Betti number bounds for fewnomial hypersurfaces via stratified Morse theory

Abstract

We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a fewnomial hypersurface in R N. In the book Fewnomials (6), A. Khovanskii gives bounds on the Betti numbers of varieties X in R n or in the positive orthant R n . These varieties include algebraic varieties, varieties defined by polynomial functions in the variables and exponentials of the variables, and also by even more general functions. Here, we consider only algebraic varieties. For these, Khovanskii bounds the sum b∗(X) of Betti numbers of X by a function that depends on n, the codimension of X, and the total number of monomials appearing (with non zero coefficients) in polynomial equations definingX. For real algebraic varieties X, the problem of bounding b∗(X) has a long history. O. A. Oleinik (8) and J. Milnor (7) used Morse theory to estimate the number of critical points of a suitable Morse function to obtain a bound. R. Thom (12) used Smith Theory to bound the mod-2 Betti numbers. This Smith-Thom bound has the form b∗(X) ≤ b∗(XC), where XC is the complex variety defined by the same polynomials as X. For instance, if X ⊂ R n is defined by polynomials of degree at most d, then Milnor's bound is b∗(X) ≤ d(2d − 1) n . If X ⊂ (R {0}) n is a smooth hypersurface defined by a polynomial with Newton polytope P , then the Smith-Thom bound is b∗(X) ≤ b∗(XC) = n! ¢ Vol(P ), where Vol(P ) is the usual volume of P. We refer to (10) for an informative history of the subject, and to the book (1) for more details. These bounds are given in term of degree or volume of a Newton polytope which are numerical deformation invariants of the complex variety XC. In contrast, the topology of a real algebraic variety depends on the coefficients of its defini ng equations, and in partic- ular, on the number of monomials involved in these equations. For instance, Descartes's rule of signs implies that the number of positive roots of a real univariate polynomial is less than its number of monomials, but the number of complex roots is equal to its degree. Khovanskii's bound can be seen as a generalization of this Descartes bound. It is smaller than the previous bounds when the defining equations have few monomials compared to their degrees. While it has always been clear that Khovanskii's bounds are unrealistically large, it appears very challenging to sharpen them. Some progress has been made recently for the number of non degenerate positive solutions to a system of n polynomials in n variables (3). Here, we bound b∗(X), when X is a fewnomial hypersurface. Suppose that X ⊂ R n is a smooth hypersurface defined by a Laurent polynomial with