# 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.

266
questions

**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

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**4**answers

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

**2**answers

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

**4**answers

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

**0**answers

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

**2**answers

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

**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 ...

**3**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**4**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**1**answer

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

**5**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**34**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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},\...