P. Pedersen, M.-F. Roy, A. Szpirglas Counting real zeroes in the multivariate case. Computational algebraic geometry, Eyssette et Galligo ed. Progress in Mathematics 109, 203-224, Birkhauser (1993). 11  Home/Search   Document Not in Database   Summary   Related Articles   Check

....this algorithm has double exponential complexity (sd) 2 O(k) Previously, this was the best algorithm for computing the Euler characteristic of general semi algebraic sets. A single exponential algorithm for computing the Euler characteristic of a smooth algebraic hypersurface is mentioned in [21]. The rest of the paper is organized as follows. In section 2 we give a brief discussion of our use of infinitesimals and the model of computation with appropriate pointers to literature. In section 3 we show how to perturb the polynomials to bring them into general position without changing the ....

....special case of computing the index of a symmetric square matrix of size k Theta k; with polynomial entries, at the real zeros of a zero dimensional system. Again, if the degrees of the polynomials in the input are bounded by d; the complexity of the subroutine is bounded by (kd) O(k) see [21]) 5.2 The algorithm for a semi algebraic set defined by one sign condition In this section, we describe an algorithm for computing the Euler characteristic of a semi algebraic set defined by one single sign condition on a family of polnomials. Using lemma 4 we can assume, without loss of ....

P. Pedersen, M.-F. Roy, A. Szpirglas Counting real zeroes in the multivariate case. Computational algebraic geometry, Eyssette et Galligo ed. Progress in Mathematics 109, 203-224, Birkhauser (1993).

....algorithm has double exponential complexity (sd) 2 O(k) Previously, this was the best algorithm for computing the Euler characteristic of general semi algebraic sets. A single exponential algorithm for computing the Euler characteristic of a smooth algebraic hypersurface is mentioned in [31]. The rest of the chapter is organized as follows. In section 2 we show how to perturb the polynomials to bring them into general position without changing the homology groups of the given semi algebraic set. In section 3 we prove our bound on the Betti numbers of closed semialgebraic sets. In ....

....the special case of computing the index of a symmetric square matrix of size k Theta k; with polynomial entries, at the real zeros of a zero dimensional system. Again, if the degrees of the polynomials in the input are bounded by d; the complexity of the subroutine is bounded by (kd) O(k) see [31]) 5.2 The algorithm for a semi algebraic set defined by one sign condition In this section, we describe an algorithm for computing the Euler characteristic of a semi algebraic set defined by one single sign condition on a family of polynomials. Using lemma 4 we can assume, without loss of ....

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P. Pedersen, M.-F. Roy, A. Szpirglas Counting real zeroes in the multivariate case, Computational algebraic geometry, Eyssette et Galligo ed. Progress in Mathematics 109, 203-224, Birkhauser (1993).

....Collins algorithm which has a double exponential complexity of (sd) 2 O(k) was the best algorithm for computing the Euler characteristic of general semi algebraic sets. A single exponential algorithm for computing the Euler characteristic of a smooth algebraic hypersurface is mentioned in [53]. The rest of the thesis is organized as follows. In chapter 2 we discuss some facts from real algebraic geometry that we will need later. In chapter 3 we describe certain essential subroutines that are going to be used repeatedly in later chapters. In chapter 4 we describe our algorithm for ....

....Z where h 0. We associate to h a symmetric bilinear form B h : A Omega A R defined by B h (f; g) Tr(L hfg ) where L hfg is the linear transformation defined above. Let the associated quadratic form, called the Hermite quadratic form, be denoted Q h . Then it is true that, Proposition 10 ([53]) SQ(Z; h) signature(Q h ) Proof: Let N be the dimension of A as a vector space and M the number of distinct points of Z R[i] I) Consider a separating element u and a basis of the vector space A of the form B = f1; u; u M Gamma1 ; M 1 ; N g. It is clear from Proposition ....

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P. Pedersen, M.-F. Roy, A. Szpirglas Counting real zeroes in the multivariate case, Computational algebraic geometry, Eyssette et Galligo ed. Progress in Mathematics 109, 203-224, Birkhauser (1993).

....the same at the parametrized points generated during the elimination phase, with the parameters specialized to y and z respectively. The quantifier free formula can then be written as Psi(Y ) y2T Psi y (Y ) In order to construct the formulae Psi y , we make use of the multivariate version ([24]) of the sign determination algorithm due to Ben Or, Kozen and Reif, in an inverse fashion. We call this the Inverse Sign Determination Subroutine. Roughly, the idea is the following. 2.3 The Inverse Sign Determination Subroutine We assume that the reader is familiar with the Sign Determination ....

....algorithm due to Ben Or, Kozen and Reif, in an inverse fashion. We call this the Inverse Sign Determination Subroutine. Roughly, the idea is the following. 2. 3 The Inverse Sign Determination Subroutine We assume that the reader is familiar with the Sign Determination Algorithm ( 27] [24], inspired by[5] Given a multivariate polynomialh, and another system of polynomials, T , having only a finite number of real zeros, denoted by Z(T ) we define the Sturm query of h with respect to T by SQ(T; h) jfx 2 Z(T ) h(x) 0gj Gammajfx 2 Z(T ) h(x) 0gj: The Sign Determination ....

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P. Pedersen, M.-F. Roy, A. Szpirglas Counting real zeroes in the multivariate case, Computational algebraic geometry, Eyssette et Galligo ed. Progress in Mathematics 109, 203-224, Birkhauser (1993).

....k . In the case k 0 = 0 we get, being slightly more specific, an algorithm evaluating the signs of a family of s polynomials at a finite set of points defined by polynomials in time sd O(k) This is a multivariate version of the Ben Or Kozen Reif algorithm improving the complexity of [13]. We plan to use the methods in this paper in order to describe a new quantifier elimination algorithm of single exponential complexity when the number of alternation of quantifiers is fixed. In this new algorithm the degrees of the polynomials in the quantifier free equivalent formula will be ....

P. Pedersen, M.-F. Roy, A. Szpirglas Counting real zeroes in the multivariate case, Computational algebraic geometry, Eyssette et Galligo ed. Progress in Mathematics 109 , 203-224, Birkhauser (1993).

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P. Pedersen, M.-F. Roy, A. Szpirglas Counting real zeroes in the multivariate case. Computational algebraic geometry, Eyssette et Galligo ed. Progress in Mathematics 109, 203-224, Birkhauser (1993). 11

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