Minimal polynomial descriptions of polyhedra and special semialgebraic sets

Details Minimal polynomial descriptions of polyhedra and special semialgebraic sets Minimal polynomial descriptions of polyhedra and special semialgebraic sets

2010-02-04

arxiv.org/abs/1002.0921v1

Mathematics

Algebraic Geometry

Scientific paper

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).

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