Equations and Complexity for the Dubois–Efroymson Dimension Theorem

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 ≥ 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 $R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ vanishing on $Z$ can be generated by polynomials of degree $\le \,\mu$. We prove the following two results: (1) There exists a polynomial $P\,\in \,R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $\le \,\mu +1$ such that $X\cap {{P}^{-1}}\left( 0 \right)$ is an irreducible algebraic subset of $X$ of codimension 1 containing $Z$. (2) Let $F$ be a polynomial in $R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $d$ vanishing on $Z$. Suppose there exists a nonsingular point $x$ of $X$ such that $F\left( x \right)\,=\,0$ and the differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then there exists a polynomial $G\,\in \,R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $\le \,\max \{d,\,\mu \,+\,1\}$ such that, for each $t\,\in \,\left( -1,\,1 \right)\,\backslash \,\{0\}$, the set $\{x\in X|F\left( x \right)+tG\left( x \right)=0\}$ is an irreducible algebraic subset of $X$ of codimension 1 containing $Z$. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.