Related papers: Computations involving symbolic powers
Large language models (LLMs) are increasingly used in situations where human values are at stake, such as decision-making tasks that involve reasoning when performed by humans. We investigate the so-called reasoning capabilities of LLMs…
It was recently shown that the theory of linear stochastic systems can be viewed as a particular case of the theory of linear systems on a certain commutative ring of power series in a countable number of variables. In the present work we…
Symbolic computation is the science of computing with symbolic objects (terms, formulae, programs, algebraic objects, geometrical objects, etc). Powerful symbolic algorithms have been developed during the past decades and have played an…
We introduce the package MacaulayPosets written for the computational algebra system Macaulay2. This package utilized the poset data type introduced in the Posets package and offers functionality for studying the Macaulay property for…
Let $R$ be a commutative Noetherian ring and let ${\bf x} :=x_1,\ldots,x_d$ be a regular $R$-sequence contained in the Jacobson radical of $R$. An ideal $I$ of $R$ is said to be a monomial ideal with respect to ${\bf x}$ if it is generated…
Functionals are an important research subject in Mathematics and Computer Science as well as a challenge in Information Technologies where the current programming paradigm states that only symbolic computations are possible on higher order…
In the spirit of "multi-culturalism", we use four kinds of computations: simulation, numeric, symbolic, and "conceptual" to explore some "games of pure chance" inspired by children board games like "Snakes and Ladders" (aka as "Chutes and…
In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the $n$-th symbolic power of a given prime ideal consists of the…
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…
We use symbolic expressions for traces of positive integer powers of a Hermitian operator (or, equivalently, coefficients of corresponding characteristic polynomial) to find solutions for the problems as follows: Factorization of…
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…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
A decomposition of any symmetric power of $\Bbb C^2\otimes\Bbb C^2\otimes\Bbb C^2$ into irreducible $sl_2(\Bbb C)\oplus sl_2(\Bbb C)\oplus sl_2(\Bbb C)$-submodules are presented. Namely, the multiplicities of irreducible summands in the…
We analyze the problem of determining Waring decompositions of the powers of any quadratic form over the field of complex numbers. Our main goal is to provide information about their rank and also to obtain decompositions whose size is as…
Learning Spaces are certain set systems that are applied in the mathematical modeling of education. We propose a suitable compression (without loss of information) of such set systems to facilitate their logical and statistical analysis.…
The main object of this work is to show how some rather elementary techniques based upon certain inverse pairs of symbolic operators would lead us easily to several decomposition formulas associated with confluent hypergeometric functions…
We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial…
Experimental mathematics is an experimental approach to mathematics in which programming and symbolic computation are used to investigate mathematical objects, identify properties and patterns, discover facts and formulas and even…
We give a description of a new Macaulay2 package called SimplicialPosets. This package provides functions for working with simplicial posets and calculating their generalized Stanley-Reisner ideals. For practical purposes, we also introduce…
A cohesive power of a structure is an effective analog of the classical ultrapower of a structure. We start with a computable structure, and consider its countable ultrapower over a cohesive set of natural numbers. A cohesive set is an…