Explicit equations and bounds for the Nakai-Nishimura-Dubois-Efroymson dimension theorem

The Nakai-Nishimura-Dubois-Efroymson dimension theorem asserts the following: "let R be an algebraically closed field or a real closed field, let X be an irreducible algebraic subset of Rn and let Y be an algebraic subset of X of codimention s>=2 (not necessarily irreducible). Then, there is an irreducible algebraic subset W of X of codimention 1 containing Y". In this paper, making use of an elementary construction, we improve this result giving explicit polynomial equations for W. Moreover, denoting by R the algebraic closure of R and embedding canonically W into projective space Pn(R), we obtain explicit upper bounds for the degree and the geometric genus of the Zariski closure of W in Pn(R). In future papers, we will use these bounds in the study of morphism space between algebraic varieties over real closed fields