We prove a lower bound on the number of ordinary conics determined by a finite point set in R 2 . An ordinary conic for S ⊂ R 2 is a conic that is determined by five points of S, and contains no other points of S. Wiseman and Wilson proved the Sylvester-Gallai-type statement that if a finite point set is not contained in a conic, then it determines at least one ordinary conic. We give a simpler proof of this statement, and then combine it with a theorem of Green and Tao to prove our main result: If S is not contained in a conic and has at most c|S| points on a line, then S determines Ω c (|S| 4 ) ordinary conics. We also give constructions, based on the group law on elliptic curves, that show that the exponent in our bound is best possible.
Suppose that nk points in general position in the plane are colored red and blue, with at least n points of each color. We show that then there exist n pairwise disjoint convex sets, each of them containing k of the points, and each of them containing points of both colors.We also show that if P is a set of n(d + 1) points in general position in R d colored by d colors with at least n points of each color, then there exist n pairwise disjoint d-dimensional simplices with vertices in P , each of them containing a point of every color.These results can be viewed as a step towards a common generalization of several previously known geometric partitioning results regarding colored point sets.
We prove that if we are given a set of points {\mathcal{P}} and set of lines {\mathcal{L}} in {\mathbb{F}_{q}^{2}} such that {|\mathcal{P}||\mathcal{L}|\gtrsim q^{8/3}} , then the set of distinct distances between points from {\mathcal{P}} and lines from {\mathcal{L}} contains a positive proportion of all distances. Using the same techniques, we also obtain a generalization of this result in higher dimensional cases.
Let F p be a prime field of order p > 2, and A be a set in F p with very small size in terms of p. In this note, we show that the number of distinct cubic distances determined by points in A × A satisfieswhich improves a result due to Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new families of expanders in four and five variables. We also give an explicit exponent of a problem of Bukh and Tsimerman, namely, we prove thatthat is not of the form g(αx + βy) for some univariate polynomial g.
We prove bounds on intersections of algebraic varieties in C 4 with Cartesian products of finite sets from C 2 , and we point out connections with several classic theorems from combinatorial geometry. Consider an algebraic variety X ⊂ C 4 of degree d, such that not all polynomials that vanish on X are of the formwhere G, H, K, L are polynomials and G and K are not constant. Let P, Q ⊂ C 2 be finite sets of size n. If X has dimension one or two, then we prove |X ∩ (P × Q)| = O d (n), while if X has dimension three, then |X ∩ (P × Q)| = O d,ε (n 4/3+ε ) for any ε > 0. Both bounds are best possible in this generality (except for the ε).These bounds can be viewed as different generalizations of the Schwartz-Zippel lemma, where we replace a product of "one-dimensional" finite subsets of C by a product of "twodimensional" finite subsets of C 2 . The bound for three-dimensional varieties generalizes the Szemerédi-Trotter theorem. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz.As corollaries of our two bounds, we obtain bounds on the number of repeated and distinct values of polynomials and polynomial maps of pairs of points in C 2 , with a characterization of those maps for which no good bounds hold. These results generalize known bounds on repeated and distinct Euclidean distances.
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