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 $\{(P(a,b),(P(a^{\prime },b^{\prime }),P(a^{\prime },b),P(a,b^{\prime })|a,a^{\prime },b,b^{\prime }\in k\}$ 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.

References in corpus (1)

• Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets