Abstract view
Equations and Complexity for the DuboisEfroymson Dimension Theorem


Published:20090601
Printed: Jun 2009
Abstract
Let $\R$ be a real closed field, let $X \subset \R^n$ be an
irreducible real algebraic set and let $Z$ be an algebraic subset of
$X$ of codimension $\geq 2$. Dubois and Efroymson proved the existence
of an irreducible algebraic subset of $X$ of codimension $1$
containing~$Z$. We improve this dimension theorem as follows. Indicate
by $\mu$ the minimum integer such that the ideal of polynomials in
$\R[x_1,\ldots,x_n]$ vanishing on $Z$ can be generated by polynomials
of degree $\leq \mu$. We prove the following two results:
\begin{inparaenum}[\rm(1)]
\item There
exists a polynomial $P \in \R[x_1,\ldots,x_n]$ of degree~$\leq \mu+1$
such that $X \cap P^{1}(0)$ is an irreducible algebraic subset of $X$
of codimension $1$ containing~$Z$.
\item Let $F$ be a polynomial in
$\R[x_1,\ldots,x_n]$ of degree~$d$ vanishing on $Z$. Suppose there
exists a nonsingular point $x$ of $X$ such that $F(x)=0$ and the
differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then
there exists a polynomial $G \in \R[x_1,\ldots,x_n]$ of degree $\leq
\max\{d,\mu+1\}$ such that, for each $t \in (1,1) \setminus \{0\}$,
the set $\{x \in X \mid F(x)+tG(x)=0\}$ is an irreducible algebraic
subset of $X$ of codimension $1$ containing~$Z$.
\end{inparaenum} Result (1) and a
slightly different version of result~(2) are valid over any
algebraically closed field also.