Let B-M : C x C -> C be a bilinear form B-M(p, q) - p(T)Mq, with an invertible matrix M is an element of C-2x2. We prove that any finite set S contained in an irreducible algebraic curve C of degree d in C determines Omega(d)(vertical bar S vertical bar(4/3)) distinct values of B-M, unless C has an exceptional form. This strengthens a result of Charalambides [1] in several ways. The proof is based on that of Pach and De Zeeuw [9], who proved a similar statement for the Euclidean distance function in R. Our main motivation for this paper is that for bilinear forms, this approach becomes more natural, and should better lend itself to understanding and generalization.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For ex
The present thesis deals with problems arising from discrete mathematics, whose proofs make use of tools from algebraic geometry and topology. The thesis is based on four papers that I have co-authored, three of which have been published in journals, and one has been submitted for publication (and also appeared as a preprint on the arxiv, and as an extendend abstract in a conference). Specifically, we deal with the following four problems: \begin{enumerate} \item We prove that if M∈C2×2 is an invertible matrix, and BM:C2×C2→C is a bilinear form BM(p,q)=pTMq, then any finite set S contained in an irreducible algebraic curve C of degree d in C2 determines Ωd(∣S∣4/3) distinct values of BM, unless C is a line, or is linearly equivalent to a curve defined by an equation of the form xk=yl, with k,l∈Z\0, and gcd(k,l)=1. \item We show that if we are given m points and n lines in the plane, then the number of distinct distances between the points and the lines is Ω(m1/5n3/5), as long as m1/2≤n≤m2. Also, we show that if we are given m points in the plane, not all collinear, then the number of distances between these points and the lines that they determine is Ω(m4/3). We also study three-dimensional versions of the distinct point-line distances problem. \item We prove the lower bound Ω(∣S∣4) on the number of ordinary conics determined by a finite point set S in R2, assuming that S is not contained in a conic, and at most c∣S∣ points of S lie on the same line (for some $0
In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer n≥3, we define e(n) to be the minimum positive integer such that any set of e(n) points in general position in the plane contains n points in convex position. In 1935, Erd\H{o}s and Szekeres proved that e(n)≤(n−22n−4)+1 and later in 1961, they obtained the lower bound 2n−2+1≤e(n), which they conjectured to be optimal. We prove that e(n)≤(n−22n−5)−(n−32n−8)+2. In a recent breakthrough, Suk proved that e(n)≤2n+O(n2/3logn). We strengthen this result by extending it to pseudo-configurations and also improving the error term. Combining our results with a theorem of Dobbins et al., we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes T'{o}th and by Pach and T'{o}th, respectively. Let c(n) (and c′(n)) denote the smallest positive integer N such that any family of N pairwise disjoint convex bodies in general position (resp., N convex bodies in general position, any pair of which share at most two boundary points) has an n members in convex position. We show that c(n)≤c′(n)≤2n+O(nlogn). Given a point set P in the plane, an ordinary circle for P is defined as a circle containing exactly three points of P. We prove that any set of n points in the plane, not all on a line or a circle, determines at least 41n2−O(n) ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large n, and characterize all point sets that come close to this minimum. We also consider the orchard problem for circles, where we determine the maximum number of circles containing four points of a given set and describe the extremal configurations. A special case of the Schwartz-Zippel lemma states that given an algebraic curve C⊂C2 of degree d and two finite sets A,B⊂C, we have ∣C∩(A×B)∣=Od(∣A∣+∣B∣). We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection ∣X∩(P×Q)∣ for a variety X⊂C4 and finite sets P,Q⊂C2. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries, we generalize the Szemer'edi-Trotter point-line incidence theorem and several known bounds on repeated and distinct Euclidean distances. We use incidence geometry to prove some sum-product bounds over arbitrary fields. First, we give an explicit exponent and improve a recent result of Bukh and Tsimerman by proving that max{∣A+A∣,∣f(A,A)∣}≫∣A∣6/5 for any small set A⊂Fp and quadratic non-degenerate polynomial f(x,y)∈Fp[x,y]. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term max{∣A+A∣,∣A⋅A∣}. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on max{∣A+B∣,∣f(B,C)∣}, for a specific type of function f. Finally, we prove that the number of distinct cubic distances generated by any small set A×A⊂Fp2 is Ω(∣A∣8/7), which improves a result of Yazici et al.
