Representing simple d-dimensional polytopes by d polynomials
Abstract
A polynomial representation of a convex d-polytope P is a finite set \p1(x),...,pn(x)\ of polynomials over Ed such that P=x ∈ dp1(x) 0 for every 1 i n. By s(d,P) we denote the least possible number of polynomials in a polynomial representation of P. It is known that d s(d,P) 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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