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98
Optimization of polynomials on compact semialgebraic sets
 SIAM J. OPTIM
"... A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxat ..."
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Cited by 57 (4 self)
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A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex
Parametrization of Semialgebraic Sets
, 1993
"... In this paper we consider the problem of the algorithmic parametrization of a d dimensional semialgebraic subset S of R n (n ? d) by a semialgebraic and continuous mapping from a subset of R d . Using the Cylindrical Algebraic Decomposition algorithm we easily obtain semialgebraic, bijective parame ..."
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In this paper we consider the problem of the algorithmic parametrization of a d dimensional semialgebraic subset S of R n (n ? d) by a semialgebraic and continuous mapping from a subset of R d . Using the Cylindrical Algebraic Decomposition algorithm we easily obtain semialgebraic, bijective
TAMENESS OF HOLOMORPHIC CLOSURE DIMENSION IN A SEMIALGEBRAIC SET
"... Abstract. Given a semianalytic set S in C n and a point p ∈ S, there is a unique smallest complexanalytic germ Xp which contains Sp, called the holomorphic closure of Sp. We show that if S is semialgebraic then Xp is a Nash germ, for every p, and S admits a semialgebraic filtration by the holomorph ..."
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Cited by 2 (1 self)
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by the holomorphic closure dimension. As a consequence, every semialgebraic subset of a complex vector space admits a semialgebraic stratification into CR manifolds. 1.
On chains of prime ideals in rings of semialgebraic functions
 Q. J. Math. XXX (2013, accepted), no. X, XXXXXX. http://qjmath.oxfordjournals.org/cgi/authordata?d=10.1093/qmath/hat048&k=3dc00fcc
"... Abstract In this work, we study the structure of nonrefinable chains of prime ideals in the (real closed) rings S(M) and S * (M) of semialgebraic and bounded semialgebraic functions on a semialgebraic set M ⊂ R m . We pay special attention to the prime zideals of S(M) and the minimal prime ideals ..."
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Cited by 3 (3 self)
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ideals of both rings. For the last, a decomposition of each semialgebraic set as an irredundant finite union of closed pure dimensional semialgebraic subsets plays a crucial role. We prove moreover the existence of maximal ideals in the ring S(M) of prefixed height whenever M is noncompact.
Computing roadmaps of semialgebraic sets on a variety
 Journal of the AMS
, 1997
"... Let R be a real closed field, Z(Q) a real algebraic variety defined as the zero set of a polynomial Q ∈ R[X1,...,Xk]andSasemialgebraic subset of Z(Q), defined by a Boolean formula with atoms of the form P < 0,P> 0,P =0withP∈P, where P is a finite subset of R[X1,...,Xk]. A semialgebraic set C ..."
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Cited by 50 (17 self)
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Let R be a real closed field, Z(Q) a real algebraic variety defined as the zero set of a polynomial Q ∈ R[X1,...,Xk]andSasemialgebraic subset of Z(Q), defined by a Boolean formula with atoms of the form P < 0,P> 0,P =0withP∈P, where P is a finite subset of R[X1,...,Xk]. A semialgebraic set
Computing Roadmaps of General SemiAlgebraic Sets
, 1993
"... In this paper we study the problem of determining whether two points lie in the same connected component of a semialgebraic set S. Although we are mostly concerned with sets S # , our algorithm can also decide if points in an arbitrary set S # R can be joined by a semialgebraic path, for any real ..."
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Cited by 54 (2 self)
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closed field R. Our algorithm computes a onedimensional semialgebraic subset ##S# of S (actually of an embedding of S in a space R for a certain real extension field R of the given field R#. ##S# is called the roadmap of S. The basis of this work is the roadmap algorithm described in [3], [4
ON THE CARDINALITY OF A SEMIALGEBRAIC SET
"... Abstract. It is shown that the cardinality of a finite semialgebraic subset over a real closed field can be computed in terms of signatures of effectively constructed quadratic forms. 1. The problem under consideration may be described as follows. Let X be a semialgebraic set over an ordered field ..."
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Abstract. It is shown that the cardinality of a finite semialgebraic subset over a real closed field can be computed in terms of signatures of effectively constructed quadratic forms. 1. The problem under consideration may be described as follows. Let X be a semialgebraic set over an ordered
Bounds for Representations of Polynomials Positive on Compact SemiAlgebraic Sets
"... . By Schmudgen's Theorem, polynomials f 2 R[X1 ; : : : ; Xn ] strictly positive on a bounded basic semialgebraic subset of R n , admit a certain representation involving sums of squares oe i from R[X1 ; : : : ; Xn ]. We show the existence of effective bounds on the degrees of the oe i by ..."
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Cited by 2 (0 self)
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. By Schmudgen's Theorem, polynomials f 2 R[X1 ; : : : ; Xn ] strictly positive on a bounded basic semialgebraic subset of R n , admit a certain representation involving sums of squares oe i from R[X1 ; : : : ; Xn ]. We show the existence of effective bounds on the degrees of the oe i
BETTI NUMBERS OF SEMIALGEBRAIC AND SUBPFAFFIAN SETS
"... Let X be a subset in [−1,1] n0 ⊂ Rn0 defined by a formula X = {x0  Q1x1Q2x2...Qνxν((x0,x1,...,xν) ∈ Xν)}, where Qi ∈ {∃, ∀}, Qi � = Qi+1, xi ∈ Rni,and Xν be either an open or a closed set in [−1,1] n0+...+nν being a difference between a finite CWcomplex and its subcomplex. We express an upper bou ..."
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Cited by 30 (9 self)
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Let X be a subset in [−1,1] n0 ⊂ Rn0 defined by a formula X = {x0  Q1x1Q2x2...Qνxν((x0,x1,...,xν) ∈ Xν)}, where Qi ∈ {∃, ∀}, Qi � = Qi+1, xi ∈ Rni,and Xν be either an open or a closed set in [−1,1] n0+...+nν being a difference between a finite CWcomplex and its subcomplex. We express an upper
Results 1  10
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98