Related papers: Solvability of Rado systems in D-sets
A system of linear forms $L=\{L_1,\ldots,L_m\}$ over $\mathbb{F}_q$ is said to be Sidorenko if the number of solutions to $L=0$ in any $A \subseteq \mathbb{F}_{q}^n$ is asymptotically as $n\to\infty$ at least the expected number of…
Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let the representation function $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s<s'$. In…
Sets of $d\times d$ matrices sharing a common invariant cone enjoy special properties, which are widely used in applications. However, finding this cone or even proving its existence/non-existence is hard. This problem is known to be…
New theorems about the existence of solution for a system of infinite linear equations with a Vandermonde type matrix of coefficients are proved. Some examples and applications of these results are shown. In particular, a kind of these…
In this work we want to prove the existence of solution for a class of fractional Hamiltonian systems given by {eqnarray*}_{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = & \nabla W(t,u(t)) u\in H^{\alpha}(\mathbb{R},…
We prove general topological Radon-type theorems for sets in $\mathbb R^d$ or on a surface. Combined with a recent result of Holmsen and Lee, we also obtain fractional Helly theorem, and consequently the existence of weak $\varepsilon$-nets…
We show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field. The argument is based on modular techniques combined with a…
This note was prepared as a handout for the MAT401 course ``Polynomial equations and fields", taught at the University of Toronto in Spring 2026. It presents a proof of a necessary condition for the solvability of algebraic equations by…
Solving a singular linear system for an individual vector solution is an ill-posed problem with a condition number infinity. From an alternative perspective, however, the general solution of a singular system is of a bounded sensitivity as…
A set $A\subseteq\mathbb N$ is called $complete$ if every sufficiently large integer can be written as the sum of distinct elements of $A$. In this paper we present a new method for proving the completeness of a set, improving results of…
The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the…
This paper illustrates the richness of the concept of regular sets of time bounds and demonstrates its application to problems of computational complexity. There is a universe of bounds whose regular subsets allow to represent several time…
In this work we prove the existence of solution for a class of perturbed fractional Hamiltonian systems given by \begin{eqnarray}\label{eq00} -{_{t}}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) - L(t)u(t) + \nabla W(t,u(t)) = f(t),…
We give a canonical partition of shifted intersecting set systems, from which one can obtain unified and elementary proofs of the Erd\H{o}s-Ko-Rado and Hilton-Milner Theorem, as well as a characterization of maximal shifted $k$-uniform…
We show that the sequence of ratios $d(n+1) / d(n)$ of consecutive values of the divisor function attains every positive rational infinitely many times. This confirms a prediction of Erd\H{o}s.
We investigate the problem of deciding whether a system of linear equations, together with divisibility conditions on the variables, has a solution over holomorphy subrings of global fields. We obtain decidability results when we allow…
There is a one-to-one correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky's enumeration of the cells of the…
Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to…
Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of…
We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of…