Related papers: A Macaulay2 package for computations with rational…
The package Binomials contains implementations of specialized algorithms for binomial ideals, including primary decomposition into binomial ideals. The current implementation works in characteristic zero. Primary decomposition is restricted…
One proves a far-reaching upper bound for the degree of a generically finite rational map between projective varieties over a base field of arbitrary characteristic. The bound is expressed as a product of certain degrees that appear…
A rational map $\phi: \mathbb{P}^1 \to \mathbb{P}^1$ along with an ordered list of fixed and critical points is called a totally marked rational map. The space of totally marked degree two rational maps, $Rat^{tm}_2$ can be parametrized by…
We construct an explicit and calculable models for rational U(2)-spectra. This is obtained by assembling seven blocks obtained in previous work: the toral part and earlier work on small toral groups. The assembly process requires detailed…
In a previous paper it was shown that a machine learning regression problem can be solved within the framework of random function theory, with the optimal kernel analytically derived from symmetry and indifference principles and coinciding…
A common challenge in scientific and technical domains is the quantitative description of geometries and shapes, e.g. in the analysis of microscope imagery or astronomical observation data. Frequently, it is desirable to go beyond scalar…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
Dynamical degrees and spectra can serve to distinguish birational automorphism groups of varieties in quantitative, as opposed to only qualitative, ways. We introduce and discuss some properties of those degrees and the Cremona degrees,…
This paper is devoted to the investigation of selected situations when the computation of projective (and other) equivalences of algebraic varieties can be efficiently solved with the help of finding projective equivalences of finite sets…
Cellular resolutions are a technique for constructing resolutions of monomial ideals by giving a cell complex labeled by monomials, or more generally, by monomial modules. This \verb|Macaulay2| package allows us to work with cellular…
This paper proposes an efficient probabilistic method that computes combinatorial gradient fields for two dimensional image data. In contrast to existing algorithms, this approach yields a geometric Morse-Smale complex that converges almost…
We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler…
Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…
We have written a computer program that implements Deligne's pullback and pushforward weight spectral sequences to compute the weight graded pieces of the rational cohomology of moduli spaces of pointed smooth curves (as well as curves of…
Remora is a higher-order, rank-polymorphic array-processing programming language, in the same general class of languages as APL and J. It is intended for writing programs to be executed on parallel hardware. We provide an example-driven…
We introduce the CpMackeyFunctors package for Macaulay2, which allows for computations with Mackey functors over a cyclic group of prime order.
The package \texttt{NumericalCertification} implements methods for certifying numerical approximations of solutions for a given system of polynomial equations. For certifying regular solutions, the package implements Smale's $\alpha$-theory…
We present the Julia package Manifolds$.$jl, providing a fast and easy-to-use library of Riemannian manifolds and Lie groups. This package enables working with data defined on a Riemannian manifold, such as the circle, the sphere, symmetric…
New invariants for 2-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems…
We define the notion of "projective" multiresolution analyses, for which, by definition, the initial space corresponds to a finitely generated projective module over the algebra $C(\btn)$ of continuous complex-valued functions on an…