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arXiv · 0709.2099

Representing simple d-dimensional polytopes by d polynomials

Gennadiy Averkov·Martin Henk
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Abstract

A polynomial representation of a convex d-polytope P is a finite set \{p_1(x),...,p_n(x)\} of polynomials over E^d such that P=\setcond{x \in \E^d}{p_1(x) \ge 0 {for every} 1 \le i \le n}. By s(d,P) we denote the least possible number of polynomials in a polynomial representation of P. It is known that d \le s(d,P) \le 2d-1. Moreover, 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.

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Gennadiy Averkov, Martin Henk. 2007-09-13. Representing simple d-dimensional polytopes by d polynomials. https://arxiv.org/abs/0709.2099

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