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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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Calculate azimuth and pitch angle from total angle and direction

I am looking for a way to compute the azimuth and pitch angles from a system where I only know the total angle and I know the circular direction of the angle. Let $a =$ azimuth angle and $b =$ pitch ...
tyobrien's user avatar
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Dimension of subspace of linear space of monomial.

Let $f_1$ and $f_2$ are homogeneous polynomials in $\mathbb{Q}[x_0,...,x_4]$ of degree $m$ and $n$ respectively. Let us fix a integer $d\geq m,n$. Let $W$ be the linear space of the collection of $...
Vector's user avatar
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Radical computations in Macaulay2

I'm trying to learn how to compute the radical of polynomial ideals in multiple variables over the real numbers in Macaulay2. From what I've gathered in some stack exchange posts([1], [2]), I just ...
User20354's user avatar
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Computing whether two finite groups are isomorphic (in C++) [closed]

I need to algorithmically compute whether two given finite groups are isomorphic. Usually I only have generators of these groups. The groups can get quite large as I'm working with subgroups of $S_{32}...
H-a-y-K's user avatar
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For some finite group $G$ assume $X,Y \subset G, X \cap Y = \emptyset$. Compute subsets of $X$ and $Y$ that generate isomorphic subgroups of $G$.

For some finite group $G$ assume $X,Y \subset G, X \cap Y = \emptyset$. I need to compute subsets $X' \subset X$ and $Y' \subset Y$ that generate isomorphic subgroups of $G$: $\langle X' \rangle \leq ...
H-a-y-K's user avatar
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For $M,C\subset S_n,|M|=|C|$ find subsets of $M$ and $C$ that generate isomorphic subgroups of $S_n$ and the isomorphism maps these subsets together

I have sets $M,C \subset S_n$, s.t. $|M| = |C| \gg 1$. Given $a \in S_n$ I can determine whether $a \in M$ but I have no way to determine whether $a \in C$, however, if we assume C is numbered for ...
H-a-y-K's user avatar
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Detect linearly dependent columns from a full-row rank matrix.

Let $A\in \mathbb{R}^{m\times n}$, with $m\leq n$, be a full-row-rank matrix. Then, there exist a collection of $n-m$ columns in $A$ that are linearly dependent on the other columns. What is the most ...
Gino's user avatar
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How to compute $\operatorname{Length}(k[[x_1,\dots,x_n]]/I)$ for some ideal $I$

Let $k$ be an algebraically closed field and $\mathcal{O}_0=k[[x_1,\dots,x_n]]$ be the ring of formal power series, $I$ be an ideal of $\mathcal{O}_0$ such that $\operatorname{Spec}(\mathcal{O}_0/I)$ ...
Peter Wu's user avatar
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Correctness of algorithm to find the number of elements of order $x$ in Symmetric Group $n$?

To find the number of elements of order $x$ in $S_n$ Generate all possible partitions of $n$ by divisors of $x$. For each partition, check if the LCM of the part lengths matches $x$. Calculate the ...
A. Random's user avatar
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Why does the F4 Algorithm return a Gröbner Basis?

I am currently trying to understand the F4 algorithm. I am working with the book „Ideals, Varieties and Algorithms“ from Cox et al and have problems with understanding their proof for the correctness ...
user1315365's user avatar
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2 answers
64 views

solving a system equations with Groebner basis

We have the equations: $$ x+y+z=3\\x^2+y^2+z^2=5 \\x^3+y^3+z^3=7 $$ Using Groebner bases techniques, i want to find : $$x^5+y^5+z^5 \\and\\ x^6+y^6+z^6$$ \ What I did was to find a Groebner basis, in ...
maths18's user avatar
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Requesting for the Reference of multiplicative Property for Resultants [duplicate]

I have learned the definition of the Resultant of two polynomials Resultant of two polynomials. Following this definition, I want to see the proof of one property described in the "Characterizing ...
Afntu's user avatar
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Resultant $\mathrm{Res}_{x}(f(x), y - g(x))$ calculation and divisibility by $f$

I have learned the definition of the Resultant of two polynomials Resultant of two polynomials. In most places, it is defined over a field. Can we similarly define it for the general ring, for example,...
Afntu's user avatar
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How to randomly construct a long subgroup chain for $S_n$

Description of the problem My task is to build an optimal method that randomly constructs a proper subgroup for a given $G \leq S_n$. Here the term "random" is applied loosely and there is ...
H-a-y-K's user avatar
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Homomorphism between vector spaces in MAGMA

I've recently started to use Magma, and I'm stuck with the following problem. Let $\mathbb{F}_q$ be a quadratic extension of $\mathbb{F}_p$, $p$ prime and let $t$ be the generator of the extension. ...
MJane's user avatar
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2 votes
1 answer
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What is the time complexity of multiplying two matrices over an arbitrary ring?

I know that the time complexity of matrix multiplication over a field is well studied (multiplying two $n \times n$ matrices can be done in $n^\omega$ field operations, where $\omega$ is the matrix ...
GHPR's user avatar
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Computationally evaluating messy symbolic sums involving geometric series

Let $t$ be a positive integer, let $p$ be a prime number, and let $q$ be a real number. I need to evaluate the sum $$ \sum_{\substack{1 \leq c \leq t \\ c \not \equiv 1 \pmod{p}}} q^{-\big((p-2)c + \...
Sebastian Monnet's user avatar
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Is there an algorithm like primary decomposition of an ideal that also assumes there are no zero divisors?

I have a large system of polynomial equations and am trying to construct an irreducible decomposition of the corresponding variety. It's large enough that feeding it to the standard radical/primary ...
Brent Baccala's user avatar
1 vote
0 answers
52 views

Why are eigenvalues in descending order when using QR Algorithm

I have been experimenting with the QR algorithm for eigenvalue computation and have come across a curious phenomenon. The resulting matrix seems to sort the eigenvalues in descending order from the ...
MPP's user avatar
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Requesting Polynomial Systems of Equations

I am teaching a course in commutative algebra, and it includes a project where the students research on a particular topic, solve a small problem and present it to the class. I usually give my ...
Grothendieck Ring's user avatar
5 votes
1 answer
82 views

Asymptotic density of certain class of finite groups (Solvable, Nilpotent, $p$-Group, etc).

I read that there is a conjecture that most groups are $2$-groups. This conjecture comes from the fact that by Higman-Sims asymptotic formula, $\#$ of $p$-group of order $p^k= p^{\frac{2}{27}k^3 + O(\...
Leon Kim's user avatar
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1 answer
169 views

Cox, Little O'Shea - Ideals, Varieties and Algorithms - Exercise 2.4.9

As the title anticipates, I have a problem with Exercise 2.4.9 by Cox et al. If $I=\langle x^{\alpha(1)},\dots,x^{\alpha(s)} \rangle$ is a monomial ideal, prove that a polynomial $f$ is in $I$ if and ...
TheWanderer's user avatar
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4 votes
1 answer
775 views

Raise a Matrix to Arbitrary Power

I have a $k\times k$ matrix $$ A_{k}= \begin{pmatrix} 1 & 1 & \cdots & 1 & 1 & 1 \\ 1 & 1 & \cdots & 1 &1 & 0\\ &\vdots & &\vdots \\ 1 & 1 &...
Apple's user avatar
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Uniqueness of quotients when reducing with Gröbner basis

Let $K$ be a field, and let $G = (g_1, \ldots, g_m)$ be a Gröbner basis in $K[x_1, \ldots, x_n]$ (i.e. $G$ is a Gröbner basis for the ideal it generates). By Adams, Loustaunau - An Introduction to ...
Adelhart's user avatar
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Is there a simple way mathematically to convert $X$ and $Y$ into $1$ where $X=Y$, and $0$ where $X \neq Y$

Given two integers, $X$ and $Y$, is there a simple mathematical expression that can be performed on them that resolves to $1$ when $X=Y$ and to $0$ when $X \neq Y$? Something that can be expressed ...
Kurt Fitzner's user avatar
2 votes
1 answer
92 views

How many non-isomorphic groups of order $5832 = 2^3 \cdot 3^6$ are there?

I'm afraid I can't provide much motivation other than personal interest. I have found David Burrell's very recent Ph.D. thesis, which identified a transcription error that resulted in an incorrect ...
Erick Wong's user avatar
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4 votes
0 answers
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Is there a way to computationally verify that the sporadic groups are simple?

I'm trying to understand the "easy" direction of the CFSG: namely, the proofs that the 18 infinite families and 26/27 sporadic groups are indeed simple. I'm working through Simple Groups of ...
Max Packer's user avatar
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49 views

Multiple differentiation of chain rule $\frac{d^n}{dx^n} f(g(x))$ computationally

The question was to find a simplified expression for $$\frac{d^n}{dx^n} f(g(x))$$ I was working to find some types of recursion relationship $$\frac{d^n}{dx^n} f(g(x))= \frac{d^{n-1}}{dx^{n-1}} (f'(g(...
ShoutOutAndCalculate's user avatar
2 votes
1 answer
238 views

How to show an ideal is prime in $F_p[x_1,...,x_4]$

Let $p$ be a prime number and $$I:=(x_1+x_2+x_3+x_4+x_1 x_2+x_1 x_3+x_1 x_4+x_2 x_3+x_2 x_4+x_3 x_4,x_1 x_2 x_3+x_1 x_2 x_4+x_1 x_3 x_4+x_2 x_3 x_4+x_1 x_2 x_3 x_4)$$ be an ideal of $F_p[x_1,x_2,x_3,...
Vector's user avatar
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0 answers
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Does ideal reduction commute with intersection?

Let $\mathbb{Z}[T_1,...,T_n]$ be the ring of polynomials over $\mathbb{Z}$, and $\mathbb{F}_p[T_1,...,T_n]$ be the ring of polynomials over $\mathbb{F}_p$, with the canonical projection $p:\mathbb{Z}[...
Vector's user avatar
  • 277
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0 answers
62 views

Structure of the Jacobson radical of the Group Algebra .

Is it possible to find the structure description of the Jacobson radical $J(FG)$ of a group algebra FG, where F and G are finite field and group respectively in GAP? I choose the group algebra $F_3D_{...
neelkanth's user avatar
  • 6,100
0 votes
1 answer
94 views

The normalized unit group using GAP.

I want the structure of The normalized unit group using GAP for the group algebra $FD_{30}$, where $F$ is a finite field with characteristic $3$ and $D_{30}$ is the dihedral group of order $30.$ I ...
neelkanth's user avatar
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1 vote
0 answers
46 views

Unit group structure GAP code. [duplicate]

I want the structure of the unit group of the group algebra $F_{3^k}D_{30}$ using GAP, where $F_{3^k}$ is any finite field of characteristic $3$ and $D_{30}$ is the dihedral group of order $30.$ I ...
neelkanth's user avatar
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0 votes
0 answers
50 views

Algorithm for the maximal isotropic subspace

Does anyone know of any algorithms that exist which can explicitly compute a maximal isotropic subspace of a diagonal quadratic form over the rationals? I have been searching through the literature, ...
scqueen's user avatar
  • 23
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0 answers
57 views

How to Compute a nonzero point $v= \langle v_x, v_y \rangle$ of Nodal Curve

This is not homework question. I am writing a research paper and studying the behaviors of complete algebraic curves and I came across this questions and I am interested in it. A nodal function is ...
holala's user avatar
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2 votes
1 answer
247 views

Orbit-Stabilizer problem for $GL(\mathbb Q,n)$ (Algorithmic approach)

The paper [1, section 1] mentions that the Orbit-Stabilizer problem is undecidable for general matrix groups. So my question is if the statement means the problem is undecidable for $G=GL(\mathbb Q,n)$...
mari's user avatar
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1 vote
0 answers
67 views

Factorization of quartic forms

Consider (complex) quartic forms of three variables that can be factored as product of quadratic forms. Being the image of a regular map from $\mathbb{P}(S_2) \times \mathbb{P}(S_2)$ to $\mathbb{P}(...
fyx1123581347's user avatar
1 vote
0 answers
26 views

Restricting Characters on Sage

I am using Sage to obtain the character table of different permutation groups using the command G.character_table(). Is there any implemented command in Sage that restricts an irreducible character (...
user1070911's user avatar
6 votes
3 answers
188 views

Sylow $2$-subgroup Mathieu Group $M_{24}$

I need to compute the Sylow $2$-subgroup of the Mathieu Group $M_{24}$. Unfortunately, this is hard to identify with a machine as it is of order $2^{10}$ and therefore not on the GAP library. I have ...
user1070911's user avatar
6 votes
0 answers
72 views

Computing block systems for non-transitive permutation groups.

Atkinson as well as Schönert and Seress describe methods to compute the minimal block system for transitive permutation groups; in particular in Permutation Group Algorithms by Ákos Seress, we find ...
Ingolfur's user avatar
  • 153
7 votes
1 answer
131 views

Schönert & Seress Algorithm - Computing all block systems - blocks of imprimitivity

Atkinson as well as Schönert and Seress describe methods to compute the minimal block system; in particular in Permutation Group Algorithms by Ákos Seress, we find Theorem 5.5.1 Suppose that a set S ...
Ingolfur's user avatar
  • 153
1 vote
1 answer
207 views

Find the conjugate/similar transformation matrices connecting two unitary irreducible representations corresponding to the same character.

In the GAP manual, the following description is given for the command IrreducibleRepresentationsDixon: If the option unitary is ...
Hongyi Zhao's user avatar
-1 votes
1 answer
90 views

Does a number n raised to itself (n^n) will have digits depending on the original number n? [closed]

Consider, 7^7 = 823543, the digit 3 in 823543 is repeated two times, is it possible that a number raised to itself can have a higher repetitions of any other digit or is it random. Now consider, 93^93 ...
shubham birmi's user avatar
2 votes
1 answer
305 views

Create a block diagonal matrix with different sizes of blocks in GAP

I don't know if this question has been asked before, or if this is the right site to ask it. If not, let me know about a site where can I ask, please. Problem: I want to create a block diagonal matrix ...
Alejandro Tolcachier's user avatar
2 votes
0 answers
147 views

Termination of Buchberger's algorithm for path algebras

I'm reading the paper Noncommutative Gröbner Bases, and Projective Resolutions by Edward L. Green, which presents a version of Buchberger's algorithm for path algebras. I'm trying to show that the ...
Sic Vis's user avatar
  • 658
1 vote
0 answers
130 views

Understanding Graver basis, Grobner basis, and Lawrence lifting

I'm trying to understand the relationship between Graver and Grobner basis, in particular how Graver basis can be computed via Grobner basis via Lawrence lifting. The key result appear to be Theorem 7....
user113988's user avatar
  • 2,672
1 vote
0 answers
51 views

Efficient way of simplify sum of product of multiple polynomials

Let $A \in \mathbb{R}^{n\times m}, B \in \mathbb{R}^m$. I'm trying to compute the coefficients of $n$ polynomials $C_i = (c_0^i, c_1^i, \cdots, c_{n-1}^i)$. where $\displaystyle \sum_{j=0}^{n-1} c_j^i ...
peng yu's user avatar
  • 1,271
4 votes
0 answers
205 views

Do the invariants & covariants completely characterize a projective variety up to projective equivalence?

Let $V \subseteq \mathbb{CP}^n$ be a projective variety embedded in complex projective n-space. By the nullstellensatz it is the zero-set of a finite number of homogeneous polynomials, $p_1(x_1,...x_{...
roymend's user avatar
  • 446
1 vote
0 answers
402 views

Minimal polynomial of powers of primitive element in a finite field.

Let $F=\mathbb{F}_2$ be finite field of order $2$, $f(x)$ be a minimal polynomial of degree $n$ over $F$. Let $K=F(\alpha)$, where $\alpha$ is a root of $f(x)$. My question is how to deduce the ...
PAMG's user avatar
  • 4,500
2 votes
0 answers
415 views

Reshaping a vector into a matrix - represent this as a linear map

We know that in final dimensional space every linear map can be represented by a matrix. Let $x\in\mathbb{R}^{np \times 1}$ be an $np$-dimensional vector. Then the reshape operation that transforms ...
Physics_Student's user avatar

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