An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

We give a construction of a non-degenerate polynomial F ∈ R[x, y, z] and a set A of cardinality n such that |Z(F ) ∩ (A × A × A)| ≫ n 3 2 , thus providing a new lower bound construction for the Elekes-Szabó problem. We also give a related construction for the Elekes-Rónyai problem restricted to a subgraph. This consists of a polynomial f ∈ R[x, y] that is not additive or multiplicative, a set A of size n, and a subset P ⊂ A × A of size |P | ≫ n 3/2 on which f takes only n distinct values.2000 Mathematics Subject Classification. 52C10 (26C05, 05A99) .

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In this paper we give a conditional improvement to the Elekes-Szabó problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for F ∈ Q[x, y, z] belonging to a particular family of polynomials, and any finite setsThe value of the integer s is dependent on the polynomial F , but is always bounded by s ≤ 5, and so even in the worst applicable case this gives a quantitative improvement on a bound of Raz, Sharir and de Zeeuw [24]. We give several applications to problems in discrete geometry and arithmetic combinatorics. For instance, for any set P ⊂ Q 2 and any two points p1, p2 ∈ Q 2 , we prove that at least one of the pi satisfies the bound |{ pi − p : p ∈ P }| ≫ |P | 3/5 , where • denotes Euclidean distance. This gives a conditional improvement to a result of Sharir and Solymosi [28].

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