Indiscernible arrays and rational functions with algebraic constraint

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.