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2008-07

Notes on the algebra and geometry of polynomial representations

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