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.
368
questions
5
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Expressing a element of a Matrix subgroup in terms of subgroup generators
I'm no (computational) algebraist, and my searches have been pretty unyielding (probably due to the vast amounts written on the key words), but perhaps someone may know if this is possible, and if so, ...
8
votes
2
answers
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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 ...
4
votes
0
answers
245
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Algorithm/denominators of elements of a rational affine space
I hope it's not a trivial question... Suppose I have a finite dimensional vector space $V$ over $\mathbb{Q}$ with a distinguished basis (in my case it's the $k$th graded piece of the free associative ...
8
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3
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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?
16
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3
answers
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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 ...
2
votes
0
answers
254
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Efficient computing critical points of algebraic function involved radical expression
I am interested in finding local optima of an algebraic function $f(X,Y)$. Suppose, that this expression involves radicals, for example $f(X,Y)= \frac{1}{2}(X+Y)-\sqrt{XY}$. The approach in which i am ...
1
vote
0
answers
238
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How to ask Magma to compute the induced morphisim on divisor group
Suppose Magma has computed homomorphism $h$ between function fields $F1 \to F2$. Then we have an induced homomorphism $h$ on the divisor group. Now my question is that if there's a better way to ...
7
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0
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517
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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
1
answer
527
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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 $\mathbb{A}^n$. ...
11
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1
answer
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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) &...
0
votes
1
answer
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the maximal length of a special dicksonian sequence
First, we define a sequence $t_{1},t_{2},\cdots,t_{k}$ of n-tuples dicksonian, if $\forall 1\leq i < j\leq k,$ there does not exist a non-negative n-tuple t such that
$t_{i}+t=t_{j}.$ For example, ...
13
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4
answers
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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 ...
3
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0
answers
338
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Rank of Subgroup of Elliptic Curve
I'm currently looking at two rational points $ p, q $ on an elliptic curve $E$ over $ \mathbb{Q} $. SAGE tells me that $E$ has rank 5 and no torsion, and that $p$ and $q$ both have infinite order. ...
6
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3
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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 ...
10
votes
5
answers
3k
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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 ...
43
votes
3
answers
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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 ...
4
votes
2
answers
431
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Any CAS that deals with the word problem
I'm new to the combinatorial group theory, so maybe my question is a bit naiive.
I know that the word problem is generally "unsolvable". On the other hand there are specific cases, when the problem ...
8
votes
4
answers
6k
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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 ...
1
vote
1
answer
290
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Computing the connected component without primary decomposition
Given an algebraically closed field $\mathbb{F}$ of characteristic $0$ and a closed subgroup $G$ of $GL_n(\mathbb{F})$. Let $\{g_1,\ldots, g_r\}$ be a Gr"obner basis for the correpsonding ideal $\...
4
votes
2
answers
419
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Is there a free action on a given variety?
Given a variety $V$, and a prime $p$ I want to decide if there is a free action of $\mathbb{Z}/p\mathbb{Z}$ on $V$, and to find the generator of an action if it exists. Is there a known algorithm to ...
3
votes
3
answers
343
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Computational solutions to families of systems of linear equations
Question
Does there exist a computer package that will solve families of systems of linear equations over a field of prime characteristic?
An Example
Suppose I wanted to know when the following ...
17
votes
2
answers
3k
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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 ...
8
votes
1
answer
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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 ...
5
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2
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Algorithm for Weierstrass Preparation Theorem for Formal Power Series
The Weierstrass preparation theorem for formal power series rings guarantees that if a given formal series $f(z) = \sum a_k z^k \in R[[z]]$ where $R$ is a complete local ring with maximal ideal $M$ ...
4
votes
2
answers
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Finding generators of subalgebra of polynomial algebra $K[x_1,\cdots,x_n]$ that are invariant under the action of symmetric group
Let $I =\langle f_1,\cdots,f_m\rangle \subset K[x_1,\cdots,x_n]$be an ideal,
where $f_k\in K[x_1,\cdots,x_n].$
$K[e_1,\cdots,e_n]$ the polynomial algebra generated by the elementary symmetric ...
-2
votes
1
answer
438
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Mul + div using only add/sub ? [closed]
In an algorithm book once the first example was how to compute a multiplication in a loop (only that, so I just remembered, and wanted to do it programmatically but with all operations)
...
13
votes
1
answer
888
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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 ...
7
votes
2
answers
845
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 ...
4
votes
2
answers
738
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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
index ...
6
votes
3
answers
425
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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 ...
9
votes
1
answer
676
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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 ...
6
votes
3
answers
460
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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 ...
7
votes
1
answer
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Computer power in plane geometry
I often hear that modern computer programs "may prove any theorem in elementary Euclidean geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for ...
3
votes
1
answer
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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 ...
7
votes
6
answers
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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.
17
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4
answers
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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 ...
50
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5
answers
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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 ...
8
votes
3
answers
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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
1
answer
495
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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
1
answer
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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?
2
votes
1
answer
524
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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,...
8
votes
4
answers
3k
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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
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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-...
40
votes
23
answers
16k
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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 ...
20
votes
2
answers
1k
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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 ...
14
votes
3
answers
3k
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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 ...
17
votes
6
answers
3k
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"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 ...
8
votes
3
answers
2k
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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 ...
149
votes
38
answers
38k
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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 ...
3
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2
answers
2k
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Computer program to solve a system of polynomial equations over a finite field
I have a set of polynomial equations for which I want to know the solutions (actually really the number of solutions). It would be great if I could get a computer to do it, but I'm not sure exactly ...