Related papers: The FrobeniusThresholds package for Macaulay2
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…
The probabilities for gaps in the eigenvalue spectrum of finite $ N\times N $ random unitary ensembles on the unit circle with a singular weight, and the related hermitian ensembles on the line with Cauchy weight, are found exactly. The…
With a small suitable modification, dropping the projectivity condition, we extend the notion of a Frobenius algebra to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced…
In this article, we study certain local cohomology modules over $F$-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula for computing these invariants when the $F$-pure ring is standard…
For each integer $s\geq 1$, we present a family of curves that are $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case $s = 2$, we give necessary and sufficient conditions for…
We study how certain invariants of numerical semigroups relate to the number of second kind gaps. Furthermore, given two fixed non-negative integers F and k, we provide an algorithm to compute all the numerical semigroups whose Frobenius…
The following article is an application of commutative algebra to the study of multiparameter persistent homology in topological data analysis. In particular, the theory of finite free resolutions of modules over polynomial rings is applied…
This is supplementary material for the article arxiv:0708.3250. We provide an alternative introduction of the mean Euler Characteristic, additional examples and the percolation thresholds for 2-uniform lattices.
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…
In this article, we describe the theoretical foundations of the Macaulay2 package ConnectionMatrices and explain how to use it. For a left ideal in the Weyl algebra that is of finite holonomic rank, we implement the computation of the…
The notion of a Frobenius manifold appears in relation to various topics in algebraic and analytic geometry, such and quantum cohomology, deformation of meromorphic connections, unfolding of singularities and others. In the local setting…
We provide a direct computation of the $F$-pure threshold of degree four homogeneous polynomial in two variables and, more generally, of certain homogeneous polynomials with four distinct roots. The computation depends on whether the cross…
We study some properties of restricted and associated Fubini numbers. In particular, they have the natural extensions of the original Fubini numbers in the sense of determinants. We also introduce modified Bernoulli and Cauchy numbers and…
The study of Fourier transforms of probability measures on fractal sets plays an important role in recent research. Faster decay rates are known to yield enhanced results in areas such as metric number theory. This paper focuses on…
We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities and mutual constraints are allowed in…
We study the behavior of test ideals and F-singularities in families. In particular, we obtain generic (and non-generic) restriction theorems for test ideals and non-F-pure ideals. Additionally, we study the global behavior of certain…
In this paper, we show the log canonical threshold values of the surfaces which has du Val type singularities.These surfaces can be interpreted as statistical or machine learning models. The results of $A_n, D_n, E_6, E_7$ and $E_8$ are…
The first purpose of this paper is to give the fnite transcendence of Frobenius traces for elliptic curves over $\mathbb{Q}$ without the assumption of complex multiplication (CM). This result generalizes the previous work by Luca and…
Let $p>3$ and consider a prime power $q=p^h$. We completely characterize permutation polynomials of $\mathbb{F}_{q^2}$ of the type $f_{a,b}(X) = X(1 + aX^{q(q-1)} + bX^{2(q-1)}) \in \mathbb{F}_{q^2}[X]$. In particular, using connections…
New conjectures are proposed on the numbers of FPL configurations pertaining to certain types of link patterns. Making use of the Razumov and Stroganov Ansatz, these conjectures are based on the analysis of the ground state of the…