On the elimination of algebraic inequalities (Q584335)

Let S be a locally closed semi-algebraic subset of \({\mathbb{R}}^ n\). We prove the existence of an irreducible real polynomial \(P(x_ 1,x_ 2,...,x_ n,t)\) having a real root iff \((x_ 1,x_ 2,...,x_ n)\in S\). This result generalizes an earlier result valid for closed semi- algebraic sets [the author, C. R. Acad. Sci., Paris, Sér. I 306, No.5, 265-268 (1988; Zbl 0656.14014)]. - The proof is more algebraic. First there is an explicit construction of a polynomial \(P(x_ 1,x_ 2,...,x_ n,t)\) having a real root iff all the \(x_ i\) are nonnegative. Then, using the theory of the resultant, the paper concludes with a proof of the following theorem in the particular case of S locally closed with non empty interior. Theorem: Let S be a semi-algebraic subset of \({\mathbb{R}}^ n\); S is the projection of an irreducible algebraic subset of \({\mathbb{R}}^{n+1}\) iff the Zariski closure of S is irreducible. But the complete proof of this general theorem (to appear) is not explicit.