Let $q=p^k$, where $p$ is prime, and let $M$ be a finite matroid representable over ${\Bbb{F}}_q$. Write $\chi_M(t)$ for its characteristic polynomial and $\mbox{decop}(M)$ for the least number of independent sets needed to cover its ground set. We prove that $\chi_M(q)>0$ whenever $k\ge\mbox{decop}(M)$. Geometrically, the central hyperplanes determined by any representation of $M$ fail to cover the dual of the ambient vector space. The proof rests on the Frobenius-power ideals $(X_1^{p^s},\ldots,X_n^{p^s})$, $s\ge1$. Each is preserved by every linear change of coordinates, while nonmembership records the existence of a monomial whose exponent in every variable is bounded. This permits successive normalizations of several invertible systems of linear forms without losing the exponent bounds already obtained. The coefficient form of the Combinatorial Nullstellensatz then produces a common nowhere-zero point. Finally, we test the scope of the theorem. M.~J.~Moghaddamzadeh's unpublished conjecture predicts a stronger statement over prime fields. Projective geometries show that its direct analogue fails over proper extension fields, even under the same numerical inequality.
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 $p$-normal sets, where $p$ ranges over the prime divisors of $\ell$. This generalizes an earlier theorem of Derksen and Masser (2012) on the solution set of $S$-unit equations over fields of positive characteristic. The purpose of this paper is twofold. First, we give a shorter proof of Dong and Shafrir's result, using the theorem of Derksen-Masser as a blackbox. Our proof also yields a decomposition of the zero set as a positive Boolean combination of affine transformations of zero sets of linear-exponential equations over fields. Second, we prove a multi-dimensional generalization of the Skolem-Mahler-Lech theorem over rings of finite characteristic. Specifically, we show that the zero set of every $n$-dimensional linear recurrence sequence over an $R$-module $M$ satisfying $\ell M = 0$ is the intersection of effectively computable $p$-normal sets (in $\mathbb{N}^n$), where $p$ ranges over the prime divisors of $\ell$. For example, this gives a decision procedure for whether two classical linear recurrence sequences have a common value over a ring of characteristic $p^a$ or $p^a q^b$, where $p$ and $q$ are primes.
For $n\geq 4$, let $T_n$ be the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets. Let $X_1(n)^\circ$ denote the open subscheme of the modular curve $X_1(n)$ obtained by removing the cusps corresponding to reducible N\'eron polygons. For $n \geq 10$, we give a purely algebraic and incidence-theoretic proof that, for every field $k$ with $\mathrm{char}(k)$ not dividing $n$, there is a natural bijection between $X_1(n)^\circ(k)$ and rescaling classes of $k$-realizations of $T_n$. For $k = \mathbb{C}$, this recovers a theorem of Borisov and Roulleau. We then upgrade the field-valued correspondence to an isomorphism of schemes over $\mathbb{Z}[1/n]$. The main new ingredient is a deformation-theoretic argument which allows us to verify the isomorphism on points valued in Artinian local rings. As consequences, the modular curve $X_1(n)^\circ$ acquires a natural model over $\mathbb{Z}[1/n]$ as a matroid realization space, and, for primes $p \geq 11$, the non-representability of $T_p$ over $\mathbb{Q}$ is equivalent to the prime-order case of Mazur's celebrated theorem on rational torsion points of elliptic curves. In an appendix, we explain how to upgrade the realization space of a matroid from an affine scheme over $\mathbb{Z}$ to an affine band scheme (in the sense of Baker-Jin-Lorscheid) over $\mathbb{F}_1^{\pm}$.
Let $E$ be a finite set, and let $V \subseteq \mathbb{C}^E$ be a linear subspace that is not contained in any coordinate hyperplane. The closure of $V$ in the product of projective lines $(\mathbb{P}^1)^E$ is a singular variety $Y_V$ known as the matroid Schubert variety (or arrangement Schubert variety). We use operational Chow cohomology to prove that every line bundle on $Y_V$ is the restriction of a line bundle on $(\mathbb{P}^1)^E.$ We then give combinatorial characterizations of when $Y_V$ is Gorenstein and $\mathbb{Q}$-Gorenstein, respectively. We provide examples of linear subspaces $V$ such that $Y_V$ is Gorenstein but not smooth, $\mathbb{Q}$-Gorenstein but not Gorenstein, and not $\mathbb{Q}$-Gorenstein, respectively.
Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F_q$, and let $\mathcal F\subsetneq\genfrac{[}{]}{0pt}{}{V}{k}$. The \emph{projective Ore-degree} of $\mathcal F$ is the minimum, over all $k$-subspaces $S\notin\mathcal F$, of the sum of the $\mathcal F$-degrees of the projective points contained in $S$. We prove sharp projective Ore analogues of the vector-space Erd\H{o}s--Ko--Rado and Hilton--Milner theorems. The Ore--Erd\H{o}s--Ko--Rado theorem holds for $n\ge2k+1$, with equality only for a full point-star. For nontrivial intersecting families, we determine the sharp Ore--Hilton--Milner threshold, together with the complete equality classification, when $q\ge3$ and $n\ge2k+1$, or when $q\ge2$ and $n\ge2k+2$. We further determine a sharp projective Ore-degree threshold forcing a direct-sum matching of size $s$ when $s\ge3$ and $n\ge(2s-1)k-s+4$, and derive a multicolour Ramsey consequence.
Mengyue Cao, Mei Lu, Xuyang Yan et al.· 0 citations
Let $c\phi_k(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. We prove that, for every $m\geq2$ and every $k\equiv2\pmod{2^m}$, \[ \sum_{n\geq0}c\phi_k(n)q^n\equiv\frac{\varphi(q)\,(q^2;q^2)_\infty}{(q;q)_\infty^2}\sum_{n\geq0}c\phi_{k/2}(n)q^{2n}\pmod{2^m}, \] where $\varphi(q)$ is the classical theta function. For $m=2$ this recovers a congruence of Chan, Wang, and Yang. Applied with $k=18$, we determine $c\phi_{18}(2n+1)$ modulo $16$ completely. In particular, we also prove \[ \sum_{n\geq0}c\phi_{18}(6n+1)q^n \equiv4\sum_{r\in\mathbb{Z}}q^{r(3r-1)/2}\pmod{16}, \] which proves the congruences $c\phi_{18}(30n+19)\equiv c\phi_{18}(30n+25)\equiv0\pmod{16}$ recently conjectured by Das, Nath, and Sarma (2026). It also yields further congruences modulo $16$ and a simple modulo-$8$ characterization that recovers and extends a recent congruence of those authors. As a second application we set $k=10$ and determine $c\phi_{10}(2n+1)$ modulo $8$.
Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write $\mathcal{O}_S$ for the ring of $S$-integers of $K$ and $\mathcal{O}_S^\times$ for its unit group. Let $\pi : X \rightarrow \mathbb{P}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\operatorname{Red}(\pi)$ the set of $\alpha \in \mathbb{P}^1(K)$ such that the fibre $\pi^{-1}(\alpha)$ is reducible (i.e. the Galois action on the fibre is intransitive). Hilbert's Irreducibility Theorem asserts that $\operatorname{Red}(\pi)$ is contained in a thin subset of $\mathbb{P}^1(K)$. In this paper we give an explicit description of $\mathcal{O}_S^\times \cap \operatorname{Red}(\pi)$. As an application we prove the following result inspired by a classical theorem of P\'{o}lya and Siegel. Let $p_1,\dotsc,p_s$ be rational primes. Let $f \in \mathbb{Q}[x]$.Then the following are equivalent: - There are infinitely many tuples $(e_1,\dotsc,e_s) \in \mathbb{N}^s$ such that the polynomial $f(x)-p_1^{e_1} \cdots p_s^{e_s}$ is reducible. - $f=p_1^{a_1} \cdots p_s^{a_s} g^\ell$ (with $\ell$ prime) or $f=-4 p_1^{a_1} \cdots p_s^{a_s} g^4$ for some $g \in \mathbb{Q}[x]$ and some integers $a_1,\dotsc,a_s$.
Michael Stoll, S. Siksek· 0 citations
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