Representing simple-dimensional polytopes bypolynomials

Gennadiy Averkov, Martin Henk

Abstract: A polynomial representation of a convex d-polytope P is a finite set {p 1(x), . . . , p n (x)} of polynomials over \({\mathbb {R}^d}\) such that \({P=\{x \in \mathbb {R}^d : p_i(x) \ge 0 \mbox{ for every }1 \le i \le n\}}\). Let s(d, P) be the least possible n as above. It is conjectured that s(d, P) = d for all convex d-polytopes P. We confirm this conjecture for simple d-polytopes by providing an explicit construction of d polynomials that represent a given simple d-polytope P.