We show that, for any finite field $\mathbb{F}$ and positive integer $k$, a simple $\mathbb{F}$-representable matroid $M$ with sufficiently high rank contains a rank-$k$ flat which is either independent in $M$, or is a projective or affine geometry over a subfield of $\mathbb{F}$. This generalizes a result for matroids representable over prime fields due to the first two authors. As a corollary, we obtain a Ramsey theorem for matroids representable over a finite field $\mathbb{F}$. For an integer $k$, in any $2$-colouring of the points of an $\mathbb{F}$-representable matroid of sufficiently high rank, there is a monochromatic rank-$k$ flat.
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 $t$...
We study the $q$-matroids associated with nondegenerate $\F_{q^m}$-linear near-MRD codes, which we call near-uniform $q$-matroids. Using the correspondence between rank-metric codes and their associated $q$-systems, we determine explicitly their rank functions and characterize their cyclic flats. We show that near-unif...
Let $M$ be a binary matroid of rank $r$ with no $F_7$-minor, and let $B_M^\star$ denote the maximum of its basis-generating polynomial over nonnegative weights of total mass $r$. We prove the sharp inequality $B_M^\star\leq 2^r/(r+1)$, with equality holding if and only if the nonloop simplification of $M$ is $M(K_{r+1}...
Let $\mathbb{F}_q$ be a finite field of odd characteristic with $q \equiv 2 \pmod3$. We study the directed graph defined by the Borwein--Borwein cubic arithmetic--geometric mean (AGM) over $\mathbb{F}_q$. We prove that this graph is a disjoint union of directed cycles. We associate Hessian curves with this AGM. We also...
For a line arrangement $\mathcal{A} \colon f=0$ in $\mathbb{P}^2$, let $\nu(\mathcal{A})$ be the maximal dimension of a graded piece of the Jacobian module of $f$. We study the conjecture that $\nu(\mathcal{A})$ depends only on the intersection lattice of $\mathcal{A}$. It is known that $\mathcal{A}$ is free if and onl...
If $F$ is a number field and $F = F_0 \subset F_1 \subset F_2\subset \cdots$ is a tower of Galois extensions with $\mathrm{Gal}(F_n/F)$ cyclic of order $p^n$, K. Iwasawa gave a closed formula for the growth of the $p$-part of the class number of $F_n$ whenever $n$ is large enough. We present a new proof of this formula...
Iq Souza· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
