  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)\ge 0, or p_i(x)=0 for a given list
of polynomials p_1,...,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 p_1,...,p_m.
Special cases when A is elementary closed, elementary open, a polygon, or a
polytope are considered separately.
