### 2008-07

by Averkov, G..

**Series:** 2008-07, Preprints

- MSC:
- 14P10 Semialgebraic sets and related spaces
- 52B05 Combinatorial properties (number of faces, shortest paths, etc.)

**Abstract:**

Consider a semi-algebraic set A in Rd constructed from the sets which are determinedby inequalities pi(x)> 0, pi(x)≥ 0, or pi(x)=0for a given list of polynomials p1,...,pm. We prove several statements that ﬁt into the following template. Assume that in a neighborhood of a boundary point the semi-algebraic set A can be described by an irreducible polynomial

f. Then f is a factor of a certain multiplicity of some of the polynomials p1,...,pm. Special cases when A is elementary closed, elementary open,a polygon, or a polytope are considered separately.

**Keywords:**

Irreduciblepolynomial,polygon,polytope,polynomial representation, real algebraic geometry, semi-algebraic set