We construct an explicit infinite family of simple two-dimensional Cayley complexes over $\mathbb{F}_2^n$ whose degree is polynomial in $n$ and whose nontrivial vertex-link eigenvalues lie in $[-\lambda,\lambda]$ for every fixed $\lambda>0$. For every fixed $d\ge2$, we also obtain an explicit infinite family of weighted $d$-dimensional Cayley complexes over $\mathbb{F}_2^n$ with codimension-two local spectral norm at most $1/d$ and Cayley degree $\Theta_d(n)$. Our two-dimensional construction uses evaluation at rational points of algebraic curves to produce projective direction sets and many functions affine along these directions, which may be useful for further constructions and improvements.
Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the class...
For every fixed integer $d\geq2$, we prove that the finite graphs receiving a continuous homomorphism from the standard Schreier graph $F(2^{\mathbb Z^d})$ form a $\Sigma^0_1$-complete set. This extends a theorem of Gao, Jackson, Krohne and Seward when $d=2$. For the positive part of the reduction, we show that if a gr...
In this work, we show that if $h(x)$ is an irreducible monic integer polynomial of degree $2q$ (with $q$ prime), whose defining field extension of $\mathbb{Q}$ contains a Galois extension of degree $q$, then there is a positive density of integers $n$ such that $h(n)$ is squarefree; in particular, $h(n)$ is squarefree...
We study the dimension-indexed marked-root polynomial maps introduced by Harish. For every $n\geq 4$, their lower coefficient block extends to a polynomial coordinate system, giving three-dimensional Keller maps of geometric degree $n(n-2)$ for every choice of the fixed coefficients. We determine the generic monodromy...
Let $L$ be a finite-dimensional Lie algebra over a field $F$. The comaximal graph $\Gamma(L)$ has as vertices the proper nonzero subalgebras of $L$, two of them adjacent whenever they generate $L$; its structure was previously classified for Lie algebras of dimension at most 3 over finite fields. Here we extend that wo...
David A. Towers, Y. Zuleta, Ismael Gutierrez· 0 citations
We study constructive Kempe factorizations for rational curves on $\operatorname{SO}_4(\mathbb{R})$. Motivated by motion-polynomial factorization and rational matrix curves on real classical groups, we prove that every rational curve of degree $2d$ with $d\ge1$ first factors into $d$ quadratic rational curves and then...
Yi-Fan Li, Zijia Li, Ke Ye· 0 citations
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