Minimal polynomial descriptions of polyhedra and special semialgebraic sets

TL;DR

This paper demonstrates that polyhedra and certain semialgebraic sets can be represented using a minimal number of polynomial inequalities, simplifying their algebraic descriptions.

Contribution

It introduces a method to represent d-dimensional polyhedra with exactly d polynomial inequalities and extends this to special semialgebraic sets with controlled polynomial vanishing conditions.

Findings

Polyhedra in d can be described by d polynomial inequalities.

Semialgebraic sets with limited polynomial vanishing points can be represented by s+1 polynomials.

Finiteness of vanishing points allows reduction to s polynomials.

Abstract

We show that a $d$-dimensional polyhedron $S$ in $\real^d$ can be represented by $d$-polynomial inequalities, that is, $S = \{x \in \real^d : p_0(x) \ge 0, >..., p_{d-1}(x) \ge 0 \}$, where $p_0,...,p_{d-1}$ are appropriate polynomials. Furthermore, if an elementary closed semialgebraic set $S$ is given by polynomials $q_1,...,q_k$ and for each $x \in S$ at most $s$ of these polynomials vanish in $x$, then $S$ can be represented by $s+1$ polynomials (and by $s$ polynomials under the extra assumption that the number of points $x \in S$ in which $s$ $q_i$'s vanish is finite).