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Author Parusinski, Adam ♦ Nski, Adam Parusi ♦ Szafraniec, Zbigniew
Source CiteSeerX
Content type Text
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Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Abstract . Let Y be a real algebraic subset of R m and let F : Y ! R n be a polynomial map. We show that there exist real polynomial functions g1 ; : : : ; g s on R n such that the Euler characteristic of fibres of F is the sum of signs of g i . The purpose of this paper is to give a new, self-contained and elementary proof of the following result. Theorem. Let Y be a real algebraic subset of R m and let F : Y ! R n be a polynomial map. Then there exist real polynomials g 1 (y); : : : ; g s (y) on R n such that the Euler characteristic of fibres of F is the sum of signs of g i , that is (F \Gamma1 (y)) = sgn g 1 (y) + \Delta \Delta \Delta + sgn g s (y): Our proof is based on a classical and elementary result expressing the number of real roots of a real polynomial of one variable as the signature of an associated quadratic form known already to Hermite [He1, He2] and Sylvester [Syl], see also [B], [BW], [BCR, p. 97]. In the proof we use a modern generalized version of this result...
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study
Learning Resource Type Article
Publisher Date 1998-01-01