Reducibility by polynomial functions

TL;DR

This paper investigates the structure of subsets of algebraically closed fields under polynomial preimages, exploring how polynomial functions induce a preorder relation among these sets.

Contribution

It introduces and analyzes the preorder $\, extstyleiglackslash_p ext{ on subsets of algebraically closed fields, based on polynomial preimages, providing new insights into their algebraic and combinatorial properties.

Findings

Characterization of the preorder $\, extstyleiglackslash_p$ on subsets

Conditions under which sets are related by polynomial preimages

Structural properties of subsets under polynomial transformations

Abstract

We study the preorder $\le_p$ on the family of subsets of an algebraically closed field of characteristic $0$ defined by letting $A\le_pB $ if there exists a polynomial $P$ such that $A=P^{-1}(B)$.