用d多项式表示简单d维多面体
Representing simple d-dimensional polytopes by d polynomials
摘要：
凸d-多胞形P的多项式表示是有限集\{p_1（x），.，p_n（x）\}的多项式，使得P= \setcond{x \in \E^d}{p_1(x) \ge 0 {for every} 1 \le i \le n}。通过s（d，P），我们表示P的多项式表示中的多项式的最小可能数目。已知d \le s（d，P）\le 2d-1。此外，还证明了对于所有的凸d-多面体P，s（d，P）=d.通过给出表示给定的简单d-多面体P的d多项式的显式构造，我们对简单d-多面体P证实了这一猜想.
英文摘要：
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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