Let $q=p^k$, and let $A_1,\ldots,A_{r-1}$ be invertible $n\times n$ matrices over ${\mathbb F}_q$. We prove that, if $k\ge r$, there is a vector $x$ for which \[ x,A_1x,\ldots,A_{r-1}x \] are all nowhere zero. For $r=2$ this recovers the theorem of Alon and Tarsi over nonprime finite fields. The proof tracks one monomial in the product of the coordinate forms. Frobenius powers keep every exponent below $p^r$, and finite-field polynomial nonvanishing supplies the required vector. The same method treats rectangular matrices with independent rows and prescribed forbidden values. The method also gives an extension-degree criterion for representable matroids and clarifies an unpublished prime-field conjecture of M. J. Moghaddamzadeh. Projective-geometric examples explain why the analogous field-size statement fails over proper extensions and, translated back to matrices, give lower bounds for the large-field problem.
Let $X$ be a smooth complex affine variety of dimension $n$, and let $F=(f_1,\ldots,f_r)$ be a tuple of nonzero regular functions on $X$ such that $f:=\prod_{i=1}^r f_i$ is not invertible. We study the zero loci of the Bernstein-Sato ideals $B_F^{\mathbf a}$ for nonnegative integral shifts $\mathbf a$. For a fixed log...
Let $R$ be a commutative ring and $f(a_1, \ldots, a_n) = \sum_{i=1}^k r_{i1}^{a_1} \cdots r_{in}^{a_n} m_i$ be a linear-exponential map over an $R$-module $M$. Dong and Shafrir (2026) showed that, when $\ell M = 0$ for some $\ell \in \mathbb{N}_{>0}$, the zero set of $f$ is the intersection of effectively computable $p...
For an odd prime $p$ and integers $r,q\ge 1$, we study the Butson-type complex Hadamard matrices $H_{p;r,q}(a,b)=\chi(1+p^r ab)$ on $\mathbb{Z}/p^q\mathbb{Z}$, where $\chi$ is a faithful character of the principal-unit quotient. The defining phase is the normalized $p$-adic logarithm, linear modulo $p^q$ exactly when $...
Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb{F}_p)$ be the Heisenberg group of order $p^3$ and exponent $p$. We prove $\mathsf{d}(H_{p^3})=3p-3$. The main ingredient of the proof is an order-value growth theorem. If $B$ is a noncollinear zero-sum sequence of $n$ nonzero vectors in $\mathbb{F}_p^...
We prove an incidence bound for bipartite graphs on finite subsets of $\mathbb{F}^2\times \mathbb{F}^2$ defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is $K_{k,k}$-free and its vertex classes have sizes $m$ and $n$, then it has $O_{t,k}((mn)^{2/3}+m+n+mn/p)$ edges, where $t$...
Let $q=p^h>7$ be odd, put $m=(q+1)/2$, and suppose that $i=(m-2)/2$ satisfies $\gcd(i,m)=\gcd(i+2,m)=1$. For the $\mathbf{F}_{q^2}$-maximal function field $\mathcal{F}_i=\mathbf{F}_{q^2}(x,y)$ defined by $y^m=x^i(x^2+1)$, Peter Beelen, Maria Montanucci, Jonathan Niemann, and Luciane Quoos showed that the geometric auto...
Xu Zhuang· 0 citations
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