,.

The p r o o f c a n now be c o m p l e t e d by i n d u c t i o n on t h e d e g r e e o f Ff s i n c e F^ has s m a l l e r d e g r e e t h a n F.

- Q.e.d.

Remark. I n a l l t h e s e c o n s i d e r a t i o n s we c o u l d have r e p l a c e d o u r p r o -

n-1 n¹ n^r

j e c t i v e space 3P_ by a p r o d u c t 3P_^J x . . . x 3P • , i . e . work w i t h

K R K

m u l t i f o r m s i n s t e a d o f forms. Thus Theorems 5.1 and 5.2 r e m a i n t r u e f o r m u l t i f o r m s i n s t e a d o f f o r m s .

§ 6 Comparison o f the s e t s EP(n,d) and E X ( n , d ) . 2

L o o k i n g a g a i n f o r forms F such t h a t F i s e x t r e m a l i n I o r even i n P i t i s n a t u r a l t o ask whether e v e r y F E E ( I ) a c t u a l l y l i e s i n E ( P ) . In c a s e o f a p o s i t i v e answer we w o u l d know f r o m Theorems 5.1 and 5.2 f o r any psd form' F t h a t F2 l i e s i n E ( I ) i f and o n l y i f F l i e s i n E.(P), and t h e r e l a t i o n between t h e s e t s E ( I ) and E ( P ) w o u l d be w e l l under- s t o o d .

U n f o r t u n a t e l y t h i n g s t u r n o u t t o be n o t t h a t s i m p l e . L e t us w r i t e more p r e c i s e l y P(n,d) i n s t e a d o f P(d) and I ( n , d ) i n s t e a d o f 1(d) t o

i n d i c a t e t h e number n o f v a r i a b l e s o f t h e forms under c o n s i d e r a t i o n . We ask f o r w h i c h p a i r s (n,d) w i t h n > 2, d > 2 and e v e n , t h e s e t

EI(n*,d) o f e x t r e m a l p o i n t s o f t h e cone I ( n , d ) i s c o n t a i n e d i n t h e s e t EP(n,d) o f e x t r e m a l p o i n t s o f t h e cone P ( n , d ) . The f o l l o w i n g theorem g i v e s a c o m p l e t e answer t o t h i s q u e s t i o n .

Theorem 6.1. L e t n > 2 be a n a t u r a l number' and d be an even n a t u r a l number. Then E I ( n , d ) c EP(n,d) p r e c i s e l y i n t h e f o l l o w i n g c a s e s . i ) n = 2; i i ) d < 6; i i i ) (n,d) = ( 3 , 8 ) ; i v ) (n,d) = ( 3 , 1 0 ) .

Thus the q u e s t i o n , whether E I ( n , d ) i s c o n t a i n e d i n EP(n,d) i s . answered by t h e f o l l o w i n g c h a r t :

d 2 4 6 8 10 12 14

2 / / / / / / V

3 _/ / / / _• X X

4 _{/ / /} X X X X

5 / / / ^X X X X

6 ^{/ / / X X X X}

Legend: / = p o s i t i v e answer x = n e g a t i v e answer

The r e s t o f t h e s e c t i o n i s d e v o t e d t o a p r o o f o f t h i s theorem. I f n == 2 o r d = 2 t h e n I ( n , d ) = P(n,d) and t h e r e i s n o t h i n g t o be p r o v e d , Thus we assume h e n c e f o r t h t h a t n > 3 and d ^ 4.

C o n s i d e r now t h e c a s e t h a t d = 4 o r d = 6. L e t F be a form w i t h

F € E I ( n , d ) . Suppose t h a t F does n o t l i e i n E P ( n , d ) . C a n c e l l i n g out 2 2

i n F a l l i n d e f i n i t e i r r e d u c i b l e f a c t o r s we o b t a i n a form w i t h t h e same p r o p e r t i e s , as f o l l o w s f r o m Theorem 5.1, Thus we may assume t h a t F has o n l y psd f a c t o r s . Then F c a n n o t have degree 3. Thus F i s a p s d qua- d r a t i c form. A f t e r a l i n e a r change o f c o o r d i n a t e s we have

F = + ... + x 2 2

1 r w i t h 1 < r < n. Now

F2 =s' x^ + 2 x2 ( x2 .+ .... + x2) + ( x2 + . . . + x2) 2.

2

We see t h a t F i s n o t e x t r e m a l i n I ( n , 4 ) . T h i s c o n t r a d i c t i o n p r o v e s t h a t E I ( n , d ) i s c o n t a i n e d i n EP(n,d) f o r d <• 6.

Suppose now t h a t F i s a f o r m - o f d e q r e e 4 i n n v a r i a b l e s such t h a t

2 < ~ •

F l i e s i n E I b u t n o t i n EP. I f F w o u l d c o n t a i n an i n d e f i n i t e i r r e d u c - i b l e f a c t o r t h e n t a k i n g o u t t h i s f a c t o r we w o u l d o b t a i n * a' form G w i t h G € E I ( n , d ) b u t G £ EP(n,d) f o r some d < 6 (Theorem 5.1). T h i s has 2 2

been p r o v e d t o be i m p o s s i b l e . Thus F does n o t c o n t a i n an i n d e f i n i t e f a c t o r and we may assume i n p a r t i c u l a r t h a t F i s p s d . I f F w o u l d be r e d u c i b l e t h e n F = Q,Q⁰ w i t h p s d q u a d r a t i c forms Q. and Q⁹. B u t t h e n

2 2 2 2

a l s o the f a c t o r s and Q2 o f Q^Q² would l i e i n E I (Theorem 5 . 1 ) , w h i c h means t h a t Q1 and Q2 w o u l d be s q u a r e s o f l i n e a r f o r m s . T h i s c o n t r a d i c t s t h e f a c t t h a t F has no i n d e f i n i t e f a c t o r s . Thus F must be an i r r e d u c - i b l e p o s i t i v e s e m i d e f i n i t e q u a r t i c .

I t i s known s i n c e H i l b e r t t h a t P(3,4) = 1 ( 3 , 4 ) , c f . [CL, §6] f o r an e l e m e n t a r y p r o o f i n t h e c a s e R = IR . Thus i n t h q c a s e n = 3 o u r f o r m F has t o be a sum o f s q u a r e s , b u t n o t a s q u a r e , and we o b t a i n as

2

above a c o n t r a d i c t i o n t o t h e a s s u m p t i o n t h a t F i s e x t r e m a l i n 1(3,8.).

.We have p r o v e d t h a t E I ( 3 , 8 ) i s c o n t a i n e d i n E P ( 3 , 8 ) .

Assume now t h a t F i s a f o r m i n 3 v a r i a b l e s o f d e g r e e 5 such t h a t

2

F i s e x t r e m a l i n 1(3,10)*. F c o n t a i n s an i r r e d u c i b l e f a c t o r H o f odd

2

d e g r e e , F = HG. By Theorem 5.1 t h e f o r m G i s e x t r e m a l i n I . S i n c e deg G <8 we know t h a t G i s e x t r e m a l i n P. Thus, a g a i n by Theorem 5.1, 2 2

2

t h e f o r m F i s e x t r e m a l i n P. We have p r o v e d t h a t E I ( 3 , 1 0 ) i s c o n t a i n e d i n E P ( 3 , 1 0 ) .

T h i s p r o o f works e q u a l l y w e l l o v e r a l l r e a l c l o s e d f i e l d s R, t a k i n g i n t o a c c o u n t t h e r u d i m e n t s o f [DK, §9]. No a p p e a l t o T a r s k i ' s

p r i n c i p l e i s n e c e s s a r y .

We now have v e r i f i e d a l l t h e a f f i r m a t i v e answers i n t h e c h a r t

above. To g e t a l l n e g a t i v e answers i t s u f f i c e s t o check t h a t E I ( 3 , 1 2 ) i s n o t c o n t a i n e d i n EP(3,12) and E I (4 , 8) i s n o t c o n t a i n e d i n E P( 4^/8 ) . I n d e e d , r e g a r d i n g a form F i n t h e v a r i a b l e s x^,...,x a l s o as a form i n t h e v a r i a b l e s x^ , . . ./Xn+-j / i t i s an e a s y e x e r c i s e t o p r o v e t h a t

F² 6 E I (n, d) => F² € E I (n+1 , d ) , and i t i s t r i v i a l t h a t

F² £ EP(n,d) F² £ EP(n+1 , d ) .

F u r t h e r m o r e c h o o s i n g some l i n e a r form L i n t h e v a r i a b l e s x ^ , . . . , xⁿ, i t i s e v i d e n t f r o m Theorem 5.1 t h a t

F² € EI(n,d)^T-=* F²L² € E I ( n , d+2) and

F² t EP(n,d) =* F²L² £ E P ( n , d+2)

We s h a l l now e x h i b i t a form i n E I ( 3 , 1 2 ) w h i c h i s n o t e x t r e m a l i n P ( 3 , 1 2 ) . F o r t u n a t e l y a c o u n t e r e x a m p l e f o r - ( n , d ) = (4,8) c a n be con- s t r u c t e d by s i m i l a r p r i n c i p l e s . Thus i t w i l l be s u f f i c i e n t t o d e v o t e o u r main e f f o r t s t o t h e c a s e (n,d) = ( 3 , 1 2 ) .

We s t a r t w i t h t h e t e r n a r y s e x t i c

o/ x - 4 2 ^ 4 2 ^A 4 2 - 2 2 2

S (x, y, z) = x y + y z + z x - 3x y z

i n [ C L ] . T h i s f o r m has seven z e r o s : ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) , ( 1 , 1 , 1 ) , (-1,1,1)7 (1,-1,1) and ( 1 , 1 , - 1 ) . We s h a l l l o o k a t an a u x i l i a r y form

2 2 2 2 T ( x , y , z ) . = (x y + y z - z x - x y z )

w h i c h i s chosen i n s u c h a way t h a t i t v a n i s h e s on a l l z e r o s o f S, e x c e p t (-1,1,1).

2

Theorem 6.2. L e t f ( x , y , z ) = S ( x , y , z ) + T ( x , y , z ) . Then p := f l i e s i n E I ( 3 , 1 2 ) b u t n o t i n E P ( 3 , 1 2 ) .

The f a c t t h a t p i s n o t e x t r e m a l i n P(3,12) w i l l be deduced from an easy lemma (Lemma 1 ) , and f o l l o w s by t h e way a l s o from Theorem 5.1. i , w h i l e t h e f a c t t h a t p i s e x t r e m a l i n 1(3,12) w i l l be deduced from a

d i f f i c u l t lemma (Lemma 2 ) .