Real Algebraic and Analytic Geometry
Previous   Next
78. Riccardo Ghiloni:
Explicit Equations and Bounds for the Nakai--Nishimura--Dubois--Efroymson Dimension Theorem.

e-mail:

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.

Full text, 14p.: dvi 78k, ps.gz 164k, pdf 237k.