Indiscernible arrays and rational functions with algebraic constraint

TL;DR

This paper investigates algebraic constraints on polynomials and rational functions, providing a decomposition characterization for functions with such constraints, and applies these results to indiscernible arrays in stable theories.

Contribution

It extends Tao's polynomial decomposition results to rational functions in three variables with algebraic constraints, answering a question about indiscernible arrays in model theory.

Findings

Decomposition of rational functions with algebraic constraints in three variables.

Characterization of indiscernible arrays in stable theories.

Application of polynomial decomposition to model-theoretic structures.

Abstract

Let $k$ be an algebraically closed field of characteristic zero and $P(x,y)\in k[x,y]$ be a polynomial which depends on all its variables. $P$ has an algebraic constraint if the set $\{(P(a,b),(P(a',b'),P(a',b),P(a,b')\,|\,a,a',b,b'\in k\}$ does not have the maximal Zariski-dimension. Tao proved that if $P$ has an algebraic constraint then it can be decomposed: there exists $Q,F,G\in k[x]$ such that $P(x_{1},x_{2})=Q(F(x_{1})+G(x_{2}))$, or $P(x_{1},x_{2})=Q(F(x_{1})\cdot G(x_{2}))$. In this paper we give an answer to a question raised by Hrushovski and Zilber regarding 3-dimensional indiscernible arrays in stable theories. As an application of this result we find a decomposition of rational functions in three variables which has an algebraic constraint.