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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1answer
41 views

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 [1], ...
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1answer
41 views

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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63 views

Solving a system of polynomial equation - can I trust numerical results?

To finish a proof, I need to solve a system of two polynomials with integer coefficients in two variables, $\{F_1(x,y)=0,\,F_2(x,y)=0\}$, and then show that no solutions satisfy $0<x<1$ and $y&...
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44 views

Calculating differentials between cell velocities

If I have a 2D cell-based fluid simulator, which uses velocities and pressure, how can I find the change in pressure between the neighboring cells for a cell? I might be missing something big here, ...
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0answers
53 views

Computing whether a set of polynomials cuts out a homogeneous variety

I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a homogeneous variety. My first idea is to compute the radical of the ideal $I$ that they ...
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53 views

Why is it so computationally hard to determine group isomorphism?

Finding an isomorphism requires to show that for 2 groups $G$ and $H$, there exists a bijective map $\phi : G\to H$ such that $$\phi(ab)=\phi(a)\phi(b)$$ For all $a,b \in G$. This is (probably naively)...
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0answers
22 views

How can I easily find character table of Sergeev group (finite)?

I am looking for the character table of the Sergeev group S_d for small d (say, 'd' up to 10 or up to whatever is possible). The Sergeev group $S_d$ is defined as follows: Let $\mathfrak{S}_d$ be ...
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2answers
22 views

Help with recursive function

I need some help understanting how the following conclusion was made: We have the recursive function: $ε_n=-n \cdot ε_{n-1}$ How do we come to the conclusion that $ε_n=(-1)^{n-1}\cdot n!\cdot ε_1$
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1answer
137 views

Is all group theory permutation group theory?

By Cayley's theorem every abstract group is isomorphic to some permutation group. Since the permutation group viewpoint has the advantage of considering the actions of the group on different sets, and ...
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1answer
30 views

Extending a map to a homomorphism — can this algorithm produce a false positive?

Consider the following computational problem: Let $G,G'$ be groups so that $G$ is finite and generated by $X=\left\{g_1,\ldots,g_n\right\}$. Let $f:X \to G'$ be a map. Decide whether $f$ extends to a ...
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1answer
28 views

Solving a linear system of equations with constraints

Q) I have a finite state space $S$ of size $n$ and $f:S\to \mathbb{R}$. $A,B\subset S$. $L$ is a $n\times n$ matrix such that all row sums = $0$. Also $f(A)=0$ and $f(B)=1$. I am trying to find a $...
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37 views

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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39 views

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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1answer
71 views

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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2answers
122 views

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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0answers
37 views

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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43 views

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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1answer
46 views

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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1answer
37 views

How generate a algebraicaly independent set over rational number field?

Algebraic independence. In abstract algebra, a subset ${\displaystyle S}$ of a field ${\displaystyle }$L is algebraically independent over a subfield ${\displaystyle }$K if the elements of ${\...
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1answer
39 views

normal form property of grobner basis

I am not clear on how why this proof works. I understand that for any $p$ this algorithm gives us back a $\overline{p}$ such that: $p$ and $\overline{p}$ are congruent mod $I$, and only standard ...
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1answer
35 views

Is there any software that I can use to determine whether matrix group cosets are equal?

Is there any software that I can use to determine whether matrix group cosets are equal? For instance, if I'm working with the group $SL_{2}( \mathbb{F}_{p} [[t]])$ and I want to know if $a SL_{2}( \...
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1answer
233 views

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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2answers
211 views

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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2answers
117 views

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 ...
6
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1answer
275 views

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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2answers
359 views

Boolean Polynomial

It is possible to convert a logical expression into elementary algebra. That is, there are substitutions that convert an expression such as $$ \left( \left(a \rightarrow b\right) \wedge \left(c \...
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24 views

How to manually write an element of group ring in GAP. [duplicate]

Suppose, we consider R:=GL(29); (Galois field of order $29$) G:=SmallGroup(7,1); (cyclic group of order $7$) H:=Group Ring(R, G); (this generates the group ring). Suppose that GAP is unable to fetch ...
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26 views

BPP(complexity) with binary form of number

For any language $L \subseteq \mathbb{B^{*}}$ we define language $L^{log}$ as set $\{\overline{a}\overline{b} | \overline{a} \in L, \, \overline{b} - \text{binary form of number}\,\, |\overline{a}|\}.$...
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0answers
55 views

BPP (complexity)

For any language $L \subseteq \mathbb{B^{*}}$ we define language $2 \cdot L$ as set $\{2 \cdot \overline{a} | \overline{a} \in L\}$, where $2$ in binary form equals $10$,and $\cdot$ - is ...
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1answer
47 views

Nilpotent Quotient Algorithm

Does anyone recommend me any reference on the 'Nilpotent quotient algorithm' another book other than D. Johnson "Presentation of groups"? I think that Johnson's example is a bit confusing. Thanks in ...
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1answer
344 views

How to calculate flops of matrix operations? [closed]

I am looking for a way or method to calculate flops of matrix operations, like sum or subtraction, multiplication, inverse, Singular Value Decomposition (SVD) operations and others, but I don't find ...
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0answers
48 views

Polynomial ring as a module over ring of invariants

Let $\mathbf{k}$ be a field of characteristic 0 and $S=\mathbf{k}[X_1,\cdots,X_n]$ be a polynomial algebra over $\mathbf{k}$. Let $G\subset GL_n(\mathbf{k})$ act linearly on $S$ and $R = S^G$ be the ...
2
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2answers
100 views

Subgroups of $\operatorname{GL}_2(\mathbb Z/8\mathbb Z)$

Is there some program or a location which would allow me to work and calculate with the subgroups of the group $\operatorname{GL}_2(\mathbb Z/8\mathbb Z)$?
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27 views

How to compute a unitary representation of finite group isomorphic to a given rep?

Suppose I am given some representation of a finite group: $\rho : G \to \text{GL}(n, \mathbb{C})$. I want to compute a unitary representation $\tau$ which is isomorphic to $\rho$. I know about Weyl's ...
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2answers
112 views

Algebraically independent polynomials iff linearly independent differentials

This is an exercise question in Appendix A of Introduction of Algebraic Geometry, Justin R Smith. I am looking for an intuition for the solution. if $k \rightarrow K$ is an extension of fields of ...
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0answers
89 views

Finding the numerical listed string to the center algorithm.

Suppose we have a string that occupies a length of seven characters, and the number of strings altogether is ten. We include the hamming distance of three characters. If we calculate the exact amount ...
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0answers
14 views

Proof Using Strong Reducibility

I read the following proof so many times and I could not, for the life of me, figure out how the professor went from (1) to (2). I know to prove $\overline{K} \le A$ , where $\overline{K}=\{x:\...
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0answers
31 views

Schur Index for Quaternion Algebra

I learned form this question and this answer that Schur index in GAP can be found using LoadPackage("wedderga") the functions "SchurIndex". But I am working on the field $K=\mathbb Q (\sqrt{-39})$ ...
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1answer
116 views

Cartesian Product over a list of objects in MAGMA

I'm currently trying to create the Cartesian Product of certain objects (namely: Character Tables of different finite groups). However, it seems like I don't really understand the car<$...$> ...
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0answers
156 views

Ideas for Undergrad Research Project

I'm in my second year of my maths undergrad course and hoping to do a 6 week research project this summer. I'm interested in doing a project wherein I apply a maths topic, (i.e Manifolds, Metric ...
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0answers
49 views

How to compute rational points on a projective variety in Macaulay2

I know that the package "rationalPoints" will compute rational points on an affine variety over a finite field, but I would like to do the same for varieties in projective space. Is there a built-in ...
3
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0answers
55 views

How to compute the number of equivalence classes under the relation $a\sim b\iff a=x^m b x^n$?

Let $G$ be a finite group. Fix an element $x\in G$, and denote by $\sim$ the equivalence relation on $G$ given by $a\sim b \iff \exists m,n\text{ such that }a=x^m b x^n$. Example: Let $G=\langle(123),...
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0answers
37 views

Computation with infinite Weyl algebra

My question here is very computational. My problem is in mathematical physics, so I want to ask the community what kind of software they use to do the following computation if there is any? Let $$L_{...
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2answers
116 views

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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1answer
74 views

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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0answers
44 views

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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1answer
91 views

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? ...
9
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1answer
174 views

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....
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3answers
105 views

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 "...
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2answers
118 views

Probability of ending up with a particular card on top when placing random cards in random spots

This is a probability problem I encountered today. What algorithm or equation solves the situation below generically? There is an unlimited supply of playing cards. There are 10 spots on the table to ...

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