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

Let R be a real closed field. We prove that the number of semi -algebraically connected components of a real hypersurface in Rn defined by a multi-affine polynomial of degree d is bounded by 2d-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 Rn due to Petrovskii and Oleinik, Thom and Milnor). Moreover, we show there exists c > 1, such that given a sequence (Bn)n>0 where Bn is a closed ball in Rn of positive radius, there exist hypersurfaces (Vn subset of Rn)n>0 defined by symmetric multi-affine polynomials of degree 4, such that Ei',5 bi(Vn boolean AND Bn) > cn, where bi(center dot) denotes the i-th Betti number with rational coefficients. 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. (c) 2023 Elsevier Inc. All rights reserved.