Topology of real multi-affine hypersurfaces and a homological stability property

Abstract

Let $\mathrm{R}$ be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in ${\mathrm{R}}^n$ defined by a multi-affine polynomial of degree $d$ is bounded by $2^{d-1}$. This bound is sharp and is independent of $n$ (as opposed to the classical bound of $d(2d-1{)}^{n-1}$ on the Betti numbers of hypersurfaces defined by arbitrary polynomials of degree $d$ in ${\mathrm{R}}^n$ due to Petrovski{\uı} and Ole{\uı}nik, Thom and Milnor). Moreover, we show there exists $c>1$, such that given a sequence $(B_n{)}_{n>0}$ where $B_n$ is a closed ball in ${\mathrm{R}}^n$ of positive radious, there exist hypersurfaces $(V_n{)}_{n_{>}0}$ defined by symmetric multi-affine polynomials of degree $4$, such that ${\sum }_{i\le 5}{b}_i(V_n\cap B_n)>c^n$, where $b_i(\cdot )$ denotes the $i$-th Betti number with rational coeffcients. Finally, as an application of the main result of the paper we verify a representational stability conjecture due to Basu and Riener on the cohomology modules of symmetric real algebraic sets for a new and much larger class of symmetric real algebraic sets than known before.

26 pages. Comments welcome