Related papers: Computational Methods for the Construction of a Cl…
A primary ideal in a polynomial ring can be described by the variety it defines and a finite set of Noetherian operators, which are differential operators with polynomial coefficients. We implement both symbolic and numerical algorithms to…
Noetherian operators are differential operators that encode primary components of a polynomial ideal. We develop a framework, as well as algorithms, for computing Noetherian operators with local dual spaces, both symbolically and…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
Proving statements about linear operators expressed in terms of identities often leads to finding elements of certain form in noncommutative polynomial ideals. We illustrate this by examples coming from actual operator statements and…
An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in…
We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and…
We present a package 'MixedMultiplicity' for computing mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. This enables us to find mixed volumes of convex lattice polytopes…
We count the number of submodules of an arbitrary module over a countable noetherian commutative ring. We give, along the way, a structural description of meager modules, which are defined as those that do not have the square of a simple…
The cohomology ring of a finite group, with coefficients in a finite field, can be computed by a machine, as Carlson has showed. Here "compute" means to find a presentation in terms of generators and relations, and involves only the…
In this article we give an algorithm for computing the integral closure of a reduced Noetherian ring R, in case this integral closure is finitely generated over R.
Rota-Baxter operators are an algebraic abstraction of integration. Following this classical connection, we study the relationship between Rota-Baxter operators and integrals in the case of the polynomial algebra $\mathbf{k}[x]$. We consider…
Ore operators form a common algebraic abstraction of linear ordinary differential and recurrence equations. Given an Ore operator $L$ with polynomial coefficients in $x$, it generates a left ideal $I$ in the Ore algebra over the field…
In this paper, we develop general machinery for computing the classifying ring $L^A$ of one-dimensional formal $A$-modules, for various commutative rings $A$. We then apply the machinery to obtain calculations of $L^A$ for various number…
We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related…
We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential…
This is an expository version of our paper [arXiv:1902.07384]. Our aim is to present recent Macaulay2 algorithms for computation of mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over…
The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.
In this article we present a parallel modular algorithm to compute all solutions with multiplicities of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition using…
This paper describes a method for computing all F-pure ideals for a given Cartier map of a polynomial ring over a finite field.
We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and…