Distance Configurations of Points in a Plane with a Galois group that is not Soluble

On the directions determined by a Cartesian product in an affine Galois plane

We prove that the number of directions contained in a set of the form $A \times B \subset AG(2,p)$, where $p$ is prime, is at least $|A||B| - \min\{|A|,|B|\} + 2$. Here $A$ and $B$ are subsets of $GF(p)$ each with at least two elements and $|A||B| <p$. This bound is tight for an infinite class of examples. Our main tool is the use of the Rédei polynomial with Szőnyi's extension. As an application of our main result, we obtain an upper bound on the clique number of a Paley graph, matching the current best bound obtained recently by Hanson and Petridis.

Galois射影平面、仿射平面和欧几里得网格的弧色数

The arc chromatic number for Galois projective planes, affine planes and Euclidean grids

本文研究了Galois射影平面、仿射平面和欧几里得网格的弧色数问题，确定了不同结构下的最小弧划分数及其边界条件。

关于类型1极小刚图的伽罗瓦群

On Galois groups of type-1 minimally rigid graphs

本文研究可由单边经反复Henneberg 1步构造的平面极小刚图的伽罗瓦群，证明这类群均为完全非本原群，由其所保持的划分完全确定。

Galois Groups of Composed Schubert Problems

Two Schubert problems on possibly different Grassmannians may be composed to obtain a Schubert problem on a larger Grassmannian whose number of solutions is the product of the numbers of the original problems. This generalizes a construction discovered while classifying Schubert problems on the Grassmannian of 4-planes in C^9 with imprimitive Galois groups. We give an algebraic proof of the product formula. In a number of cases, we show that the Galois group of the composed Schubert problem is a subgroup of a wreath product of the Galois groups of the original problems, and is therefore imprim…

Walks in the quarter plane, genus zero case

We use Galois theory of difference equations to study the nature of the generating series of (weighted) walks in the quarter plane with genus zero kernel curve. Using this approach, we prove that the generating series do not satisfy any nontrivial (possibly nonlinear) algebraic differential equation with rational coefficients.

On the nature of the generating series of walks in the quarter plane

In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many of the recent results about these series, but also to go beyond them. For instance, we give for the first time hypertranscendency results, {\it i.e.}, we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational coefficients.

大步长象限游走轨道上的伽罗瓦结构

A Galois structure on the orbit of large steps walks in the quadrant

本文为四分之一平面大歩长游走的轨道赋予伽罗瓦群作用，推广了游走群概念，并据此发展伽罗瓦方法证明不变量与解耦的存在性，首次证明大步长加权模型的代数性，验证BBMM猜想并发现新代数模型。

Algebraic approach to the completeness problem for $(k,n)$-arcs in planes over finite fields

In a projective plane over a finite field, complete $(k,n)$-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with $k$ close to the order of the plane, is considered a hard problem. A natural candidate to be a small $(k,n)$-arc with few characters is the set $Ω(\mathcal{C})$ of the points of a plane curve $\mathcal{C}$ of degree $n$ (containing no linear components) such that some line meets $\mathcal{C}$ transversally in the plane, i.e. in $n$ pair…