Using computers 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.

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4
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2answers
534 views

Indexed tensor manipulation CAS

hello. I am looking for tensor manipulation software that would allow me: declare indices declare results of contraction (or simplification rules) allow algebraic simplifications and expansion ...
5
votes
3answers
375 views

Automatic proving some expression is positive

Is there any automated (i.e., some algorithm) to prove that a certain algebraic expression is always non-negative in some range ? If so, is there any implementation you would suggest? My concrete ...
7
votes
1answer
473 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 ...
5
votes
2answers
340 views

Complexity of high-order differentiation

Let $g(x) = \exp(f(x))$. Assuming numerical (or symbolic) values of $f(x), f'(x), f''(x), \ldots, f^{(n)}(x)$ are known, is there a way to compute $g'(x), g''(x), \ldots g^{(n)}(x)$ (or even the ...
0
votes
0answers
730 views

Efficient quadratic residue mod 2^32

I want to determine if a value is a quadratic residue mod $2^{32}$. I've developed a very fast pre-screening method based on a Bloom Filter that identifies quadratic residues for mod $2^7=128$ in ...
5
votes
1answer
613 views

Computer power in plane geometry

I often hear that modern computer programs "may prove any theorem in elementary Eucledian geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for arbitrary or ...
3
votes
1answer
607 views

Adjoint/transpose of wavelet transform

I'm using a wavelet transform in Matlab, so I think of it as a black-box. I'll represent it here as $W(x)$. There's a reconstruction function as well, which I'll write as $W^\dagger(y)$. I can ...
6
votes
6answers
1k views

What are you using for symbolic computation?

What are the pluses and minuses of different software packages? Anything new worth checking out? I'm especially interested in open source packages.
12
votes
4answers
973 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 ...
21
votes
5answers
6k 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 ...
6
votes
3answers
1k views

Computing only the order of Galois group (not the group itself).

My question is related to this one: Computing the Galois group of a polynomial. I was wondering if there is a faster algorithm just to compute the order of the group rather than the group itself. ...
5
votes
1answer
357 views

software for computations on flag varieties in arbitrary characteristic

Is there any software that will compute cohomology of vector bundles (or just line bundles) on flag manifolds? The only one I know of is Macaulay2, via the Schubert2 package, but it works with what ...
7
votes
1answer
480 views

Are there any sofware packages for computing Picard numbers?

Are there any computer algebra systems (e.g. Macaulay2 og singular) that allows one to compute the Picard number (i.e. the rank of the Neron-Severi group) of a given variety?
1
vote
1answer
361 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 ...
7
votes
4answers
2k views

Basis for modular forms of half-integral weight.

Given a character $\chi$ and $k$ odd how can one compute a basis for the space of modular forms $M_\frac{k}{2}(\Gamma_0(4),\chi)$. By compute a basis I mean, finding the beginning of the Fourier ...
5
votes
2answers
603 views

Software for computing multi-graded Hilbert series

The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights (1,1,-1,-1) is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series $\frac{1 - ...
32
votes
20answers
7k views

Open source mathematical software.

I want some recomendation on which software I should install on my computer, an open source program for general abstract mathematical purposes (as opposed to applied mathematics). I would likely use ...
15
votes
2answers
695 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 ...
12
votes
3answers
2k 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
6answers
2k 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 ...
6
votes
3answers
1k 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 ...
63
votes
33answers
15k 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 ...
6
votes
1answer
587 views

Choosing a fast computer algebra system that works in characteristic p?

Hi all, I want to compute in $\mathbb{F}_q (x)((y))$ i.e. a Laurent series ring over the rational functions over $\mathbb{F}_q$. The computations are fairly basic, but they involve raising to the qth ...
3
votes
1answer
373 views

Decomposition of modules using computer packages

I am interested in computing direct sum decomposition of modules over some quotients of polynomial rings over a field (do not care much about the field at this point). Does any one know a package ...
10
votes
2answers
357 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? ...
0
votes
2answers
105 views

Properties of adjacent submatrixes [closed]

Hi! I've encountered a matrix problem when designing an algorithm, which I cannot seem to figure out. I have a (square) matrix with the following properties: j<k → aij<aik, aji<aki ...
23
votes
4answers
2k 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 ...
2
votes
4answers
2k views

Symmetrical Presentation of 4-Dimensional Rotation Matrix

This question is not urgent; just a matter of curiosity... It is relatively easy to generate an arbitrary 3D or even 4D rotation matrix using conjugation (i.e. YXY−1) of orthogonal rotations. I ...
7
votes
5answers
3k 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 ...
9
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 ...
1
vote
4answers
370 views

CAS for finding closed form solutions to PDEs and SDEs?

Are there any specialized Computer Algebra Systems (or packages for these) for finding closed form solutions to a) partial differential equations, b) stochastic differential equations? If yes, what ...
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 ...
5
votes
4answers
427 views

Sections of a divisor on elliptic curve

I'm interested in producing explicit bases for the sections of a line bundle on an embedded genus 1 curve. Let me restrict to the first case that I don't know how to do, so that I can be as concrete ...
9
votes
8answers
2k 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 ...
82
votes
66answers
14k 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 ...
5
votes
4answers
380 views

Software for rigorous optimization of real polynomials

I am looking for software that can find a global minimum of a polynomial function in R^n over a polyhedral domain (given by some linear inequalities say). The number of variables n is not more than a ...
5
votes
3answers
552 views

Is there a software package that does Schubert Calculus computations?

Is there a good software package for doing computations in the cohomology ring of Grassmannians? Things like, I can write down a polynomial in, in fact, special Schubert classes, but it's one where ...
9
votes
5answers
581 views

is there a good computer package for working with bicomplexes?

I'm interested in working with bicomplexes of modules over polynomial rings, specifically tensoring them together, and the operation of taking cohomology in one direction, and then the other. Is ...
6
votes
1answer
514 views

Is there a good computer package for working with complexes over non-commutative rings?

I'm interested in doing computations with certain non-commutative rings, most of which involve taking derived tensor products. Does anyone know of a computer algebra package which will find ...