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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8
votes
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
2answers
190 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 ...
9
votes
2answers
429 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 ...
1
vote
1answer
87 views

Problem while multiplying under a set of relators [closed]

I have defined $S_4$ (Symmetric group of order 4), and with the base field $Z_5$, groupring $Z_5S_4$ is constructed. Then I have taken two elements of this group ring and I want to multiply them to ...
4
votes
1answer
139 views

Is there a point in 6-dimensional space satisfying these polynomial inequalities?

I would like to know if there is a point $(a, b, p, q, x, y) \in [0,1]^6$ satisfying the following collection of inequalities. $b \ge a$ $q \ge p$ $y \ge x$ $a \ge p \ge a^2$ $b \ge q \ge b^2$ $p \ge ...
2
votes
0answers
102 views

Computing whether a set of polynomials cuts out a projective variety

I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a projective variety, i.e. whether the radical of the ideal $I$ that they generate is homogeneous....
1
vote
0answers
92 views

Saturated ideals in computational algebra

Let $R$ a commutative ring with one and $I, J \triangleleft R$ two ideals. The saturated ideal $I^{sat}_J$ with respect $J$ is the ideal $$(I : J^\infty )= \cup_{n \geq 1} (I:J^n)$$ where $(I:J^n)= \{...
7
votes
1answer
190 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 ...
0
votes
0answers
94 views

Stabilizers in the action of $\mathrm{GL}(n, \mathbb Z)$ on $\mathbb Z^n$

How can we calculate effectively the subgroups of $G: = \mathrm{GL}(n, \mathbb Z)$ which fix pointwise a given submodule $S$ of $\mathbb Z^n$ in the action of $G$ on $\mathbb Z^n$ by left ...
5
votes
4answers
5k 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 ...
6
votes
2answers
127 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 $A$ ...
21
votes
4answers
870 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 ...
2
votes
0answers
67 views

Affine projection of polynomials for a given set of points

(Not sure this question fits here, I will remove it in case it doesn't) Let $F_{\mathrm{ML}}[x_1, \ldots,x_n]$ denote the set of multilinear polynomials over a finite field $F=\mathbb{F}_q$ (i.e. a ...
3
votes
2answers
292 views

Alternate descriptions of finite fields

The finite field of order $p^n$ is isomorphic to $(\mathbb Z/p \mathbb Z)[X]/(P)$, where $P$ is an irreducible polynomial in $(\mathbb Z/p \mathbb Z)[X]$ of degree $n$. This describes every finite ...
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 ...
3
votes
0answers
61 views

Frobenius algebras of small dimensions

In Classification of commutative Frobenius algebras , Jeremy Rickard showed that there are infinitely many commutative (local without loss of generality) Frobenius algebras of vector space dimension ...
7
votes
0answers
107 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 ...
0
votes
0answers
54 views

Low rank approximation

Can we solve low rank approximation problem by using concept of Gröbner basis? I was trying to find it by Macaulay2 but didn't find the answer. I was trying to do by toric ideals as for them Gröbner ...
5
votes
0answers
98 views

Rank of matrix over UFD polynomial ring

I have a matrix, $M$, size of approximately $20\times 25$, over a polynomial ring $\mathbb{Q}[x_1, \cdots, x_n]$ and a number $r$ such that I would like to test whether or not there exist values for ...
28
votes
1answer
1k views

Algebraic dependency over $\mathbb{F}_{2}$

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$ such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall i\in[n]:f_{i}(a)=a_{i}$....
11
votes
1answer
392 views

Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials

Let $n>1$ and $p$ be an odd prime with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. Notice that, since $p-1 \mid n-1$, we have $T^p - T \mid T^n-T$ in $\mathbb{F}_p[T]...
4
votes
0answers
164 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 ...
0
votes
1answer
95 views

IntersectInP bug of Macaulay2 [closed]

I am trying to use the intersectInP command in Macaulay2, inside package ReesAlgebra. However, I tried to follow the exact code in the user-guide, but it doesn't run in my Ubuntu app (of win 10). Can ...
15
votes
4answers
3k 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.
5
votes
2answers
175 views

Obtaining quiver and relations for finite p-groups

Given a finite field $K$ with $p$ elements and a finite $p$-group $G$, is there a way to obtain the quiver and relations of $KG$ with GAP (and its package QPA)? Since $KG$ is local, the quiver should ...
4
votes
1answer
139 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 ...
2
votes
0answers
45 views

Determining the symmetry group of a system of ODEs as tensor product decomposition

Suppose you have a linear system of ODEs in implicit form represented by an $n \times n$ matrix $M$ and you are trying to identify a subgroup of $\mathrm{GL}(n)$ that is the symmetry group of this ...
5
votes
2answers
498 views

Finding relations between invariant polynomials

Suppose I have an action of a linear reductive group ($GL(2,\mathbb{C})^2$ in this case) on a complex vector space (of dimension $16$) and I want to compute explicitly the ring of invariants of this ...
2
votes
1answer
128 views

Intersection of a $\mathbb{Q}$-affine space with $\mathbb{Z}^n$

Let $E$, a $\mathbb{Q}$-affine space of arbitrary dimension included in $\mathbb{Q}^n$. Is it possible to check efficiently if $E \cap \mathbb{Z}^n$ is empty or not? If is an hard problem could give ...
2
votes
0answers
67 views

CAS for finite-dimensional complex representations of $S_n$

Does there exist a computer algebra system that can work with finite-dimensional complex representations of the symmetric groups on finitely many letters? It should have the following functionality: (...
1
vote
0answers
62 views

Is there a workable numerical method for determining the center of a circle through three points? [closed]

I'm a 73-year-old engineer struggling with numerically implementing a math problem. I am working on a kinematic linkage project that generates motion paths (as long sequences of x,y coordinates) of ...
3
votes
1answer
336 views

Special linear Diophantine system - is it solvable in general?

Background: An equivalent question was asked on MSE almost two years before this post now. It was never fully resolved. - Here, we are asking if further progress can be made. Motivation Solving this ...
0
votes
0answers
137 views

Quantifier elimination with no free variables and real polyhedral inequalities

In this introductory blog post https://cstheory.blogoverflow.com/2011/11/something-you-should-know-about-quantifier-elimination-part-i/ it is mentioned in the very last line that "I do not know ...
3
votes
1answer
308 views

Lower bound for polyhedral real quantifier elimination

All known examples for double exponential lower bounds for real quantifier elimination involves polynomial inequalities with degree $>1$. Is there an example of double exponentiality with ...
1
vote
0answers
68 views

Degree bounds on coordinates of points in a zero-dimensional variety

Let $S = \{f_1, \dots, f_s \in \mathbb{Q}[x_1, \dots, x_n]\}$ have a zero-dimensional nullset $V \subset \mathbb{C}^n$, and suppose that each $f_i$ has total degree at most $d$. Is there a shared ...
43
votes
3answers
4k 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 ...
11
votes
1answer
1k views

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) &...
16
votes
5answers
4k 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 ...
21
votes
1answer
1k views

Main open computational problems in quantifier elimination?

A language is said to have quantifier elimination if every first-order-logic sentence in the language can be shown to be equivalent to a quantifier-free sentence, i.e., a sentence without any $\forall$...
5
votes
0answers
109 views

Stable equivalence and stable Auslander algebras

Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules. Recall that the Auslander algebra of $A$ is $End_A(M)$ and the ...
2
votes
0answers
57 views

Constructing stable equivalences for finite dimensional algebras

Given a finite dimensional (non-selfinjective) algebra $A$. Is there a method (for example using QPA) to construct algebras stable equivalent to $A$? Such a thing is easily possible for derived ...
128
votes
34answers
31k 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 ...
5
votes
2answers
301 views

What are the most general methods for solving equations in closed form with Lambert W?

What are the most general methods for solving equations with help of Lambert W function or with a generalization of Lambert W function in closed form? I gave a method in MSE here. Which algorithms ...
21
votes
2answers
777 views

What is currently feasible in invariant theory for binary forms?

When Paul Gordan became a professor in 1875 he could show the binary form in any degree has some finite complete system of (general linear) invariants, but he could not actually give a complete system ...
4
votes
0answers
78 views

Deciding whether two algebras are derived equivalent

Given two finite dimensional quiver algebras $A$ and $B$ (over a nice field in case that helps, for example a finite field). Question: Can an there be a finite algorithm that decides whether $A$ ...
4
votes
2answers
213 views

On Auslander algebras

Given a connected quiver algebra $A$ that is representation finite, the Auslander algebra $B_A$ of $A$ is defined as the endomorphism ring of the direct sum of each indecomposable $A$-module. It is ...
5
votes
1answer
139 views

Ideals of commutative Frobenius algebras

Given a finite dimensional commutative (connected=local) Frobenius algebra $A$ over a field $K$. Question 1: Does $A$ have only finitely many ideals? (the answer should be no in the non-commutative ...
4
votes
1answer
230 views

Higher roots modulo prime complexity best algorithm

Given integers $a,\ell$ and prime $p$ we need to find the roots of the algebraic equation $x^\ell\equiv a\bmod p$. We know there are at most $\ell$ such $x$. What is the best method to find all such ...
1
vote
2answers
238 views

Finding all submodules

Given a finite dimensional local commutative algebra over a finite field $K$ and a finite dimensional module $M$. What is the fastest/best way to obtain all submodule from $M$ using a Computer algebra ...
0
votes
1answer
98 views

Quantifier elimination and where is this quantified convex program in the polynomial hierarchy?

I have a quantified convex program of the form that I need to solve $$\exists(x_{1,1},\dots,x_{1,n})\in\mathbb R^n\quad\forall(x_{2,1},\dots,x_{2,n})\in\mathbb R^n$$ $$\vdots$$ $$\exists(x_{2t-1,1},\...

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