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

Abstract

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 27 e 5 3 + 88 90 ∏ i = 0 k − 1 ( 2 i ( n − 1 ) + 1 ) positive solutions and 27 e 10 3 + 88 90 ∏ i = 0 k − 1 ( 2 i ( n − 1 ) + 1 ) non-degenerate non-zero real solutions. This dramatically improves F. Bihan and F. Sottile’s bounds of e 2 + 3 4 2 k 2 n k and e 4 + 3 4 2 k 2 n k 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.