Related papers: Another method to solve the grasshopper problem (t…

In this expository note we show how combinatorial Nullstellensatz by N. Alon naturally appears in solutions of elementary problems. Simple ideas gradually and naturally appear in such solutions, thus bringing a reader to generalizations.…

A newly-generalized problem from a problem initially thought for the Mathematical Olympiad and the methods to solve it.

We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime…

The aim of this essay is to better understand the Grasshopper Problem on the surface of the unit sphere. The problem is motivated by analysing Bell inequalities, but can be formulated as a geometric puzzle as follows. Given a white sphere…

This paper is a supplement to a talk for mathematics teachers given at the 2016 LSU Mathematics Contest for High School Students. The paper covers more details and aspects than could be covered in the talk. We start with an interesting…

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of…

Efficient algorithms for many problems in optimization and computational algebra often arise from casting them as systems of polynomial equations. Blum, Shub, and Smale formalized this as Hilbert's Nullstellensatz Problem $HN_R$: given…

We introduce and physically motivate the following problem in geometric combinatorics, originally inspired by analysing Bell inequalities. A grasshopper lands at a random point on a planar lawn of area one. It then jumps once, a fixed…

In this work, we establish near-linear and strong convergence for a natural first-order iterative algorithm that simulates Von Neumann's Alternating Projections method in zero-sum games. First, we provide a precise analysis of Optimistic…

Complementarity problems often permit distinct solutions, a fact of major significance in optimization, game theory and other fields. In this paper, we develop a numerical technique for computing multiple isolated solutions of…

This article introduces a pedagogical method for {\it solving combinatorial problems} that frequently involve structures that are unfamiliar or less familiar. Indeed, an indirect method has been proposed in order to evade any possible…

These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one…

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which…

Mathematical problems and their solutions of the Fourth International Students' Olympiad in cryptography NSUCRYPTO'2017 are presented. We consider problems related to attacks on ciphers and hash functions, cryptographic Boolean functions,…

This paper studies the underlying combinatorial structure of a class of object rearrangement problems, which appear frequently in applications. The problems involve multiple, similar-geometry objects placed on a flat, horizontal surface,…

We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method…

Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3-colorable, hamiltonian, etc.) if and only if a related system…

Real Nullstellensatz is a classical result from Real Algebraic Geometry. It has recently been extended to quaternionic polynomials by Alon and Paran. The aim of this paper is to extend their Quaternionic Nullstellensatz to matrix…

The planar grasshopper problem, originally introduced in (Goulko & Kent 2017 Proc. R. Soc. A 473, 20170494), is a striking example of a model with long-range isotropic interactions whose ground states break rotational symmetry. In this work…

We introduce a new form of the polynomial method based on what we call "shift operators," which we use to give efficient and intuitive new proofs of results previously shown using a wide range of polynomial methods, including Alon's…