# Questions tagged [computational-algebra]

Computational algebra is an area of algebra that seeks efficient algorithms to answer fundamental problems concerning basic algebraic objects (groups, rings, fields, etc.). For questions about generic computer algebra systems, use [tag:computer-algebra-systems].

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### Practical algorithm to calculate power subgroup of a polycyclic group

I am looking for a practical algorithm to calculate the power subgroup $G^n := \langle g^n \mid g \in G \rangle$ of a (possibly infinite) polycyclic group $G$. A theoretical algorithm is given in , ...
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### Using GAP to find coset representatives [closed]

Given a finitely generated group $G$ and a normal subgroup of finite index $K$, how can I use GAP to find a list of coset representatives, and also show that two coset representatives are equal?
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### Smallest possible size of matrix A with row space S

Suppose S has a 6 dimensional subspace of nine-dimensional space R^9. (a) What is the smallest possible size of a matrix A that has row space S? (b) What is the smallest possible size of a matrix B ...
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### Are there established algorithms for working with towers of low-degree algebraic extensions?

I'm interested in doing 'computational ruler-and-compass' construction simulations along the lines of Euclidea and similar tools. Because the constructions can get rather involved, I'd like to be able ...
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### Decide if certain polynomial is in an Ideal

Let $I$ be the ideal $I = (x^3y-x^2y^2,x^3z+z^2yx,x^2-xz) \subset \mathbb{Q}[x,y,z]$. I have to decide if $x$ is part of $I$ or $\sqrt I$. My first take was computing the Groebnerbasis $G$ of $I$ by ...
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### Q: how to describe these results by a descendants tree in gap

I wrote an implement to find the "fullyInvariantGroups" in GAP and the results appeared as below: ...
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### Factoring matrices over $\mathbb{Z}_k$ for $k$ composite?

Say you have some matrix $C\in\mathbb{Z}_k^{n\times m}$, where $k$ is composite, and say the rank of $C$ is $r$. Moreover, say that you have some prior knowledge that $C$ can be written as $C = AB$, ...
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### Finding lcm and gcd of two ideals

I was studying Gröbner bases and I wanted to find $\operatorname{lcm}$ and $\gcd$ of two ideals $\langle x_1^2 + x_2x_3^2 - x_3^2\rangle$ and $\langle x_1x_2+x_2^2-x\rangle$. I know I should find a ...
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### How does GAP calculate 2-closure?

GAP software has a method for calculating the two closure of a (permutation) group? how does it do that calculation?
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### Sum of determinants of block submatrices

I have a $2n \times 2n$ matrix, $M$. I view it a block matrix, of $n^2$ blocks, each of shape $2\times 2$. Computing the determinant of $M$ is easy by conventional methods. I could also look at ...
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### Macaulay2: How to compute the remainder when dividing a polynomial by a set of polynomials (in some order)?

I'm writing Buchberger's Criterion in a program in Macaulay2 to check whether or not the set of polynomials I have form a Grobner basis for the ideal it generates. However, I have not been able to ...
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### Program to find intersection of subgroups of free groups

As the title says, I am working on examples for a research project I'm doing, and I need a way to efficiently calculate the intersection of subgroups of a free group (say, of rank 2). Are there any ...
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### Discriminant of homogeneous polynomials

Let $f$ be a homogeneous polynomial in variables $x,y,z$. Suppose that the sum of coefficients of $\frac{\partial^i f}{\partial x^i}$ is $0$ for each $0 \leq i \leq r$. I believe that, in this ...
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### How can I find an isomorphism between two representations?

Suppose I have two representations (over $\mathbb{C}$) of a finite group $G$, $\rho : G \to GL(V)$ and $\tau : G \to GL(W)$. If I am given that these representations are isomorphic, i.e. there exists ...
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### Library for visualizing computation graph

Does anyone know a tool or library (preferable JavaScript) that can visualize an equation as a computational graph, such that for example the sigmoid function with inputs $\mathbf{w}$, $\mathbf{x}$ ...
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### Near-Near-MDS codes

I am trying to understand the codes which are not maximum distance separable but are at a distance of 2 from being Maximum Distance Separable. I have trying to find articles specifically related to ...
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### Computing geodesics on pseudo-riemannian manifolds

Consider a pseudo-riemannian manifold $M$ with a metric tensor $g$. Now, given two points $p_1, p_2$ in $M$, how do I compute (as in, programatically compute) the geodesic between these two points? ...
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### Is there a group with $2$ generators having exactly $17$ subgroups of index three?

I recently saw a fun problem from a past qualifying exam from Stanford. It is Problem 10, part (b) in this document. I will screenshot the problem and its solution here: My question is the following....
### What does it mean that field $\mathbb{F}_{p^n}$ “contains” the prime field $\mathbb{Z}_p$?
I have read in few books (example Computational Number Theory, page 77) that any extension field $\mathbb{F}_{p^n}$ "contains" as a subfield the prime field $\mathbb{Z}_p$? What exactly does "...