Speaker: Elad Levi
(Hebrew University):
Title:
Rational functions with algebraic constraints

Abstract:
A polynomial P(x,y) over an algebraically closed field k has an algebraic constraint if the set

{(P(a,b),(P(a#,b#),P(a#,b),P(a,b#)|a,a#,b,b##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#k[x] such that P(x,y)=Q(F(x)+G(y)), or P(x,y)=Q(F(x)#G(y)). We will discuss the generalisation of this result to rational functions with 3-variables and show the connection to a problem raised by Hrushovski and Zilber regarding 3-dimensional indiscernible arrays.