Notes on algebra and geometry of polynomial representations

2008/05/05 by Averkov, Gennadiy
#14P10 #52B05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.0805.0522

Abstract

Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)≥ 0, or p_i(x)=0 for a given list of polynomials p₁,...,p_m. 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,...,p_m. Special cases when A is elementary closed, elementary open, a polygon, or a polytope are considered separately.

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