Submission date: 24 June 2015.

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)⋅ 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.

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Full text arXiv 1506.07489: pdf, ps.