相关论文: Bases for Riemann-Roch spaces of one point divisor…
The Riemann -Rock theorem plays a central role in the theory of Riemann surfaces with applications to several branches in Mathematics and Physics. Suppose $X$ ia a compact Riemann surface of genus $g$ and $P \in X$. By the Riemann-Roch…
We prove a Riemann-Roch theorem for real divisors on edge-weighted graphs over the reals, extending the result of Baker and Norine for integral divisors on graphs with multiple edges.
We study the algebraic rank of a divisor on a graph, an invariant defined using divisors on algebraic curves dual to the graph. We prove it satisfies the Riemann-Roch formula, a specialization property, and the Clifford inequality. We prove…
We explicitly describe the set of gaps and the Weierstrass semigroup at a totally ramified place of degree one on a Kummer extension defined by the affine equation $y^m = f(x)$ over $K$, an algebraic extension of $\mathbb{F}_q$, where…
We introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be ``liftable'' under finite morphisms, paving…
We present an algorithm to compute the Weierstrass semigroup at a point P together with functions for each value in the semigroup, provided P is the only branch at infinity of a singular plane model for the curve. As a byproduct, the method…
In this article we explicitly determine the Weierstrass semigroup at any place of some $\mathbb{F}_{q^2}$-maximal Fermat function fields $\mathcal{F}_m$, namely for $m=(q+1)/2$ and $m=(q+1)/3$. These famous function fields arise as Galois…
The function $h^0$ for a number field is analogous to the dimension of the Riemann-Roch spaces at divisors on an algebraic curve. We provide a method to compute this function for number fields with unit group of rank at most 2, even with…
The Weierstrass semigroups and pure gaps can be helpful in constructing codes with better parameters. In this paper, we investigate explicitly the minimal generating set of the Weierstrass semigroups associated with several totally ramified…
For addressing optimisation tasks on finite dimensional quantum systems, we give a comprehensive account of the foundations of gradient flows on Riemannian manifolds including new developments: we extend former results from Lie groups such…
We compute generators and relations for the section ring of a rational divisor on an elliptic curve. Our technique generalizes the work of O'Dorney (in genus zero) and Voight--Zureick-Brown (for specific divisors arising from the study of…
A linearized function field $F$ can be viewed as a Galois extension of a rational function field $K(x)$. For a totally ramified place $Q$ of degree one in $F/K(x)$, we give a unified description of the set $G(Q)$ of gaps at $Q$. As a…
This paper surveys and illustrates geometric methods for constructing normal bases allowing efficient finite field arithmetic. These bases are constructed using the additive group, the multiplicative group and the Lucas torus. We describe…
We give effective bounds for the class number of any algebraic function field of genus $g$ defined over a finite field. These bounds depend on the possibly partial information on the number of places on each degree $\leq g$. Such bounds are…
In this work, we use the notion of ``symmetry'' of functions for an extension $K/L$ of finite fields to produce extensions of a function field $F/K$ in which almost all places of degree one split completely. Then we introduce the notion of…
The zero divisor of the theta function of a compact Riemann surface $X$ of genus $g$ is the canonical theta divisor of Pic${}^{(g-1)}$ up to translation by the Riemann constant $\Delta$ for a base point $P$ of $X$. The complement of the…
In this paper, we study configurations of three rational points on the Hermitian curve over $\mathbb{F}_{q^2}$ and classify them according to their Weierstrass semigroups. For $q>3$, we show that the number of distinct semigroups of this…
We improve known estimates for the number of points of bounded height in semigroup orbits of polarized dynamical systems. In particular, we give exact asymptotics for generic semigroups acting on the projective line. The main new ingredient…
We give bounds on the degree of generation and relations of section rings associated to arbitrary $\mathbb{Q}$-divisors on projective spaces of all dimensions and Hirzebruch surfaces. For section rings of effective $\mathbb{Q}$-divisors on…
We present an algorithm to compute bases for the spaces L(G), provided G is a rational divisor over a non-singular absolutely irreducible algebraic curve, and also another algorithm to compute the Weierstrass semigroup at P together with…