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 codimension s ≥ 2 (not necessarily irreducible). Then, there is an irreducible algebraic subset W of X of codimension 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 the 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.