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2007-31

Representing simple d-dimensional polytopes by d polynomials

by Averkov, Gennadiy, Henk, Martin.


Series: 2007-31, Preprints

MSC:
14P05 Real algebraic sets
52B11 $n$-dimensional 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.

Keywords:
Metric Geometry; Algebraic Geometry