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Few distinct distances implies no heavy lines or circles

Publications associées (45)

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Algebraic and topological methods in combinatorics

Adrian Claudiu Valculescu

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 o ...

EPFL2017

Frobenius techniques in birational geometry

Zsolt Patakfalvi

This is a survey for the 2015 AMS Summer Institute on Algebraic Geometry about the Frobenius type techniques recently used extensively in positive characteristic algebraic geometry. We first explain the basic ideas through simple versions of the fundamenta ...

2017

From Tarski's Plank Problem to Simultaneous Approximation

János Pach, Andrei Kupavskii

A slab (or plank) is the part of the d-dimensional Euclidean space that lies between two parallel hyperplanes. The distance between the these hyperplanes is called the width of the slab. It is conjectured that the members of any infinite family of slabs wi ...

Mathematical Assoc Amer2017

Erdos Distinct Distances Problem and Extensions over Finite Spaces

Van Thang Pham

In this thesis we study a number of problems in Discrete Combinatorial Geometry in finite spaces. The contents in this thesis are structured as follows: In Chapter 1 we will state the main results and the notations which will be used throughout the thesis. ...

EPFL2017

On the existence of ordinary triangles

Sebastian Urban Stich, Radoslav Fulek, Márton Naszódi

Let P be a finite point set in the plane. A \emph{c-ordinary triangle} in P is a subset of P consisting of three non-collinear points such that each of the three lines determined by the three points contains at most c points of P. Motivated by a question o ...

Elsevier2017

Distinct Values Of Bilinear Forms On Algebraic Curves

Frank de Zeeuw, Adrian Claudiu Valculescu

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/ ...

Univ Calgary, Dept Math & Statistics2016

Fibre Products of Supersingular Curves and the Enumeration of Irreducible Polynomials with Prescribed Coefficients

Robert Granger

For any positive integers

n\geq 3, r\geq 1

we present formulae for the number of irreducible polynomials of degree

n

over the finite field

\mathbb{F}_{2^r}

where the coefficients of

x^{n-1}

x^{n-2}

and

x^{n-3}

are zero. Our proofs involve coun ...

Elsevier2016

An Inner-Product Calculus for Periodic Functions and Curves

Michaël Unser, Daniel Andreas Schmitter, Anaïs Laure Marie-Thérèse Badoual

Our motivation is the design of efficient algorithms to process closed curves represented by basis functions or wavelets. To that end, we introduce an inner-product calculus to evaluate correlations and

L _{ 2 }

distances between such curves. In partic ...

2016

Computational Aspects of Jacobians of Hyperelliptic Curves

Alina Dudeanu

Nowadays, one area of research in cryptanalysis is solving the Discrete Logarithm Problem (DLP) in finite groups whose group representation is not yet exploited. For such groups, the best one can do is using a generic method to attack the DLP, the fastest ...

EPFL2016

On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves

János Pach, Gábor Tardos

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least ...

Cambridge Univ Press2016