# 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.

268
questions

**9**

votes

**3**answers

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 \...

**9**

votes

**5**answers

617 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 ...

**9**

votes

**0**answers

172 views

### When is cohomology of a finitely presented dg-algebra computable?

Given a smooth affine variety $X$ defined over $\mathbb{Q}$, its singular cohomology is isomorphic to the algebraic de Rham cohomology, which is the cohomology of the complex $\Omega_X^0\to\Omega_X^1\...

**8**

votes

**3**answers

748 views

### Web interface for GAP (or other computer algebra system dealing with finite groups)?

GAP is computer algebra system which allows to make calculations with finite groups. (See wikipedia link for an example).
Is there web interface for it ? (I cannot google it.)
Or may be some other ...

**8**

votes

**6**answers

2k views

### computer algebra system for polynomial algebras over finite fields

Is there a computer algebra system that can do arithmetic over polynomial algebras over finite fields where I can specify the extension?
Exempli gratia, if $f(x), g(x) \in \mathbb{F}_p[\mu]/(m(\mu))[...

**8**

votes

**3**answers

2k 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.
...

**8**

votes

**4**answers

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 ...

**8**

votes

**1**answer

683 views

### How to compute with the Stark conjectures?

I would like a convenient basis for the elements of a fixed abelian extension $E$ of a real quadratic field $\mathbb{Q}(\sqrt{d})$. The accepted answer to this MO question suggests that the Stark ...

**8**

votes

**2**answers

289 views

### Computer Algebra Systems that support variable sized matrices

I'm familiar with sympy, the matlab symbolic package, reduce, and have tried out a few other computer algebra systems. However, as far as I can tell, none of them seem to be able to do algebra on ...

**8**

votes

**3**answers

437 views

### Computing determinants of matrices of linear forms

Suppose we have three $n \times n$ matrices $A$, $B$, $C$ with floating point entries. We would like to compute the polynomial $\det (xA+yB+zC)$. At least in Mathematica, and I think in all computer ...

**8**

votes

**2**answers

931 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 ...

**8**

votes

**1**answer

195 views

### Is there a good computer program for searching for endomorphisms between finite algebras which make diagrams commute? Is this problem NP-complete?

Let $(X,*),(Y,*),(Z,*)$ be finite algebras. The binary operations $*$ are not required to satisfy any identities though I am interested in the special case where $*$ is associative. Suppose that $f:X\...

**8**

votes

**2**answers

194 views

### Is there a CAS that can solve a given system of equations in a finite group algebra $kG$?

Let $k$ be a finite field with char$(k)=p>0$. Let $G$ be a finite group.
Consider the group algebra $kG$.
I would like to solve a given system of equations in $kG$.
Question:
Is there a computer ...

**8**

votes

**1**answer

809 views

### Efficient SVD of a matrix without some of the columns

I have a matrix $A \in \mathbb{R}^{p \times q}$ of rank $r$ and its SVD decomposition, i.e,
$$
A = U S V^\top,
$$
where $U \in \mathbb{R}^{p \times r}$ and $V \in \mathbb{R}^{q \times r}$ are ...

**8**

votes

**1**answer

1k views

### Algorithm to find exponential map of differential operators acting on function

I am trying to write a computer program which computes the action of the exponential of a differential operator on a function, for any given differential operator.
Examples:
$\exp(\varepsilon \...

**8**

votes

**1**answer

824 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 ...

**8**

votes

**1**answer

3k views

### Constructing a unitary matrix

Setting:
Given a set of $n\times n$ matrices $A_i$, I would like to find a linear combination of these matrices $Q = \sum_i A_i x_i$ with $x_i$ a set of complex numbers, such that $Q$ is unitary: $Q^{...

**8**

votes

**1**answer

170 views

### Algebraic Shifting Computer Code

Is anyone aware of computer code that will algebraically shift a simplicial complex (as in this Kalai paper)? Ideally, I am looking for an implementation that can run in something like Sage or ...

**7**

votes

**6**answers

2k views

### Computation of a minimal polynomial

It is relatively easy (but sometimes quite cumbersome) to compute the minimal polynomial of an algebraic number $\alpha$ when $\alpha$ is expressible in radicals. For example, the simple query
"...

**7**

votes

**6**answers

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.

**7**

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 ...

**7**

votes

**4**answers

489 views

### Software for rigorous optimization of real polynomials

I am looking for software that can find a global minimum of a polynomial function over a polyhedral domain (given by, say, some linear inequalities) in $\mathbb R^n$. The number of variables, $n$, is ...

**7**

votes

**3**answers

286 views

### Computer algebra system for Weyl algebra computations

Does anyone have a suggestion for the best computer program to perform calculations in the 2nd Weyl algebra?

**7**

votes

**3**answers

1k views

### Is there a MAGMA function to calculate the absolutely irreducible components of an algebraic curve defined over the rationals?

Given a curve defined over the rationals, is it computationaly possible to find all its absolutely irreducible components?
Is there an implementation of this in the MAGMA program?

**7**

votes

**1**answer

643 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?

**7**

votes

**3**answers

3k views

### Using MAGMA for Group Theory

I've just started a PhD in Group Theory and need to use the computer programme MAGMA. I wonder if anyone could help me with a couple of (probably very basic things).
I need to produce a Hasse diagram ...

**7**

votes

**2**answers

929 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 - abcd}{(1-ac)...

**7**

votes

**1**answer

581 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 ...

**7**

votes

**1**answer

191 views

### Computing homology of subvarieties of Euclidean spaces by persistent homology

Let $M$ be a submanifold of the Euclidean space $\mathbb{R}^n$. Let $G$ be a finite group acting on $M$ freely. I want to compute the homology (or even the cohomology ring) of $M/G$.
Suppose the ...

**7**

votes

**0**answers

108 views

### Classification of Frobenius algebras of small dimensions

Despite (commutative) Frobenius algebras over a field $K$ being a very popular class of algebraic objects, it seems no attempt of classification (up to $K$-algebra isomorphism) for them has been ...

**7**

votes

**0**answers

293 views

### Computer Algebra solution for simplicial resolutions for André-Quillen cohomology

Hello,
I would like to experiment with André-Quillen (co)homology. Especially for singular rings.
A key problem is that the construction of a simplicial resolution of a ring seems to require a rather ...

**7**

votes

**0**answers

490 views

### Where can I find tables of dual canonical basis vectors?

Leclerc (arXiv:math/0209133) has given us an algorithm for computing the dual canonical basis of the upper part of a quantised enveloping algebra.
Now presumably this algorithm has been implemented ...

**6**

votes

**5**answers

1k views

### Computer algebra system for calculation of characteristic polynomial of sparse matrix

I have a $n \times n$ matrix, for which i need to calculate the characteristic polynomial. The matrix is over $GF(2)$, and $n \approx 10^4$. However the matrix is very sparse, with around $ n $ non ...

**6**

votes

**1**answer

259 views

### A centralised website for computational attempts in graph theory and metric geometry?

The set of questions below stems from this question.
1) does a website exist that contains (at least links to) code and data files, with the aim to centralise computational results in graph theory ...

**6**

votes

**3**answers

2k views

### Differential Geometry/General Relativity Computer Algebra

Hi,
could anybody recommend a CAS suited to DG/GR applications such as computation of connection coefficients or generating (and possibly solving) PDEs for, for example, an unknown metric of given ...

**6**

votes

**1**answer

686 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 ...

**6**

votes

**2**answers

474 views

### (Efficient) computation of symmetric powers of square matrices

I'm looking for software that can compute symmetric powers of medium-size square (say rational, 100 by 100) matrices, and ideally can do so efficiently if the matrix is sparse enough. I haven't found ...

**6**

votes

**2**answers

767 views

### Bounds on remainder term of power series of elementary functions

This is mainly a question about the remainder term of power series for elementary functions.
I'm very interested in aspects of calculating or computing elementary operations and functions, by which I ...

**6**

votes

**1**answer

185 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 ...

**6**

votes

**1**answer

263 views

### Check irreducibility of an explicit polynomial, without computer

I have a polynomial of degree 8 in 6 variables given explicitly by
$$ (\sqrt{1+(x_1+x_2+x_3)^2+(y_1+y_2+y_3)^2}+\sqrt{1+x_1^2+y_1^2}+\sqrt{1+x_2^2+y_2^2}+\sqrt{1+x_3^2+y_3^2})\times\text{the other ...

**6**

votes

**1**answer

447 views

### Using math software to show that the following groups are infinite?

I would like to show that the following finitely presented group in 3 generators $P, Q, R$ is infinite in certain cases:
$$P^p, Q^q, R^r, (PQ)^2, (QR)^2, (PQR)^2, (QR^{r/2+1})^a (RQR^{r/2})^b$$
For ...

**6**

votes

**2**answers

2k views

### Computing the fixed field of an automorphism of a function field

Let say we have a function field $k(x,y)$ defined by $f(x,y)$ over $k$, with $\sigma \in Aut(k(x,y)/k)$ and. Suppose, I'm not that out of luck, so that either of $\prod \sigma^i(x)$ or $\sum \sigma^i(...

**6**

votes

**2**answers

143 views

### Is the matrix positive definite given the Gauss-Seidel method converges?

I know that the Gauss-Seidel method converges given that the matrix you want to solve is symmetric positive definite. However, I'm wondering if the "converse" of the statement is true. Namely, if $A$ ...

**6**

votes

**1**answer

488 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 $\mathbb{A}^n$. ...

**6**

votes

**1**answer

1k views

### Quantum Group Calculations in Mathematica

I'm trying to learn how to do algebraic manipulations in Mathematica but not finding the help very helpful. I'm going to ask about a specific quantum group example related to a previous question of ...

**6**

votes

**1**answer

296 views

### Calculating the Ext-algebra with a computer

Given a finite dimensional quiver algebra $A$ over an arbitrary field and a module $M$ of finite injective dimension or finite projective dimension.
Let $B$ be the Ext algebra of $M$, that is $B:=\...

**6**

votes

**1**answer

237 views

### Problem on triangles

Let $T\subset \mathbb{R}^2$ be any triangle and $T^t$ a deformation of $T$. Call $l_1,l_2,l_3$ the squares of the lengths of the sides of $T$ and $l_1^t,l_2^t,l_3^t$ the squares of the lengths of the ...

**6**

votes

**3**answers

419 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 ...

**6**

votes

**1**answer

312 views

### Computing algebraic properties of trace fields, as given by SnapPy

SnapPy can tell you the trace field of a hyperbolic $3$-manifold (which is awesome), but it specifies the field by outputting:
the minimal polynomial of the field over $\mathbb{Q}$, and
a decimal ...

**6**

votes

**1**answer

118 views

### Software for computing equivariants

If $\Gamma$ is a finite group with action on two vector spaces $\mathbb R^n$ and $\mathbb R^m$ denoted by $\gamma_n$ and $\gamma_m$ respectively, the fundamental equivariants are the polynomials $f: \...