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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149 votes
38 answers
38k 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 ...
50 votes
5 answers
14k views

The unification of Mathematics via Topos Theory

In her paper The unification of Mathematics via Topos Theory, Olivia Caramello says "one can generate a huge number of new results in any mathematical field without any creative effort". Is ...
Roy Maclean's user avatar
  • 1,140
21 votes
5 answers
6k 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 ...
warsaga's user avatar
  • 1,186
65 votes
2 answers
23k views

Does there exist a complete implementation of the Risch algorithm?

Is there a generally available (commercial or not) complete implementation of the Risch algorithm for determining whether an elementary function has an elementary antiderivative? The Wikipedia article ...
Timothy Chow's user avatar
  • 78.1k
10 votes
5 answers
3k 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 ...
Andrew's user avatar
  • 103
12 votes
1 answer
967 views

Where to publish a long classification?

Suppose that the classification of some mathematical (say algebraic) notions requires (say) 70 pages. Let clarify that (say) 90% of the pages are used to write the result itself, whereas only 10% are ...
9 votes
2 answers
523 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 ...
Alejandro Tolcachier's user avatar
5 votes
2 answers
860 views

Given a formal power series ,decide whether there exists a polynomial the series satisfies and if it exists,how to write it down?

Given a formal power series $$y(x)=\sum_{i=0}^{\infty} a_i x^i$$ Is there an algorithm that decides whether there exists a polynomial$$ P(x,y)=p_n(x)y^n+p_{n-1}(x)y^{n-1}+\cdots+p_0(x)=0,p_j(x)\in F[x]...
XL _At_Here_There's user avatar
5 votes
2 answers
327 views

Reliability of ILP approach to number-theoretic optimization

This question is inspired by the recent answer, where @RobPratt proposed to use integer linear programming (ILP) for solving a number-theoretic optimization problem. I will consider a very similar ...
Max Alekseyev's user avatar
3 votes
4 answers
820 views

Compute the two-fold partial integral, where the three-fold full integral is known

I have the following trivariate ($\rho_{11}, \rho_{22}, \mu$) function \begin{equation} 4 \mu ^{3 \beta +1} \rho_{11}^{3 \beta +1} \left(-\rho_{11}-\rho_{22}+1\right){}^{3 \beta +1} \rho_{22}^{3 \...
Paul B. Slater's user avatar
43 votes
3 answers
5k 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 ...
Timothy Chow's user avatar
  • 78.1k
28 votes
4 answers
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 ...
TerronaBell's user avatar
  • 3,039
26 votes
5 answers
6k 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(\...
pavpanchekha's user avatar
  • 1,461
24 votes
5 answers
11k 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 ...
Nathan McKenzie's user avatar
24 votes
7 answers
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 ...
Melanie Matchett Wood's user avatar
21 votes
4 answers
983 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 ...
Gordon Royle's user avatar
  • 12.3k
20 votes
2 answers
1k 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 ...
fan's user avatar
  • 315
20 votes
5 answers
3k views

How can you find an integer coefficient polynomial knowing its values only at a few points (but requiring the coefficients be small)?

Example: How can you guess a polynomial $p$ if you know that $p(2) = 11$? It is simple: just write 11 in binary format: 1011 and it gives the coefficients: $p(x) = x^3+x+1$. Well, of course, this ...
Alexander Chervov's user avatar
15 votes
4 answers
4k 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.
google's user avatar
  • 151
12 votes
1 answer
670 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 ...
Timm von Puttkamer's user avatar
11 votes
5 answers
6k 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 ...
Rhubbarb's user avatar
  • 524
11 votes
3 answers
900 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 ...
Garfield's user avatar
  • 262
8 votes
4 answers
6k views

Computational algebra: where?

I'm on my last semester of a math B.Sc. and about to start studying for a math M.Sc in the same institute. It now seems like a good time to start thinking of a PhD. I'm interested in both algebra and ...
8 votes
3 answers
2k views

Rational exponential expressions

Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by: The leaves 1 and $x$ for $x$ drawn from a class of variables; and Closed under the binary ...
Charles Stewart's user avatar
8 votes
2 answers
1k views

Homological computations

Suppose I have a group acting on some Hadamard manifold, and I want to understand as much as possible about the (co)homology of the quotient. In my case I can find a fundamental domain for the action ...
Igor Rivin's user avatar
  • 95.6k
6 votes
1 answer
527 views

Does a variety contain a cartesian product of two curves?

We are given an affine variety $V\subset \mathbb{A}^n\times\mathbb{A}^n$, and wish to know if it contains a product of the form $C_1\times C_2$, where $C_1$ and $C_2$ are two curves in $\mathbb{A}^n$. ...
Boris Bukh's user avatar
  • 7,746
6 votes
1 answer
254 views

Algebraization of Bayesian networks?

The algebraization of classical propositional logic is Boolean algebra. Bayesian networks are a generalization of classical propositional logic with probability truth-values. What is the ...
YKY's user avatar
  • 508
4 votes
1 answer
176 views

On nearly Frobenius algebras

Let $A$ be a quiver algebra over a field $k$ with multiplication $m$. By https://arxiv.org/pdf/1705.10222.pdf definition 6, $A$ is called nearly Frobenius in case there exists a (non-zero? Was this ...
Mare's user avatar
  • 25.8k
4 votes
0 answers
410 views

Computing the volume of intersection between a ball and a box

$C$ is the set of vectors which are coordinate-wise less than $\overline{c}\in [-1,1]^d$ and greater than $\underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex in $d$ that ...
Christian Chapman's user avatar
2 votes
1 answer
524 views

Efficient derivation of null space of large symbolic matrices?

Hi all, I'm wondering if anyone is aware of an efficient mechanism by which to derive the null space of a "large" symbolic matrix. Here, large means on the order of 10^2 rows, not necessarily square,...
Paul's user avatar
  • 21