Questions tagged [computer-algebra]

Using computer-aid approach to solve algebraic problems. Questions with this tag should typically include at least one other tag indicating what sort of algebraic problem is involved, such as ac.commutative-algebra or rt.representation-theory or ag.algebraic-geometry.

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128
votes
34answers
31k views

Computer Algebra Errors

In the course of doing mathematics, I make extensive use of computer-based calculations. There's one CAS that I use mostly, even though I occasionally come across out-and-out wrong answers. After ...
110
votes
71answers
18k views

Most helpful math resources on the web

What are really helpful math resources out there on the web? Please don't only post a link but a short description of what it does and why it is helpful. Please only one resource per answer and let ...
53
votes
2answers
3k views

Reasons to prefer one large prime over another to approximate characteristic zero

Background: In running algebraic geometry computations using software such as Macaulay2, it is often easier and faster to work over $\mathbb F_p = \mathbb Z / p\mathbb Z$ for a large prime $p$, rather ...
43
votes
3answers
4k views

Is there a systematic method for differentiating under the integral sign?

This MO question by Tim Gowers reminded me of a question I've wondered about for some time. In the delightful book Surely You're Joking, Mr. Feynman!, Feynman praises the technique of differentiating ...
40
votes
23answers
14k views

Open source mathematical software

I want some recommendation on which software I should install on my computer. I'm looking for an open source program for general abstract mathematical purposes (as opposed to applied mathematics). I ...
38
votes
5answers
11k views

The unification of Mathematics via Topos Theory

When the paper The unification of Mathematics via Topos Theory by Olivia Caramello, says "one can generate a huge number of new results in any mathematical field without any creative effort." is this ...
28
votes
1answer
1k views

Algebraic dependency over $\mathbb{F}_{2}$

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$ such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall i\in[n]:f_{i}(a)=a_{i}$....
27
votes
4answers
3k views

Can Gröbner bases be used to compute solutions to large, real-world problems?

In particular, suppose I have an affine algebraic variety over $\mathbb{R}^n$ described by generators of a radical ideal $I$ and I want to find (perhaps not all of the) points in the variety. Several ...
25
votes
0answers
765 views

Where to submit this work with several unusual features?

I appreciate that questions about where to submit are generally considered off-topic, but I hope that the unusual features of the present case may make it acceptable. I have put a monograph on github ...
24
votes
7answers
3k views

Computer package for representation theory of the symmetric group

Is there a computer algebra package in which I can compute the following for representations of a specific symmetric group (e.g. $S_7$): (1) $V \otimes W$ (2) $S_\lambda V$, where $S_\lambda$ is a ...
23
votes
5answers
9k views

Fastest Algorithm to Compute the Sum of Primes?

Can anyone help me with references to the current fastest algorithms for counting the exact sum of primes less than some number n? I'm specifically curious about the best case running times, of ...
23
votes
1answer
620 views

two's and three's survive in gcd of Lagrange

Lagrange's four square_theorem states that every positive integer $N$ can be written as a sum of four squares of integers. At present, let's focus only on positive integer summands; that is, $N=a_1^2+...
22
votes
3answers
4k views

Minimal polynomial of cos(π/n)

I know that $\cos(\pi/n)$ is a root of the Chebyshev polynomial $(T_n + 1)$, in fact it is the largest root of that polynomial, but often that polynomial factors. For example, if $n = 2 k$ then $\cos(\...
21
votes
4answers
871 views

Is there a way of canonically labelling permutation groups?

When working with large numbers of graphs, a canonical labelling routine is essential as, after the initial cost of canonically labelling each graph, it permits isomorphism checks to be replaced with ...
21
votes
2answers
777 views

What is currently feasible in invariant theory for binary forms?

When Paul Gordan became a professor in 1875 he could show the binary form in any degree has some finite complete system of (general linear) invariants, but he could not actually give a complete system ...
21
votes
1answer
1k views

Main open computational problems in quantifier elimination?

A language is said to have quantifier elimination if every first-order-logic sentence in the language can be shown to be equivalent to a quantifier-free sentence, i.e., a sentence without any $\forall$...
19
votes
2answers
911 views

Where to publish computer computations

In a paper I developed some theory; some of the applications require extensive computations that are not part of the paper. I wrote a Mathematica notebook. I want to publish a PDF and .nb version ...
17
votes
6answers
3k views

“Oldest” bug in computer algebra system?

The goal of this question is to find an error in a computation by a computer algebra system where the 'correct' answer (complete with correct reasoning to justify the answer) can be found in the ...
16
votes
3answers
1k views

What to do when your research runs into a computationally challenging problem?

Occasionally, but more frequently lately, I would like to perform some hard computations. As an example, yesterday the following question came up: What is the projective dimension of the edge ideal ...
16
votes
2answers
593 views

From polynomial ideal over $\mathbb{Q}$ to polynomial ideal over $\mathbb{Z}$

Is there an algorithm to compute, given a polynomial ideal $I\subset \mathbb{Q}[x_1,\dotsc,x_n]$, the ideal $I\cap \mathbb{Z}[x_1,\dotsc,x_n]$ in $\mathbb{Z}[x_1,\dotsc,x_n]$? The input and output ...
16
votes
5answers
4k views

Fast computation of a Groebner basis. What is possible?

I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or ...
15
votes
4answers
3k views

Program for computing group cohomology

Is there any computer program with which I can compute the group cohomology H^n(G,V) for a group G acting linearly on a vector space? I mainly care about infinite groups.
15
votes
4answers
1k views

An experiment on random matrices

A bit unsure if my use/mention of proprietary software might be inappropriate or even frowned upon here. If this is the case, or if this kind of experimental question is not welcome, please let me ...
15
votes
1answer
514 views

Computer software for periods

Kontsevich and Zagier define a period as an integral of a rational function (over $\mathbb{Q}$) defined on a $\mathbb{Q}$-semialgebraic set. They conjecture that if two periods are equal, then the ...
14
votes
2answers
2k views

Compute Lie algebra cohomology

Is there a computer algebra system that is able to compute the Lie algebra cohomology in a given representation? What if the Lie algebra is finite dimensional? In my case I would like to be able to ...
13
votes
3answers
3k views

Computing (on a computer) higher ramification groups and/or conductors of representations.

I am supervising an undergraduate for a project in which he's going to talk about the relationship between Galois representations and modular forms. We decided we'd figure out a few examples of weight ...
13
votes
1answer
848 views

Computational Question about finite local rings:

Let $(A,\mathfrak{m})$ be a local Artinian ring with finite residue field, which I'm happy to assume is $\mathbf{F}_3$. (In particular, $A$ has finitely many elements.) I would like to do some ...
13
votes
1answer
547 views

How can I tell if a variety is normal?

Suppose $R$ is a subalgebra of ${\mathbb C}[x_1,...,x_N]$ generated by polynomials $p_1,...,p_k.$ I know that ${\mathbb C}[x_1,...,x_N]$ is the integral closure of $R$. Is there an algorithm to ...
12
votes
4answers
4k views

how to determine whether an ideal is prime or not by an algorithm

Given polynomials $f_{1},\cdots,f_{n}\in \mathbb{C}[x_{1},\cdots,x_{m}]$, do we have an algorithm to determine whether the ideal $I=(f_{1},\cdots,f_{n})$ is prime ideal or not? Of course, we assume ...
12
votes
1answer
1k views

An efficient isomorphism between finite fields

Let $p$ be a prime number. Let $f$ and $g$ be irreducible polynomials over $\mathbb{F}_p$, both of degree $n$. We know that factor-rings $\mathbb{F}_p[x]/(f)$ and $\mathbb{F}_p[x]/(g)$ are isomorphic ...
12
votes
4answers
2k views

Is Gauss-Seidel guaranteed to converge on *semi* positive definite matrices?

I know that the Gauss-Seidel method is guaranteed to converge given that the matrix you want to solve is positive definite. I've looked at the proofs of convergence, and it appears that one cannot ...
12
votes
2answers
450 views

Ideal Membership without Certificate?

I have a homogeneous ideal $I=\langle f_1,\ldots,f_r\rangle$ of the polynomial ring $\mathbb C[X_1,\ldots,X_n]=:R$ where each of the $f_i$ is actually over $\mathbb Z$. My computations are usually ...
12
votes
2answers
564 views

Faster multiplication with a restricted set of multiplicands?

Let $A$ be a set of $k>1$ distinct elements from a semigroup. We wish to compute the product $$ p=b_1 b_2 \cdots b_n$$ where each $b_i\in A$. Clearly $n-1$ multiplications suffice to compute $p$; ...
11
votes
9answers
2k views

Is there a non self-referencing non-computable function?

I've seen in college that some functions are not computable. The proof for that was the case of Halt(x,y) function. The thing is, the proof used a very artificial (IMHO) case which is evaluating ...
11
votes
1answer
437 views

Representing field elements in a computer

I'm wondering if there is existing terminology to describe fields $F$ with the properties below. I don't have a completely precise description of the concept I have in mind, but hopefully this will be ...
11
votes
1answer
392 views

Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials

Let $n>1$ and $p$ be an odd prime with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. Notice that, since $p-1 \mid n-1$, we have $T^p - T \mid T^n-T$ in $\mathbb{F}_p[T]...
11
votes
1answer
528 views

Verify that a group is hyperbolic via computer algebra

I would like to know whether there is some computer algebra software that can be used to verify if a group, given by a finite presentation, is hyperbolic (in the sense that it terminates with "yes" if ...
11
votes
1answer
372 views

How do computer algebra packages like Sagemath implement rank of a matrix

I am not sure if this is the right place to ask this question, but I believe there will be people here who do computations on computer algebra packages like Sage in their work. I have been using ...
11
votes
1answer
1k views

A 2F1 Hypergeometric identity from a Feynman integral

Using two different approaches to evaluating the dimensionally regularized ($d=4-2\epsilon$ dimensional Euclidean space), equal mass ($x=m^2$), 2-loop vacuum Feynman diagram $$ \begin{align} I(x) &...
10
votes
5answers
2k views

Multipolynomial resultants

We know that the resultant of two polynomials can be computed as the determinant of their Sylvester matrix ( http://en.wikipedia.org/wiki/Sylvester_matrix ). How do we compute the resultant of more ...
10
votes
5answers
5k views

Finding minimal or canonical expressions for Boolean truth tables

This is not an urgent question, but something I've been curious about for quite some time. Consider a Boolean function in n inputs: the truth table for this function has 2n rows. There are uses of ...
10
votes
2answers
702 views

Computing in quantum groups

I'd be interested in doing some computations in quantum groups $ U_q(\mathfrak g)$ that are conceptually simple (``is this element 0"?, and $\mathfrak g = sl_5$), but are somewhat lengthy to do by ...
10
votes
2answers
383 views

Counting points on varieties of low codimension

The graduate students here at MIT have been thinking about questions like the following: Over $\mathbb{F}\_q$, how many symmetric matrices are there with nonzero determinant and $0$'s on the diagonal? ...
10
votes
3answers
788 views

Algorithms in Invariant Theory

Let $V$ be a polynomial representation of the general linear group $\Gamma:=\DeclareMathOperator{\Gl}{Gl}\Gl_n(\newcommand{\C}{\mathbb C}\C)$. In chapter 4.6 of his book "Algorithms in Invariant ...
10
votes
2answers
403 views

Monomial orderings in noncommutative setting

An ordering of monomials in the free associative algebra $k\langle x_1,\ldots,x_n\rangle$ is called a monomial ordering (EDIT: it seems that an equally common term used in this context is "term ...
9
votes
8answers
5k views

Which computer algebra system should I be using to solve large systems of sparse linear equations over a number field?

This is related to Noah's recent question about solving quadratics in a number field, but about an even earlier and easier step. Suppose I have a huge system of linear equations, say ~10^6 equations ...
9
votes
2answers
429 views

A “subtle” isomorphism testing problem: $\mathbb{Z}\ltimes_{A} \mathbb{Z}^5\cong \mathbb{Z}\ltimes_{B}\mathbb{Z}^5$ or not?

EDIT: I've made a mistake with the matrices. Now it is corrected. A couple of days ago I asked this question. There, answerers gave me excellent hints to solve that case and others too. But I've found ...
9
votes
1answer
638 views

Mathematical software for computing in integral group rings of discrete groups?

I'm doing computations in the integral group ring of a discrete group, in particular the discrete Heisenberg group. In this case elements are integral combinations of monomials $x^k y^m z^n$, where ...
9
votes
2answers
3k views

How to do integrals involving two Bessel functions and another function?

I often encounter the integrals in the following form: $\int_0^\infty{\rm Bessel}(ax)\cdot{\rm Bessel}(bx)\cdot f(cx)dx$, where Bessel can be $J$, $N$, $H^{(1)}$, $H^{(2)}$, $I$, or $K$; and $f(x)$ ...
9
votes
3answers
1k views

Is there a stable algorithm for polynomial division (in several variables)?

Suppose you have a homogeneous ideal $I$ inside the algebra $\mathbb{C}[x_1,...,x_d]$ of complex polynomials in $d$-variables. Can one find a basis for $I$, say $\{f_1,...,f_k\}$, such that every $h \...

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