Notes on algebra and geometry of polynomial representations

TL;DR

This paper explores the algebraic and geometric properties of polynomial representations of semi-algebraic sets, focusing on boundary behavior and polynomial factorization in various special cases.

Contribution

It provides new insights into the relationship between boundary descriptions and polynomial factorizations for semi-algebraic sets.

Findings

Boundary points are characterized by polynomial factors.

Polynomial factors relate to the local description of semi-algebraic sets.

Special cases like polygons and polytopes are analyzed separately.

Abstract

Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)\ge 0, or p_i(x)=0 for a given list of polynomials p_1,...,p_m. We prove several statements that fit into the following template. Assume that in a neighborhood of a boundary point the semi-algebraic set A can be described by an irreducible polynomial f. Then f is a factor of a certain multiplicity of some of the polynomials p_1,...,p_m. Special cases when A is elementary closed, elementary open, a polygon, or a polytope are considered separately.