Reducibility by polynomial functions

Abstract

We study the preorder ${\le }_{\mathrm{Pol}}$ (and the associated equivalence relation ${\equiv }_{\mathrm{Pol}}$) on the family of subsets of an algebraically closed field k of characteristic 0, defined by letting $A{\le }_{\mathrm{Pol}}B$ iff there exists a polynomial P such that $A=P^{-1}\left(B\right)$. We concentrate mainly on the finite subsets of k and prove that the ${\equiv }_{\mathrm{Pol}}$-equivalence classes of sets of a given finite cardinality form an affine algebraic variety; inside these varieties, we compute in particular the dimension of the set of ${\equiv }_{\mathrm{Pol}}$-classes that have less representatives than the generic ones and the dimension of the set of ${\equiv }_{\mathrm{Pol}}$-classes that are comparable with a given ${\equiv }_{\mathrm{Pol}}$-class.

We also show that, in a specified sense, very many ${\equiv }_{\mathrm{Pol}}$-classes are ${\le }_{\mathrm{Pol}}$-maximal (or ${\le }_{\mathrm{Pol}}$-maximal under the class of singletons, for ${\equiv }_{\mathrm{Pol}}$-classes of finite sets).