Notes on the algebra and geometry of polynomial representations

Gennadiy Averkov

Abstract: The paper deals with a semi-algebraic set $A$ in $\real^d$ constructed by the inequalities $p_i(x)>0$, $p_i(x) \ge 0$, and $p_i(x) = 0$ for a given list of polynomials $p_1,\ldots,p_m$, and presents 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 3$p_1,\ldots,p_m$. Special attention is paid to the case when $A$ is a polytope.

Keywords: irreducible polynomial, polygon, polytope, polynomial representation, real algebraic geometry, semi-algebraic set

Classification (MSC2000): 14P10, 52B11

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