Related papers: Logarithmic A-hypergeometric series
This paper is the third in a series exploring Frobenius's method for $A$-hypergeometric systems. Frobenius's method is a classical technique for constructing logarithmic series solutions of differential equations by perturbing exponents of…
The Frobenius method can be used to compute solutions of ordinary linear differential equations by generalized power series. Each series converges in a circle which at least extends to the nearest singular point; hence exponentially fast…
In this paper, following [6], we continue to develop the perturbing method of constructing logarithmic series solutions to a regular A-hypergeometric system. Fixing a fake exponent of an A-hypergeometric system, we consider some spaces of…
Using Hahn series, one can attach to any linear Mahler equation a basis of solutions at 0 reminiscent of the solutions of linear differential equations at a regular singularity. We show that such a basis of solutions can be produced by…
In this paper, we discuss the computational approach to the results established by Okuyama and Saito. Although their results are often difficult to compute, we prove that, when the negative support of a fake exponent $v$ with respect to a…
We study the solutions of irregular A-hypergeometric systems that are constructed from Gr\"obner degenerations with respect to generic positive weight vectors. These are formal logarithmic Puiseux series that belong to explicitly described…
Let consider $n$ natural numbers $a\_1 ,\ldots , a\_{n} $. Let $S$ be the numerical semigroup generated by $a\_1 ,\ldots , a\_{n} $. Set $A=K[t^{a\_1}, \ldots , t^{a\_n}]=K[{x\_1}, \ldots , {x\_n}]/I$. The aim of this paper is:…
Recently, Kedlaya proves certain formula describing explicitly the Frobenius structure on a hypergeometric equation. In this paper, we give a generalization of it. In our case, the Frobenius matrix is no longer described by p-adic gamma…
The Frobenius method can be used to represent solutions of ordinary differential equations by (generalized) power series. It is useful to have prior knowledge of the coefficients of this series. In this contribution we demonstrate that the…
We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…
We describe an algorithm for computing, for all primes $p \leq X$, the mod-$p$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive in time quasilinear in $X$. This combines the Beukers--Cohen--Mellit trace formula…
We construct Frobenius structures on the $\mathbb{C}^{\times}$-bundle of the complement of a toric arrangement associated with a root system, by making use of a one-parameter family of torsion free and flat connections on it. This gives…
A submonoid A of N^d has a natural order defined by a <= a + b for elements a and b of A. The Frobenius complex is the order complex of an open interval of A with respect to this order. In this paper, the homotopy type of the Frobenius…
The main subject of this paper is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods…
This paper proposes a new, visual method to study numerical semigroups and the Frobenius problem. The method is based on building a so-called reduction graph, whose nodes usually correspond to monogenic semigroups, and whose edges can have…
We study second order and third order linear differential equations with analytic coefficients under the viewpoint of finding formal solutions and studying their convergence. We address some untouched aspects of Frobenius methods for second…
For a fixed positive integer $e$, we describe an algorithm for computing, for all primes $p \leq X$, the mod-$p^e$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive over $\mathbb{Q}$ in time quasilinear in $X$.…
An affine monoid is an additive monoid which is cancellative, pointed and finitely generated. An affine monoid $\Lambda$ has the partial order defined by $\lambda \le \lambda + \mu$. The Frobenius complex is the order complex of an open…
In this paper, using the theory of the so-called fractional calculus we show that it is possible to easily obtain the solutions for the confluent hypergeometric equation. Our approach is to be compared with the standard one (Frobenius)…
A construction theorem for Frobenius manifolds with logarithmic poles is established. This is a generalization of a theorem of Hertling and Manin. As an application we prove a generalization of the reconstruction theorem of Kontsevich and…