Real Algebraic and Analytic Geometry
Submission: 2004, January 13.
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 $\R^n$ and let $Y$ be an algebraic subset of $X$ of codimension $s \geq 2$ $($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 $\cR$ the algebraic closure of $\R$ and embedding canonically $W$ into the projective space $\PP^n(\cR)$, we obtain explicit upper bounds for the degree and the geometric genus of the Zariski closure of $W$ in $\PP^n(\cR)$. In future papers, we will use these bounds in the study of morphism space between algebraic varieties over real closed fields.
Keywords and Phrases: Dimension theorems, Irreducible algebraic subvarieties, Upper bounds for the degree of algebraic varieties, Upper bounds for the geometric genus of algebraic varieties.
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