K. Schmudgen, The K-moment problem for compact semi-algebraic sets, Math. Ann. 289, 203-206, 1991  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(u ## j (x) where u # i (x) is defined as # (u # i ) # x and analogously for u ## j (x) It is proved in [11] that program (7) is, in fact, the semidefinite dual of (5) thus, weak semidefinite duality implies that p # t p # . 10) A result in real algebraic geometry (Schmudgen [21], Putinar [20] Jacobi and Prestel [7] asserts that, when F is compact (and satisfies some additional technical condition) then every positive polynomial on F has a decomposition (6) This result implies the asymptotic convergence of the lower bounds # # t , p # t to p # as t goes to infinity ....

K. Schmudgen. The K-moment problem for compact semi-algebraic sets. Mathematische Annalen, 289:203--206, 1991.

....at the expense of fixing some of the variables, to search for the best possible # for the given degree. In the case of basic compact semialgebraic sets, i.e. compact sets of the form K = f 1 (x) 0, f s (x) 0 , a stronger version of the Positivstellensatz, due to Schmudgen [Sch91] can be applied. It says that a polynomial f(x) that is strictly positive on K, actually belongs to the cone generated by the f i . The Positivstellensatz presented in Theorem 4.6 only guarantees in this case the existence of g, h in the cone such that fg = 1 h. An important computational drawback ....

K. Schmudgen. The k-moment problem for compact semialgebraic sets. Math. Ann., 289:203-- 206, 1991.

....K S is compact and r = f # . Proof. By definition of f # , f 0 on K S . Suppose # 0 is given. Then f f # # 0 on K S so, by Putinar s Theorem, f f # # M S , i.e. f #. The result follows from this, using Proposition 1.1. Following standard terminology [6] 9] 12][14][15] we say that the moment problem holds for M S if for each L K S there exists a positive Borel measure on K S such that #f # R[x] L(f) f d. 1.4 Proposition. The following are equivalent: 1) The moment problem holds for M S . f # . Proof. 1) R[x] Suppose # R, f ....

K. Schmudgen, The K-moment problem for compact semialgebraic sets, Math. Ann. 289 (1991), 203-- 206.

....in 1920. For n = 1 and F = R , a sequence y = y i ) i#0 is a F moment sequence if and only if both y and e 1 y = y i 1 ) i#0 are positive semidefinite, a result shown by Stieltjes in 1894. When F is a compact semi algebraic set in R , i.e. where g # are polynomials, Schmudgen [Sc91] shows that y is a F moment sequence if and only if y and g y are positive semidefinite for any product g = g i 1 . g i k of distinct polynomials among g # (# = 1, m) Reformulating Corollary 3 as a moment result in a semigroup. In fact, the result from Corollary 3 can also be ....

K. Schmudgen. The K-moment problem for compact semi-algebraic sets. Mathematische Annalen, 289:203--206, 1991.

....measure of c. We say that a multisequence of moments is determinate if it has precisely one representing measure. The literature concerning the full F moment problem (not necessarily complex) is extensive and it is still growing (see for instance [6, 7, 18, 19, 1, 23, 3, 22, 4, 25, 16, 30] and [8, 26, 27, 15, 28, 5, 29] where semi algebraic F s are considered) The truncated F moment problem has intensively been studied since early 90 s mostly by Curto and Fialkow (cf. 2, 20, 21, 11, 12, 10, 13, 14] In 1994 R. E. Curto asked a question whether the truncated F moment problem is more general than the full ....

K. Schmudgen, The K-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), 203--206.

....y = y i ) i0 is a F moment sequence if and only if both y and e 1 y = y i 1 ) i0 are positive semidefinite, a result shown by Stieltjes in 1894. When F is a compact semi algebraic set in R n , i.e. F = fx 2 R n j g (x) 0 for = 1; mg (43) where g are polynomials, Schmudgen [Sc91] shows that y is a F moment sequence if and only if y and g y are positive semidefinite for any product g = g i 1 : g i k of distinct polynomials among g ( 1; m) Reformulating Corollary 3 as a moment result in a semigroup. In fact, the result from Corollary 3 can also be ....

K. Schmudgen. The K-moment problem for compact semi-algebraic sets. Mathematische Annalen, 289:203--206, 1991.

....Proof. Immediate from the above analysis, using Corollary 4.3 and Theorem 6.1. # Note: This extends the result in [10, Cor. 3.1] It also extends the corresponding result of Jacobi and Prestel for p = 1 in [7, Th. 4. 4] Both of these latter results, in turn, extend the basic result of Schmudgen [19] [21] Using Corollary 5.3 instead of Corollary 4.3, we also have a corresponding result for cylinders: 7.2 Corollary. Suppose there exist positive integers k, # such that kp # # n i=0 x 2 i # 0 holds on the set # # X S p(#) #= 0 . Then, for f # A[1 p] Y ] the following are ....

....notation: We define R subalgebras B i of A[1 p] inductively by B 0 = A[1 p] and B i 1 = the ring of elements of B i which are geometrically bounded on XM i where M i : M[1 p 2 ] # B i . The B i are a certain poor man s version of the iterated holomorphy rings defined in [1] and [19]. 6 Also, as in the proof of Theorem 2.2, B : f # A[1 p] # an integer k such that k f, k f # M[1 p 2 ] the ring of arithmetically bounded elements, and M # : M[1 p 2 ] # B. Thus A[1 p] B 0 # B 1 # # B and there are canonical restriction maps XM = XM[1 p 2 ] ....

K. Schmudgen, The K-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), 203--206.

....the expense of fixing some of the variables, to search for the best possible for the given degree. In the case of basic compact semialgebraic sets, i.e. compact sets of the form K = fx 2 R n ; f 1 (x) 0; f s (x) 0g, a stronger version of the Positivstellensatz, due to Schmudgen [Sch91] can be applied. It says that a polynomial f(x) that is strictly positive on K, actually belongs to the cone generated by the f i . The Positivstellensatz presented in Theorem 4.6 only guarantees in this case the existence of g; h in the cone such that fg = 1 h. An important computational drawback ....

K. Schmudgen. The k-moment problem for compact semialgebraic sets. Math. Ann., 289:203-- 206, 1991.

....FOR REPRESENTATIONS OF POLYNOMIALS POSITIVE ON COMPACT SEMI ALGEBRAIC SETS ALEXANDER PRESTEL Dedicated to Murray Marshall on the occasion of his 60 th birthday In this paper we deal with effectivity problems in connection with the following theorem proved by K. Schmudgen in [Sch]. Let R [X] R [X 1 ; X n ] be the ring of real polynomials in X 1 ; X n . For h 1 ; hm 2 R [X] the set S(h) S(h 1 ; hm ) fa 2 R n jh 1 (a) 0; hm (a) 0g is a basic closed semi algebraic subset of R n . Schmudgen s Theorem states that if S(h) is ....

K. Schmudgen: The K-moment problem for compact semi-algebraic sets. Math. Ann 289, 203-206 (1991).

....version of Schmudgen s Theorem over nonarchimedean real closed fields, and then applying the Compactness and Completeness Theorem from Model Theory. 1. Statement of the Results In this paper we deal with effectivity problems in connection with the following theorem proved by K. Schmudgen in [Sch]. Let R[X] R[X 1 ; Xn ] be the ring of real polynomials in X 1 ; Xn . For h 1 ; hm 2 R[X] the set S(h) S(h 1 ; hm ) fa 2 R n j h 1 (a) 0; hm (a) 0g is a basic closed semi algebraic subset of R n . Schmudgen s Theorem states that if S(h) is ....

K. Schmudgen: The K-moment problem for compact semi-algebraic sets. Math. Ann. 289, 203-206 (1991).

....to the condition: b(x) r(x)a(x) is a sum of squares; 4.14) which clearly implies that (4.13) holds. In the case of basic compact semialgebraic sets, i.e. compact sets of the form K = fx 2 R n ; f 1 (x) 0; f s (x) 0g, a stronger version of the Positivstellensatz, due to Schmudgen [81] can be applied. It says that a polynomial f(x) that is strictly positive on K, actually belongs to the cone generated by the f i . The Positivstellensatz presented in Theorem 4.4 only guarantees in this case the existence of g; h in the cone such that fg = 1 h. Example 4.9 To illustrate the ....

K. Schmudgen. The k-moment problem for compact semialgebraic sets. Math. Ann., 289:203--206, 1991.

....Open problems Let K be the basic closed semi algebraic set in R n defined by some finite set of polynomial inequalities g 1 0; g s 0 and let T be the preordering in the polynomial ring R [X] R [X 1 ; X n ] generated by g 1 ; g s . For K compact, Schmudgen proves in [10] that: The K Moment Problem has a positive solution. y) 8 f 2 R [X] f 0 on K ) 8 real ffl 0, f ffl 2 T . In the present paper, we consider the status of ( and (y) when K is not compact. At the same time, we consider a third property: z) 8 f 2 R [X] f 0 on K ) 9 q 2 T such that 8 ....

.... ) n , c 1 ; c m 2 R , and m 1. For g 2 R [X] g = P a k X k , g(E)p : Z ) n R is defined by g(E)p( X k a k p(k ) The condition that L 2 K lin S corresponds exactly to the condition that functions g e (E)p, e 2 f0; 1g s are positive definite, see [8] or [10]. Consequently, is equivalent to the assertion that every non zero function p : Z ) n R with g e (E)p positive definite for all e 2 f0; 1g s comes from a positive Borel measure on R n supported by K S . Schmudgen s 1991 paper [10] settles the Moment Problem in the compact case. ....

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K. Schmudgen, The K-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), 203--206.

....This is the so called Kadison Dubois theorem. It is used in real algebra, e.g. by Becker [1] in his study of sums of 2n th powers and the real holomorphy ring of a field. In [2] Becker and Schwartz give a short elementary proof in the commutative case. In his solution of the moment problem in [8], Schmudgen gives a representation of polynomials strictly positive on a bounded basic closed semi algebraic set in R n . Putinar [7] gives a criterion for linear representations to exist. Jacobi and Prestel [5] show how Schmudgen s representation can be improved and determine exactly when the ....

K. Schmudgen, The K-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), 203--206.

....is given which holds for any basic closed semialgebraic set in R n (compact or not) The proof is an extension of Wormann s proof. The Positivstellensatz, proved by G. Stengle in [12] is a standard tool in real algebraic geometry; see [2] 5] 7] In his solution of the K moment problem in [11], Schmudgen proves a surprisingly strong version of 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 ....

....the set f0; 1g r , where each s e is a sum of squares in R[V ] A basic version of the Positivstellensatz [7, Lemma 7. 5] asserts that, for any f 2 R[V ] f 0 on K iff (1 s)f = 1 t for some s; t 2 T: For more comprehensive formulations of the Positivstellensatz, see [2] 5] 7] In [11], Schmudgen proves, for K compact and f 2 R[V ] f 0 on K ) f 2 T or, equivalently, f 0 on K iff f ffl 2 T for any rational ffl 0: We refer to this latter result as the archimedean Positivstellensatz. Schmudgen s proof uses methods from functional analysis. In [13] 14] Wormann gives an ....

K. Schmudgen, The K-moment problem for compact semialgebraic sets, Math. Ann. 289 (1991), 203--206.

....of higher level for real algebraic geometry. In 1991 K. Schm udgen proved with functional analytic methods a basic semialgebraic result. Namely assume that our xed semialgebraic subset S is compact. Then any polynomial f 2 R[X] which is positive on S is already contained in the preordering P ([24]) In 1996 the second author found a purely algebraic proof of this result ( 26] Its main ingredient is the representation theorem of Kadison Dubois. In this paper we will show that it is precisely this theorem which allows us to generalize the results in the quadratic case to preorderings of ....

K. Schmudgen: The K-moment problem for compact semi-algebraic sets, Math Ann. 289 (1991), 203-206.

.... Thomas Jacobi Alexander Prestel 1 Introduction Let K = S(p 1 ; p m ) be a compact basic closed semi algebraic subset of R n , i.e. p 1 ; p m are polynomials from R[X] R[X 1 ; X n ] and S(p 1 ; p m ) fa 2 R n j p 1 (a) 0; p m (a) 0g: In [Sch], Schm udgen has shown that every p 2 R[X] strictly positive on K, i.e. p(a) 0 for all a 2 S, has a representation p = X p 1 1 p m m (1.1) where = 1 ; m ) 2 f0; 1g m and 2 P R[X] 2 = set of sums of squares in the ring R[X] In [Pu] Putinar ....

K. Schm udgen: The K-Moment Problem for Compact Semi-Algebraic Sets, Mathematische Annalen 289 (1991), 203-206 13

....g 1 ; g k . Assume that S(g 1 ; g k ) is bounded. Then for every f 2 R[X 1 ; X n ] which is strictly positive on S(g 1 ; g k ) there exist sums of polynomial squares s e 2 R[X 1 ; X n ] e 2 N k ) such that f = P e2N k s e g e1 1 g em k (see [Sch] and [W] Be W] Schm udgens proof is based on the Positivstellensatz and makes essential use of functional analytic techniques. By re ning the functional analytic approach of Schm udgen, Putinar later gave in [Pu] a criterion to decide whether a linear representation is possible: Suppose ....

K. Schm udgen: The K-Moment Problem for Compact Semi-Algebraic Sets, Mathematische Annalen 289 (1991), 203-206

....that BT (A) A. Then (1) SBT (A) A. 2) For any f 2 A, f 0 on SperT (A) f 2 T . Note. 2) is a strengthening of (1) In [8] Wormann gives a short elementary proof of (1) The fact that (1) implies (2) is a general result which follows from the Kadison Dubois Theorem; see [2] 8] See [7] for Schmudgen s original statement and proof and for the connection of this to the K moment problem for Borel measures on compact semi algebraic sets. It is natural to ask if, in the above result, the assumption that A is a finitely generated R algebra is necessary. Recently, Monnier [6] has ....

K. Schmudgen, The K-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), 203--206.

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K. Schmudgen, The K-moment problem for compact semi-algebraic sets, Math. Ann. 289, 203-206, 1991

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K. Schm udgen, The K-moment problem for compact semi-algebraic sets, Math. Ann., 289 (1991), pp. 203--206.

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K. Schmudgen. The K-moment problem for compact semi-algebraic sets. Mathematische Annalen, 289:203--206, 1991.

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K. Schmudgen. The K-moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), 203--206.

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K. Schmudgen. The k-moment problem for compact semi-algebraic sets. Math. Ann., (2):203--206, 1991.

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K. Schmudgen. The k-moment problem for compact semialgebraic sets. Mathematische Annalen, 289:203--206, 1991.

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K. Schmudgen. The k-moment problem for compact semialgebraic sets. Math. Ann., 289:203--206, 1991.

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K. Schmudgen. The K-moment problem for compact semi-algebraic sets. Mathematische Annalen, 289:203--206, 1991.

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K. Schmudgen. The K-moment problem for compact semi-algebraic sets. Mathematische Annalen, 289:203--206, 1991.

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