Canonical Forms of $2 \times 2 \times 2$ and $2 \times 2 \times 2 \times 2$ Symmetric Tensors over Prime Fields

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   Ian Seong · 2024 · Combinatorics · AI summary available

   Abstract

   Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $n,k$ denote integers with $n>2k\geq 6$. Let $V$ denote a vector space over $\mathbb{F}_{q}$ that has dimension $n$. The vertex set of the Grassmann graph $J_q(n,k)$ consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are adjacent whenever their intersection has dimension $k-1$. Let $\partial$ denote the path-length distance function of $J_q(n,k)$. Pick vertices $x,y$ of $J_q(n,k)$ such that $1<\partial(x,y)<k$. Let $\text{Stab}(x,y)$ denote the subgroup of $GL(V)$ that stabilizes both $x$ and $y$. In this paper, we investigate the orbits of $\text{Stab}(x,y)$ acting on the local graph $\Gamma(x)$. We sh…

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   Hery Randriamaro · 2023 · Combinatorics · AI summary available

   Abstract

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   Abstract

   In this paper, we present a generalization of the Askey-Wilson relations that involves a projective geometry. A projective geometry is defined as follows. Let $h>k\geq 1$ denote integers. Let $\mathbb{F}_{q}$ denote a finite field with $q$ elements. Let $\mathcal{V}$ denote an $(h+k)$-dimensional vector space over $\mathbb{F}_{q}$. Let the set $P$ consist of the subspaces of $\mathcal{V}$. The set $P$, together with the inclusion partial order, is a poset called a projective geometry. We define a matrix $A\in \text{Mat}_{P}(\mathbb{C})$ as follows. For $u,v\in P$, the $(u,v)$-entry of $A$ is $1$ if each of $u,v$ covers $u\cap v$, and $0$ otherwise. Fix $y\in P$ with $\dim y=k$. We define a d…

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   Yu-Yen Chien · 2017 · Combinatorics · Not summarized yet

   Abstract

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   Matthew Kwan, Lisa Sauermann · 2019 · Combinatorics · AI summary available

   Abstract

   Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of $f$ can be as large as about $1/\sqrt{n}$, but still poorly understood is the "inverse" question of characterising the algebraic and arithmetic features $f$ must have if it has point probabilities comparable to this bound. In this paper we prove some results of an algebraic flavour, showing that if $f$ has point probabilities much larger…

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   Proof of the Kakeya set conjecture over rings of integers modulo square-free $N$

   Manik Dhar, Zeev Dvir · 2020 · Combinatorics · Not summarized yet

   Abstract

   A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a set containing a line in each direction. We show that, when $N$ is any square-free integer, the size of the smallest Kakeya set in $(\mathbb{Z}/N\mathbb{Z})^n$ is at least $C_{n,\epsilon} N^{n - \epsilon}$ for any $\epsilon$ -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime $N$. We also show that the case of general $N$ can be reduced to lower bounding the $\mathbb{F}_p$ rank of the incidence matrix of points and hyperplanes over $(\mathbb{Z}/p^k\mathbb{Z})^n$.

7. Related Not relevant

   Joint equidistributions of mesh patterns 123 and 321 with symmetric and antipodal shadings

   Shuzhen Lv, Philip B. Zhang · 2024 · Combinatorics · Not summarized yet

   Abstract

   It is well known that the number of 123-avoiding and 321-avoiding permutations is the same, and these numbers correspond to the Catalan numbers. However, patterns 123 and 321 are not equidistributed. In the context of mesh patterns, patterns formed by the permutations 123 and 321 with identical shadings are sometimes jointly equidistributed. In this paper, we prove 20 joint equidistributions of mesh patterns 123 and 321 with symmetric shadings, and 36 joint equidistributions of the same patterns with antipodal shadings.

8. Related Not relevant

   A characterization of a class of maximum nonlinear functions

   Doreen Hertel, Alexander Pott · 2005 · Combinatorics · AI summary available

   Abstract

   Maximum nonlinear functions on finite fields are widely used in cryptography because the coordinate functions have large distance to linear functions. More precisely, the Hamming distance to the characteristic functions of hyperplanes is large. One class of maximum nonlinear functions are the Gold power functions We characterize these functions in terms of the distance of their coordinate functions to characteristic functions of subspaces of codimension 2.