Real Algebraic and Analytic Geometry |

Explicit Equations and Bounds for the Nakai--Nishimura--Dubois--Efroymson Dimension Theorem.

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Submission: 2004, January 13.

*Abstract:
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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