Bounds on the number of solutions of polynomial systems and the Betti numbers of real piecewise algebraic hypersurfaces

摘要

This paper, based on Bihan and Sottile’s method which reduces a polynomial system to its Gale dual system and then bounds the number of solutions of this Gale system, proves that a real coefficient polynomial system with n equations and with n variables involving n+k+1 monomials has fewer than 27e53+8890∏i=0k−1(2i(n−1)+1) positive solutions and 27e103+8890∏i=0k−1(2i(n−1)+1) non-degenerate non-zero real solutions. This dramatically improves F. Bihan and F. Sottile’s bounds of e2+342k2nk and e4+342k2nk respectively. Using the new upper bound for positive solutions, we establish restrictions to the sum of the Betti numbers of real piecewise algebraic hypersurfaces and real piecewise algebraic curves. A new bound on the number of compact components of algebraic hypersurfaces in R>n is also given.