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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9
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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 \...
9
votes
5answers
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
0answers
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
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3answers
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
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6answers
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
3answers
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
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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 ...
8
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1answer
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
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2answers
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
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3answers
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
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2answers
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
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1answer
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
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2answers
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
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1answer
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
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1answer
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
1answer
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
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1answer
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
1answer
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
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6answers
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
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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.
7
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3answers
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
4answers
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
3answers
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
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3answers
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
1answer
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
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3answers
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
2answers
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
1answer
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
1answer
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
0answers
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
0answers
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
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0answers
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
5answers
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
1answer
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
3answers
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
1answer
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
2answers
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
2answers
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
1answer
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
1answer
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
1answer
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
2answers
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
2answers
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
1answer
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
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1answer
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
1answer
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
1answer
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
3answers
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
1answer
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
1answer
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: \...