arXiv subjects

Mieke Wessel

Publications and source records attributed to Mieke Wessel.

4 recordsLinked to original sources

arXiv2026

High dimensional Riemann--Roch spaces in linear spaces with small squares

Let $F$ be a function field over an algebraically closed field $K$ and $S$ a finite dimensional $K$-subspace of $F$. The square of $S$ is spanned by all products of pairs of elements in $S$. We conjecture that if $\dim S^2 \leq 3 \dim S - 4$, then $S^2$ must contain a Riemann--Roch space of dimension at least $2 \dim S - 1 + g$, where $g$ is the genus of $F$. This generalizes a theorem of Freiman from additive combinatorics, stating that small sumsets must contain long arithmetic progressions. We prove our conjecture in the case that $S$ is contained in a Riemann--Roch space of dimension at most $3/2\dim S + 1$. For the proof we study the annihilator of $S$ and introduce the notion of weight for linear forms.

math.NT↗

arXiv2025

Solving quadratic forms in restricted variables with the circle method

Let $f(\mathbf x)$ be a non-singular quadratic form with sufficiently many mixed terms and $t$ an integer. For a sequence of weights $\mathcal A$ we study the number of weighted solutions to $f(\mathbf x) = t$. In particular, we give conditions on both $\mathcal A$ and $f$ such that we can use the circle method to count such solutions of bounded height.

math.NT↗

arXiv2024

A graph-theoretic proof of Cobham's Dichotomy for automatic sequences

We give a new graph-theoretic proof of Cobham's Theorem which says that the support of an automatic sequence is either sparse or grows at least like $N^α$ for some $α> 0$. The proof uses the notions of tied vertices and cycle arboressences. With the ideas of the proof we can also give a new interpretation of the rank of a sparse sequence as the height of its cycle arboressence. In the non-sparse case we are able to determine the supremum of possible $α$, which turns out to be the logarithm of an integer root of a Perron number.

math.CO↗

arXiv2024

On Freiman's Theorem in a function field setting

We prove some new instances of a conjecture of Bachoc, Couvreur and Zémor that generalizes Freiman's $3k-4$ Theorem to a multiplicative version in a function field setting. As a consequence we find that if $F$ is a rational function field over an algebraically closed field $K$ and $S \subset F$ a finite dimensional $K$-vector space such that $\dim S^2 = 2\dim S + 1$, then the conjecture holds.

math.NT↗