Abstract: 

Consider a finite set F of one-variable polynomials of bounded degree over the complex numbers. Let X be a finite subset of the complex numbers and F*X be the collection of f(x) for f in F and x in X. Suppose the sizes of F and X are comparable in the sense that |F| is bounded above and below by some fixed powers of |X|. We want to ask when it is possible that |F*X|<|X|^{1+e} for very small e>0? The goal of this talk is to answer this question by considering a generalization of the Elekes-Szabó's Theorem. This is joint work with Martin Bays.

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