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LetÌýF(x,y,z)Ìýbe a real trivariate polynomial of constant degree, and letÌýA,B,CÌýbe threeÌýsets of real numbers, each of sizeÌýn. How many roots canÌýFÌýhave onÌýA x B x C?Ìý This question has been studied by Elekes and Rónyai and then by Elekes and Szabó about 15 years ago.ÌýOne of their striking results is that, for the special case whereÌýF(x,y,z) = z-f(x,y), eitherÌýFÌývanishesÌýatÌýo(n2)Ìýnumber of points ofÌýA x B x C, or else f must have one of the special formsÌýf(x,y) = h(p(x)+q(y))Ìý orÌýf(x,y) = h(p(x)q(y)), for univariate polynomialsÌýp, q, h. In the talk I will discuss several recent results, in which the analysis is greatly simplified, and the bounds become sharp: IfÌýFÌýdoes not have a special form, the number of roots is at mostÌýO(n11/6). The results also hold over the complex field.Ìý This setup arises in various Erdös-type problems in combinatorial geometry, and the result provides a unified tool for their analysis. If time allows, I will discuss an application of this kind to the following problem: GivenÌýnÌýpoints lying on aÌýd-dimensional algebraic variety inÌýR^D, show that there always exists a large subsetÌýS, such that all the distances spanned by pairs of points ofÌýSÌýare distinct. This is a variant of a question of Erdos from 1957, studied recently by Conlon et al. Based onÌýjointÌýworks with Micha Sharir, Jozsef Solymosi, and Frank de Zeeuw.