Related papers: Computations with rational maps between multi-proj…
We provide new examples of integrable rational maps in four dimensions with two rational invariants, which have unexpected geometric properties, as for example orbits confined to non algebraic varieties, and fall outside classes studied by…
In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or…
Symbolic powers are a classical commutative algebra topic that relates to primary decomposition, consisting, in some circumstances, of the functions that vanish up to a certain order on a given variety. However, these are notoriously…
Some polynomials $P$ with rational coefficients give rise to well defined maps between cyclic groups, $\Z_q\longrightarrow\Z_r$, $x+q\Z\longmapsto P(x)+r\Z$. More generally, there are polynomials in several variables with tuples of rational…
{\tt AbstractSimplicialComplexes.m2} is a computer algebra package written for the computer algebra system {\tt Macaulay2} \cite{M2}. It provides new infrastructure to work with abstract simplicial complexes and related homological…
We describe the main functions of the Macaulay2 package Quasidegrees. The purpose of this package is to compute the quasidegree set of a finitely generated A-graded module presented as the cokernel of a monomial matrix. We provide examples…
We introduce the CpMackeyFunctors package for Macaulay2, which allows for computations with Mackey functors over a cyclic group of prime order.
We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation.
We present the Mathematica package MultivariateResidues, which allows for the efficient evaluation of multivariate residues based on methods from computational algebraic geometry. Multivariate residues appear in several contexts of…
Let $X,Y$ be two irreducible subvarieties of the projective space $\mathbb{P}^n$, and $d\geq 1$ an integer number. The main result of this paper is an algorithm to construct {\bf explicitly}, in terms of $d$ and the ideals defining $X$ and…
We introduce the Macaulay2 package HomologicalShiftIdeals. It allows to compute the homological shift ideals of a monomial ideal, and to check the homological shift properties, including having linear resolution, having linear quotients, or…
Multimodal multi-objective problems (MMOPs) commonly arise in real-world problems where distant solutions in decision space correspond to very similar objective values. To obtain all solutions for MMOPs, many multimodal multi-objective…
We survey some general-purpose symbolic software packages that implement algorithms from enumerative and analytic combinatorics. Software for the following areas is covered: basic combinatorial objects, symbolic combinatorics, P\'olya…
In this paper, we introduce complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces. A key property of these maps is their orientation awareness. More specifically, we…
We prove that holomorphic maps from an open subset of a complex smooth projective curve to a complex smooth projective rationally simply connected variety can be approximated by algebraic maps for the compact-open topology. This theorem can…
We give an overview of our philosophy of pictures in mathematics. We emphasize a bi-directional process between picture language and mathematical concepts: abstraction and simulation. This motivates a program to understand different…
The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections…
We highlight some features of the SimplicialComplexes package in Macaulay2.
We provide algorithms for symbolic integration of hyperlogarithms multiplied by rational functions, which also include multiple polylogarithms when their arguments are rational functions. These algorithms are implemented in Maple and we…
Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order piecewise polynomials, multilevel schemes, or other types…