Representing simple d-dimensional polytopes by d polynomials

TL;DR

This paper proves that simple convex d-polytopes can be represented exactly by d polynomials, confirming a conjecture and providing explicit constructions for such polynomial representations.

Contribution

It confirms the conjecture that simple d-polytopes can be represented by d polynomials and offers an explicit construction method for these representations.

Findings

Simple d-polytopes can be represented by exactly d polynomials.

The paper provides an explicit construction for polynomial representations.

It confirms the conjecture s(d,P)=d for simple d-polytopes.

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.