Mathematika最新文献

Let $$ and $$ be natural numbers greater or equal to 2. Let $$ be a homogeneous polynomial in $$ variables of degree $$ with integer coefficients $$, where $$ denotes the inner product, and $$ denotes the Veronese embedding with $$. Consider a variety $$ in $$, defined by $$. In this paper, we examine a set of integer vectors $$, defined by

We prove that for any $$, there exists a constant $$ such that the following is true. Let $$ be an infinite sequence of bipartite graphs such that $$ and $$ hold for all $$. Then, in any $$-edge-coloured complete graph $$, there is a collection of at most $$ monochromatic subgraphs, each of which is isomorphic to an element of $$, whose vertex sets partition $$. This proves a conjecture of Corsten and Mendonça in a strong form and generalises results on the multi-colour Ramsey numbers of bounded-degree bipartite graphs. It also settles the bipartite case of a general conjecture of Grinshpun and Sárközy.

In this article, we examine the Poissonian pair correlation (PPC) statistic for higher dimensional real sequences. Specifically, we demonstrate that for $$, almost all $$, the sequence $$ in $$ has PPC conditionally on the additive energy bound of $$. This bound is more relaxed compared to the additive energy bound for one dimension as discussed in [Aistleitner, El-Baz, and Munsch, Geom. Funct. Anal. 31 (2021), 483–512]. More generally, we derive the PPC for $$ for almost all $$. As a consequence we establish the metric PPC for $$ provided that all of the $$ are greater than two.

Given a graph on $$ vertices with $$ edges, each of unit resistance, how small can the average resistance between pairs of vertices be? There are two very plausible extremal constructions — graphs like a star, and graphs which are close to regular — with the transition between them occurring when the average degree is 3. However, in this paper, we show that there are significantly better constructions for a range of average degree including average degree near 3. A key idea is to link this question to a analogous question about rooted graphs — namely ‘which rooted graph minimises the average resistance to the root?’. The rooted case is much simpler to analyse that the unrooted, and the one of the main results of this paper is that the two cases are asymptotically equivalent.

In this paper, we study the integer solutions of a family of Fermat-type equations of signature $$, $$. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant $$ such that if $$, there are no solutions $$ of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory.

We extend classical estimates for the vector balancing constant of $$ equipped with the Euclidean and the maximum norms proved in the 1980s by showing that for $$ and $$, given vector families $$ with $$, one may select vectors $$ with

The conjectured squarefree density of an integral polynomial $$ in $$ variables is an Euler product $$ which can be considered as a product of local densities. We show that a necessary and sufficient condition for $$ to be 0 when $$ is a polynomial in $$ variables over the integers, is that either there is a prime $$ such that the values of $$ at all integer points are divisible by $$ or the polynomial is not squarefree as a polynomial. We also show that generally the upper squarefree density $$ satisfies $$.

Let $$ be integers. Using a fragmentation technique, we characterise $$-tuples $$ of non-empty families of partitions of $$ such that it suffices that an order-$$ tensor has bounded $$-rank for each $$ for it to have bounded $$-rank. On the way, we prove power lower bounds on suitable products of diagonal tensors, providing a qualitative answer to a question of Naslund.

Let $$ be a positive integer. Let $$ be a henselian local ring with residue field $$ of $$th level $$. We give some upper and lower bounds for the $$th Waring number $$ in terms of $$ and $$. In large number of cases, we are able to compute $$. Similar results for the $$th Waring number of the total ring of fractions of $$ are obtained. We then provide applications. In particular, we compute $$ and $$ for $$ and any prime $$.

We investigate polytopes inscribed into a sphere that are normally equivalent (or strongly isomorphic) to a given polytope $$. We show that the associated space of polytopes, called the inscribed cone of $$, is closed under Minkowski addition. Inscribed cones are interpreted as type cones of ideal hyperbolic polytopes and as deformation spaces of Delaunay subdivisions. In particular, testing if there is an inscribed polytope normally equivalent to $$ is polynomial time solvable. Normal equivalence is decided on the level of normal fans and we study the structure of inscribed cones for various classes of polytopes and fans, including simple, simplicial, and even. We classify (virtually) inscribable fans in dimension 2 as well as inscribable permutahedra and nestohedra. A second goal of the paper is to introduce inscribed virtual polytopes. Polytopes with a fixed normal fan $$ form a monoid with respect to Minkowski addition and the associated Grothendieck group is called the type space of $$. Elements of the type space correspond to formal Minkowski differences and are naturally equipped with vertices and hence with a notion of inscribability. We show that inscribed virtual polytopes form a subgroup, which can be nontrivial even if $$ does not have actual inscribed polytopes. We relate inscribed virtual polytopes to routed particle trajectories, that is, piecewise-linear trajectories of particles in a ball with restricted directions. The state spaces gives rise to connected groupoids generated by reflections, called reflection groupoids. The endomorphism groups of reflection groupoids can be thought of as discrete holonomy groups of the trajectories and we determine when they are reflection groups.