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Gábor Hegedüs

Publications and source records attributed to Gábor Hegedüs.

At least 37 records · Page 2Linked to original sources
arXiv2020

An upper bound for the size of $s$-distance sets in real algebraic sets

In a recent paper Petrov and Pohoata developed a new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics and Sylvester's Law of Inertia for real quadratic forms. As an application, they gave a simple proof of the Bannai-Bannai-Stanton bound on the size of $s$-distance sets (subsets $\mbox{$\cal A$}\subseteq {\mathbb R}^n$ which determine at most $s$ different distances). In this paper we extend their work and prove upper bounds for the size of $s$-distance sets in various real algebraic sets. This way we obtain a novel and short proof for the bound of Delsarte-Goethals-Seidel on spherical $s$-distance sets and a generalization of a bound by Bannai-Kawasaki-Nitamizu-Sato on $s$-distance sets on unions of spheres. In our arguments we use the method of Petrov and Pohoata together with some Gröbner basis techniques.

math.CO↗
arXiv2020

A new proof of a generalization of Gerzon's bound

In this paper we give a short, new proof of a natural generalization of Gerzon's bound. This bound improves the Delsarte, Goethals and Seidel's upper bound in a special case. Our proof is a simple application of the linear algebra bound method.

math.CO↗
arXiv2018

A new upper bound for the size of $s$-distance sets in boxes

Let $q,d\geq 2$ be integers. Define $$ J(q,d):=\frac 1q \Big( \min_{0<x<1} \frac{1-x^q}{1-x} x^{-\frac{q-1}{d}}\Big). $$ Let $\mbox{$\cal G$}\subseteq {\mathbb R}^n$ be an arbitrary subset. We denote by $d(\mbox{$\cal G$})$ the set of (non-zero) distances among points of $\mbox{$\cal G$}$: $$ d(\mbox{$\cal G$}):=\{d( p_1, p_2):~ p_1, p_2\in \mbox{$\cal G$}, p_1\ne p_2\}. $$ Our main result is a new upper bound for the size of $s$-distance sets in boxes. More concretely, let $A_i\subseteq \mathbb R$, $|A_i|=q\geq 2$ be subsets for each $1\leq i\leq n$. Consider the box $\mbox{$\cal B$}:=\prod_{i=1}^n A_i\subseteq {\mathbb R}^n$. Suppose that $\mbox{$\cal G$}\subseteq \mbox{$\cal B$}$ is a set such that $|d(\mbox{$\cal G$})|\leq s$. Let $d:=\frac{n(q-1)}{s}$. Then $$|\mbox{$\cal G$}|\leq 2(qJ(q,d))^n.$$ We use Tao's slice rank bounding method in our proof.

math.CO↗
arXiv2017

A new exponential upper bound for the Erdős-Ginzburg-Ziv constant

Naslund used Tao's slice rank bounding method to give new exponential upper bounds for the Erdős--Ginzburg-Ziv constant of finite Abelian groups of high rank. In our short manuscript we improve slightly Naslund's upper bounds. We extend Naslund's results and prove new exponential upper bounds for the Erdős--Ginzburg-Ziv constant of arbitrary finite Abelian groups. Our main results depend on a conjecture about Property D.

math.NT↗
arXiv2017

A new upper bound for the size of a sunflower-free family

We combine here Tao's slice-rank bounding method and Gröbner basis techniques and apply here to the Erdős-Rado Sunflower Conjecture. Let $\frac{3k}{2}\leq n\leq 3k$ be integers. We prove that if $\mbox{$\cal F$}$ be a $k$-uniform family of subsets of $[n]$ without a sunflower with 3 petals, then $$ |\mbox{$\cal F$}|\leq 3{n \choose n/3}. $$ We give also some new upper bounds for the size of a sunflower-free family in $2^{[n]}$.

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arXiv2017

A note on linear Sperner families

In an earlier work we described Gröbner bases of the ideal of polynomials over a field, which vanish on the set of characteristic vectors $\mathbf{v} \in \{0,1\}^n$ of the complete $d$ unifom set family over the ground set $[n]$. In particular, it turns out that the standard monomials of the above ideal are {\em ballot monomials}. We give here a partial extension of the latter fact. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials. A set family is a linear Sperner system if the characteristic vectors satisfy a linear equation $a_1v_1+\cdots +a_nv_n=k$, where $0<a_q\leq a_2\leq \cdots \leq a_n$ and $k$ are integers. As an application, we confirm a conjecture of Frankl for linear Sperner systems.

math.CO↗
arXiv2015

A generalization of the Erdős-Ko-Rado Theorem

Our main result is a new upper bound for the size of k-uniform, L-intersecting families of sets, where L contains only positive integers. We characterize extremal families in this setting. Our proof is based on the Ray-Chaudhuri--Wilson Theorem. As an application, we give a new proof for the Erdős-Ko-Rado Theorem, improve Fisher's inequality in the uniform case and give an uniform version of the Frankl-Füredi conjecture .

math.CO↗
arXiv2015

Shattering bounds for tuple systems

Let $\mbox{ V}(n,d,q)$ stand for the $q$--ary Hamming spheres. Let $\mbox{ C}\subseteq (q)^n$ denote a tuple system such that $\mbox{ C}\subseteq \cup_{i=0}^s \mbox{ V}(n,d_i,q)$, where $d_1<\ldots <d_s$. We give here a general upper bound on the size of a shattered sets of the tuple system $\mbox{ C}$.

math.CO↗
arXiv2015

A Bollobás-type theorem for affine subspaces

Let $W$ denote the $n$-dimensional affine space over the finite field $\mathbb F_q$. We prove here a Bollobás-type upper bound in the case of the set of affine subspaces. We give a construction of a pair of families of affine subspaces, which shows that our result is almost sharp.

math.CO↗
arXiv2015

From the Fundamental Theorem of Algebra to Kempe's Universality Theorem

This article provides a gentle introduction for a general mathematical audience to the factorization theory of motion polynomials and its application in mechanism science. This theory connects in a rather unexpected way a seemingly abstract mathematical topic, the non-unique factorization of certain polynomials over the ring of dual quaternions, with engineering applications. Four years after its introduction, it is already clear how beneficial it has been to both fields.

math.RA↗
arXiv2015

Ehrhart polynomial roots of reflexive polytopes

Recent work has focused on the roots z of the Ehrhart polynomial of a lattice polytope P. The case when Re(z) = -1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when dim(P) <= 7. We also consider the "half-strip condition", where all roots z satisfy -dim(P)/2 <= Re(z) <= dim(P)/2-1, and show that this holds for any reflexive polytope with dim(P) <= 5. We give an example of a 10-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi--Shibata in dimension 34.

math.CO↗
arXiv2015

An explicit Ramsey graph

Explicit construction of Ramsey graphs has remained a challenging open problem for a long time. Frankl--Wilson \cite{FW}, Alon \cite{A} and Grolmusz \cite{G2} gave the best explicit constructions of graphs on $m$ vertices with no clique or independent set of size $m^{(1+o(1))\frac{1}{4}\frac{\log m}{\log \log m}}$. We describe here an explicit construction which produces for every integer $m>1$ a graph on at least $m^{(1+o(1))\frac{1}{3}\frac{\log m}{\log \log m}}$ vertices containing neither a clique of size $m$ nor an independent set of size $m$. In the proof we use the polynomial subspace method and some character theory of the complete symmmetric group.

math.CO↗
arXiv2014

Balancing Sets of Vectors

Let $n$ be an arbitrary integer, let $p$ be a prime factor of $n$. Denote by $ω_1$ the $p^{th}$ primitive unity root, $ω_1:=e^{\frac{2πi}{p}}$. Define $ω_i:=ω_1^i$ for $0\leq i\leq p-1$ and $B:=\{1,ω_1,...,ω_{p-1}\}^n$. Denote by $K(n,p)$ the minimum $k$ for which there exist vectors $v_1,...,v_k\in B$ such that for any vector $w\in B$, there is an $i$, $1\leq i\leq k$, such that $v_i\cdot w=0$, where $v\cdot w$ is the usual scalar product of $v$ and $w$. Gröbner basis methods and linear algebra proof gives the lower bound $K(n,p)\geq n(p-1)$. Galvin posed the following problem: Let $m=m(n)$ denote the minimal integer such that there exists subsets $A_1,...,A_m$ of $\{1,...,4n\}$ with $|A_i|=2n$ for each $1\leq i\leq n$, such that for any subset $B\subseteq [4n]$ with $2n$ elements there is at least one $i$, $1\leq i\leq m$, with $A_i\cap B$ having $n$ elements. We obtain here the result $m(p)\geq p$ in the case of $p>3$ primes.

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arXiv2013

The Theory of Bonds II: Closed 6R Linkages with Maximal Genus

In this paper, we introduce a method that allows to produce necessary conditions on the Denavit--Hartenberg parameters for the mobility of a closed linkage with six rotational joints. We use it to prove that the genus of the configuration curve of a such a linkage is at most five, and to give a complete classification of the linkages with a configuration curve of degree four or five. The classification contains new families.

math.AG↗
arXiv2013

Four-Pose Synthesis of Angle-Symmetric 6R Linkages

We use the recently introduced factorization theory of motion polynomials over the dual quaternions for the synthesis of closed kinematic loops with six revolute joints that visit four prescribed poses. Our approach admits either no or a one-parametric family of solutions. We suggest strategies for picking good solutions from this family.

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arXiv2013

The Theory of Bonds: A New Method for the Analysis of Linkages

In this paper we introduce a new technique, based on dual quaternions, for the analysis of closed linkages with revolute joints: the theory of bonds. The bond structure comprises a lot of information on closed revolute chains with a one-parametric mobility. We demonstrate the usefulness of bond theory by giving a new and transparent proof for the well-known classification of overconstrained 5R linkages.

math.AG↗
arXiv2012

Factorization of Rational Curves in the Study Quadric and Revolute Linkages

Given a generic rational curve $C$ in the group of Euclidean displacements we construct a linkage such that the constrained motion of one of the links is exactly $C$. Our construction is based on the factorization of polynomials over dual quaternions. Low degree examples include the Bennett mechanisms and contain new types of overconstrained 6R-chains as sub-mechanisms.

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