EXTENDING THE ARCHIMEDEAN POSITIVSTELLENSATZ TO THE NON-COMPACT CASE M. Marshall Abstract. A generalization of Schmudgen's Positivstellensatz is given

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1 EXTENDING THE ARCHIMEDEAN POSITIVSTELLENSATZ TO THE NON-COMPACT CASE M. Marshall Abstract. A generalization o Schmudgen's Positivstellensatz is given which holds or any basic closed semialgebraic set in R n (compact or not). The proo is an extension o Wormann's proo. The Positivstellensatz, proved by G. Stengle in [12], is a standard tool in real algebraic geometry see [2] [5] [7]. In his solution o the K-moment problem in [11], Schmudgen proves a surprisingly strong version o the Positivstellensatz in the compact case. Schmudgen's result has since been extended and improved in various ways see [1] [4] [6] [9] [10]. In the present paper we describe an extension in another direction, to the non-compact case. Let V be an algebraic set in R n. The coordinate ring R[V ]ov is the ring o all polynomial unctions : V! R. R[V ] is generated as an R-algebra by x 1 ::: x n where x i : V! R denotes the i-th coordinate unction. For any nite subset S = 1 ::: r g o R[V ], let K = K S be the basic closed semialgebraic set in V dened by the r inequalities i 0, i =1 ::: r,i.e., K = a 2 V j i (a) 0 i=1 ::: rg and let T = T S denote the preordering P o R[V ] generated by 1 ::: r, i.e., the set o all unctions o the orm = e s e e 1 ::: e r 1 r e = (e 1 ::: e r ) running through the set 0 1g r, where each s e is a sum o squares in R[V ]. A basic version o the Positivstellensatz [7, Lemma 7.5] asserts that, or any 2 R[V ], > 0 on K i (1 + s) =1+t or some s t 2 T: For more comprehensive ormulations o the Positivstellensatz, see [2] [5] [7]. In [11], Schmudgen proves, or K compact and 2 R[V ], > 0 on K ) 2 T or, equivalently, 0 on K i + 2 T or any rational >0: We reer to this latter result as the archimedean Positivstellensatz. Schmudgen's proo uses methods rom unctional analysis. In [13] [14] Wormann gives an algebraic proo. As one might expect, both proos rely heavily on the Positivstellensatz Mathematics Subject Classication. Primary 12D15 14P10 Secondary 44A60. This research was unded in part by NSERC o Canada. Part o the work was done while the Author was on sabbatical leave at Universidad Complutense in Madrid. This part was unded by Universidad Complutense. Typeset by AMS-TEX

2 As is well-known, the Positivstellensatz remains true with R replaced by an arbitrary real closed eld. This is not true or the archimedean Positivstellensatz. On the other hand, it is possible to extend the archimedean Positivstellensatz so as to include the case where K is not compact. This is the content o the present paper. The idea is to replace the constant unction 1 by any unction p 2 1+T which grows suciently rapidly on K in the sense that there exists integers M k 0such that Mp k x i ;Mp k holds on K, i = 1 ::: n. Such a unction p always exists (see Note 1.5 below) and, or any such p, we prove (see Corollary 3.1 below) that 0 on K,9an integer m 0 such that 8 rational >0 9 an integer ` 0 such that p`( + p m ) 2 T: O course, i K is compact then we can take p = 1 and what we have then is exactly the archimedean Positivstellensatz. For another (more complicated) variation o this same result, but with with ` > 0' replacing ` 0', see Corollary 3.2 below. Our proo ollows closely the orm o Wormann's proo given in [13] [14]. The Author wishes to thank C. Andradas and E. Becker, or looking at early versions o the paper, and or providing useul comments. 1. Extension o a result o Wormann. Throughout, A denotes a commutative ring with 1 and Sper(A) denotes the real spectrum o A, i.e., the set o all orderings o A [2] [5] [7]. A preordering o A is a subset T o A satisying T + T T TT T and a 2 2 T or all a 2 A: I T is a preordering in A, Sper T (A) denotes the set o orderings o A lying over T. For simplicity, we always assume Q A. In his proo o the archimedean Positivstellensatz in [13] [14], Wormann proves the ollowing result in the case B = R. Our proo is an easy generalization o Wormann's proo. 1.1 Theorem. Suppose B is a subring o A such that A is nitely generated over B, say A = B[x 1 ::: x n ], and T is apreordering o A such that, on Sper T (A), each x i is bounded by an element o B \ T. Then, or each a 2 A there exists b 2 B \ T such that b a 2 T. Recall that a preprime o A is a subset T o A such that T + T T TT T Q + T and ; 1 =2 T: 1.2 Note. For any preprime T o A, the set C = a 2 A j9b 2 B \ T such that b a 2 T g is a subring o A. This ollows rom the identity b 1 b 2 a 1 a 2 = 1 2 (b 1 a 1 )(b 2 ; a 2 )+ 1 2 (b 1 a 1 )(b 2 + a 2 ): 2

3 Any preordering is a preprime. I T is a preordering and b ; a 2 2 T, b 2 B \ T, then b T and b a =(b ; a2 )+(a 1 2 )2 2 T. Thus, i a 2 2 C then a 2 C. More generally, i b ; (a ::: + a 2 k ) 2 T or some b 2 B \ T, then b ; a2 i 2 T so a i 2 C or i =1 ::: k. Also, since b 2 ; b 2 = 0 2 T, b 2, and consequently b, lies in C or any b 2 B, sob C. Proo o Theorem 1.1. Let z = x ::: + x 2 n. By our assumption, there exists c 2 B \ T such that c ; z > 0 on Sper T (A). By the (abstract) Positivstellensatz, there exists p q 2 T such that (1 + p)(c ; z) = 1+q. Let T 0 = T +(c ; z)t. Then c ; z 2 T 0 so, by Note 1.2 (applied to T 0 ), or every element a 2 A, there exists d 2 B such that d ; a 2 T 0,so(d ; a)(1 + p) 2 T. In particular, there exists d 2 B such that (d ; p)(1 + p) 2 T. Adding ( d ; 2 p)2, this yields (d + d2 ) ; p 2 T. 4 Multiplying this by c 2 T and adding (1 + p)(c ; z) 2 T, and pz 2 T, this yields c(1 + d 2 )2 ; z 2 T. Since c(1 + d 2 )2 2 B \ T,we are done, using Note 1.2 again (this time applied to T ). 1.3 Corollary. Suppose A is a nitely generated R-algebra, say A = R[x 1 ::: x n ], T is a preordering o A, and p is an element o 1+T such that Mp k x i on Sper T (A), i = 1 ::: n, or some integers k M 0. Then, or each 2 A, there exist integers k M 0 such that Mp k 2 T: Proo. Take B = R[p]. The hypothesis implies that each x i is bounded by an element o B on Sper P T (A), so, by Theorem 1.1, there exists g 2 B P such that k g 2 T,say g = r j=0 jp j k. Take M to be any integer satisying M jr j=0 jj. Then, since p ; 1 2 T, Mp k ; g 2 T so, adding, Mp k 2 T. 1.4 Corollary. Suppose A is the coordinate ring o an algebraic set V R n, T = T S or some nite subset S o A, and p is an element o 1+T such that Mp k jx i j holds on K S, i = 1 ::: n, or some integers k M 0. Then, or each 2 A, there exist integers k M 0 such that Mp k 2 T: Proo. By Tarski's Transer Principle (e.g., see [5]), since the inequality Mp k x i holds on the set K S, it holds on the bigger set Sper T (A). 1.5 Note. (1) Such an element p always exists, e.g., take p =1+ (2) Depending on the nature o Sper T (A), there may bea\better"choice or p. For example: I each x i is 0 on Sper T (A) and P x i ; 1 2 T, we can take p = P x i. I each x i is bounded on Sper T (A), we can take p =1. 3 nx i=1 x 2 i :

4 (3) The integers k M such that Mp k 2 T are dicult to estimate. I we know Mp k x i 2 T, i =1 ::: n, the identity in Note 1.2 yields NM d p kd 2 T where d is the degree o (viewed as a polynomial in x 1 ::: x n ) and N is anyinteger the sum o the absolute values o the coecients. This estimate also applies to preprimes. (4) I the R-algebra A is not nitely generated, Corollary 1.3 may break down in various ways. In the ring o global real analytic unctions on a real analytic maniold, or example, one does not normally expect exp(p) to be bounded by a polynomial in p. The preordering considered in [8] provides another example: 1.6 Example. Take A to be the polynomial algebra over R in countably many variables X 1 X 2 ::: and let T be the preordering in A generated by the elements X i and (1 ; X i )(1 + X i+1 ) i 1: Then 1 X i 0 holds on Sper T (A), so any p 2 1+T satises the hypothesis o Corollary 1.3. On the other hand, p does not satisy the conclusion. There is some least integer ` 0 such that p 2 R[X 1 ::: X`]. Then, or any 2 A, i Mp k ; 2 2 T or some integers k M 0 then 2 R[X 1 ::: X`]. The proo is the same as the proo o [8, Prop. 1]. T = [ n1t n where T n denotes the preordering in R[X 1 ::: X n ] generated by X 1 ::: X n and (1 ; X i )(1 + X i+1 ), i = 1 ::: n; 1. I =2 R[X 1 ::: X`], then Mp k ; 2 2 T n nt n;1, n>`. By [8, Lemma 2], the leading coecient omp k ; 2, viewing Mp k ; 2 as a polynomial in X n,is;g 2,whereg is the leading coecient o, and ;g 2 2 S n;1, the preordering in R[X 1 ::: X n;1] generated by X i,1; X i, i =1 ::: n ; 1. Since S n;1 \;S n;1 = 0g [8, Lemma 1], this is impossible. 2. The Kadison-Dubois Theorem. The other ingredient in Wormann's proo is the Kadison-Dubois Theorem. For T a preprime o A, let X(T ) denote the set o ring homomorphisms : A! R such that (T ) 0. I A is the coordinate ring o a real algebraic set V, X(T ) is naturally identied with the set a 2 V j (a) 0 or all 2 T g: I T = T S, the preordering generated by some nite set S in A, this is just the set K S considered at the beginning. The part o the Kadison-Dubois Theorem that we work with is the ollowing: 2.1 Theorem. Suppose T is a preprime o A which is Archimedean, i.e., or each 2 A, there exists an integer M 0 such that M 2 T. Then, or any 2 A, the ollowing are equivalent: (1) () 0 or each 2 X(T ). (2) + 2 T or each rational >0. Proo. See [3]. We use a certain sel-strengthening o Theorem 2.1: 4

5 2.2 Theorem. Suppose T is a preprime o A, and p is an element o 1+T such that, or each 2 A, there exists integers k M 0 such that Mp k 2 T. Then, or any 2 A, the ollowing are equivalent: (1) () 0 or each 2 X(T ). (2) For any suciently large integer m and or each rational >0, there exists an integer ` 0 such that p`( + p m ) 2 T. Note: I p =1this is just Theorem 2.1. Proo. Consider the localization A[ 1 p ]oa at p and the extension T [ 1 p ]ot to A[ 1 p ], and let C = 2 A[ 1 p ] j there exists an integer N 0 such that N 2 T [1 p ]g: Thus the preprime T 0 := T [ 1 p ] \ C in C is Archimedean. Note: 1 1 p 2 T [ 1 p ] (since p 2 1+T ), so 1 p 2 T 0. Also, or each 2 A, p k 2 C or some k 0, so C[p] =A[ 1 p ]. Also, we have X(T ) ' X(T [ 1 p ]),! X(T 0 ) the latter map being restriction. The image o X(T [ 1 p ]) in X(T 0 ) consists o those in X(T 0 ) satisying ( 1 p ) 6= 0(so ( 1 p ) > 0.) Now x 2 A and pick m so large that p 2 C. I (1) holds, then ( m;1 p m ) 0 or all 2 X(T 0 ) so, by Theorem 2.1, p m + 2 T 0 or all rational >0 and, clearing ractions, p`( + p m ) 2 T or suciently large ` 0. Conversely, i (2) holds, then dividing by p m+`, p m + 2 T 0 or all rational >0 so, by Theorem 2.1, ( p m ) 0 or all 2 X(T 0 ). Clearly this implies () 0 or all 2 X(T ). 3.3 Remark. (1) It is possible to generalize Theorem 2.2, replacing the set p k j k 0g by any multiplicative set in 1 + T. (2) I the image o X(T ) is dense in X(T 0 ), we can replace m by m ; 1 in the above argument. (3) I A = R[x 1 ::: x n ], the ring C in the proo can be replaced by a ring which is nitely generated over R. For example, we can x k M 0 so that Mp k x i 2 T, i =1 ::: n and take C = R[ x 1 p k ::: x n p k 1 p ]: P d Also, i we decompose as = j=0 j(x 1 ::: x n ) where j is homogeneous o degree j, then so we can take m = kd +1. nx p kd = j=0 1 p k(d;j) j( x 1 p k ::: x n p k ) 2 C The ollowing example, pointed out to the Author by E.Becker, is closely related to the proo o the theorem o Polya given in [13] [14]. 5

6 2.4 Example. Let A to be the polynomial ring R[X 1 ::: X n ], T the preprime in A generated by R + X 1 ::: X n, P X i ; 1 and take p = P X i. Then p 2 1+T and p X i 2 T, i =1 ::: n, sothehypothesis o Theorem 2.2 is satised. By the above remark, we can take X(T ) is identied with X(T 0 ) is identied with C = R[ X 1 p ::: X n p 1 p ]: a 2 R n j a i 0 X a i 1g (b c) 2 R n+1 j b i 0 X b i =1 1 c 0g and the image o X(T )inx(t 0 ) is identied with (b c) 2 R n+1 j b i 0 X b i =1 1 c>0g: The image o X(T ) is dense in X(T 0 ). By the above remark, i d = deg(), then p d 2 C, and we can take m = d. Thus we have the ollowing: 2.5 Corollary. Let T P P be the preprime in the polynomial ring R[X 1 ::: X n ] generated by R +, X 1 ::: X n and X i ; 1, and let p = X P i. Then a polynomial 2 R[X 1 ::: X n ] o degree d is non-negative on the set a 2 R n j a i 0 a i 1g i or all rational >0 there exists an integer ` 0 such that p`( + p d ) 2 T. 3. Extension o the archimedean Positivstellensatz. We return now to the geometric set-up considered at the beginning. i.e., V is an algebraic set in R n with coordinate ring R[V ] = R[x 1 ::: x n ], S = 1 ::: r g is a nite subset o R[V ], T = T S, the preordering o R[V ] generated by 1 ::: r, and K = K S = a 2 V j i (a) 0 i=1 ::: rg: 3.1 Corollary. Suppose p 2 1+T is chosen so that there exist integers k M 0 such that Mp k jx i j holds on K, i =1 ::: n. Then, or any 2 R[V ], the ollowing are equivalent: (1) is non-negative on K. (2) For any suciently large integer m and or all rational > 0, there exists an integer ` 0 such that p`( + p m ) 2 T. Proo. By Corollary 1.4, we are in a position to apply Theorem 2.2. Since X(T )is identied with K, this gives us what we want. 3.2 Corollary. Suppose p 2 1+T is chosen so that there exist integers k M 0 such that Mp k jx i j holds on K, i =1 ::: n. Then, or any 2 R[V ], the ollowing are equivalent: (1) is strictly positive on K. (2) There exists k 0 and a rational >0 such that p k on K. (3) There exists k 0 and a rational 1 > 0 such that or any suciently large integer m and or all rational 2 > 0, there exists an integer ` 0 such that p`(p k ; p m ) 2 T. 6

7 Proo. (2) ) (1) is clear, and (2), (3) is immediate rom Corollary 3.1, so it only remains to check (1) ) (2). By the Positivstellensatz, r =1+s or some r s2 T. Choose k L so large that Lp k r on K. Then Lp k r =1+s 1, so p k 1 L on K. O course, there are always plenty o choices or p see Note 1.5. Also, i K is compact, one can take p = 1, and what we have then is just the archimedean Positivstellensatz. 3.3 Remark. Corollary 1.4 carries over, suitably generalized, to an arbitrary real closed eld R. The results in Section 3 cannot be so extended (unless R is archimedean) because they depend, in an essential way, on the Kadison-Dubois Theorem. The Kadison-Dubois Theorem tells only about the meaning o the statement ` 0' on X(T ). In the non-archimedean situation, X(T ) =, so this has nothing much to do with the basic closed set K. Reerences 1. F. Acquistapace, C. Andradas, F. Broglia, The strict Positivstellensatz or global analytic unctions and the moment problem or semianalytic sets, preprint. 2. C. Andradas, L. Brocker, J. M. Ruiz, Constructible sets in real geometry, Ergeb Math., Springer, Berlin, Heidelberg, New York, E. Becker, N. Schwartz, Zum Darstellungssatz von Kadison-Dubois, Arch. Math. 39 (1983), 421{ R. Berr, T. Worman, Positive polynomials and tame preorderings, Math. Zeit. (to appear). 5. J. Bochnak, M. Coste, M.F. Roy, Geometrie Algebrique Reelle, Ergeb Math., Springer, Berlin, Heidelberg, New York, T. Jacobi, A representation theorem or certain partially ordered commutative rings, Math. Zeit. (to appear). 7. T.Y. Lam, An introduction to real algebra, Rky. Mtn. J. Math. 14 (1984), 767{ M. Marshall, A real holomorphy ring without the Schmudgen property, Canad. Math. Bull. (to appear). 9. J.P. Monnier, Anneaux d'holomorphie et Positivstellensatz archimedien, Manuscripta Math. (to appear). 10. M. Putinar, Positive polynomials on compact semi-algebraic sets, Ind. Univ. Math. J. 42 (1993), 969{ K. Schmudgen, The K-moment problem or compact semialgebraic sets, Math. Ann. 289 (1991), 203{ G. Stengle, A Nullstellensatz and a Positivstellensatz in semialgebraic geometry, Math. Ann. 207 (1974), 67{ T. Wormann, Short algebraic proos o theorems o Schmudgen and Polya, preprint. 14. T. Wormann, Strikt positive Polynome in der semialgebraischen Geometrie, PhD Thesis, Dortmund, Department o Mathematics & Statistics, University o Saskatchewan, Saskatoon, SK Canada, S7N 0W0 address: marshall@math.usask.ca 7