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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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 ...
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1
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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(\...
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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 ...
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Raise a Matrix to Arbitrary Power
I have a $k\times k$ matrix
$$
A_{k}=
\begin{pmatrix}
1 & 1 & \cdots & 1 & 1 & 1 \\
1 & 1 & \cdots & 1 &1 & 0\\
&\vdots & &\vdots \\
1 & 1 &...
- 59
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Uniqueness of quotients when reducing with Gröbner basis
Let $K$ be a field, and let $G = (g_1, \ldots, g_m)$ be a Gröbner basis in $K[x_1, \ldots, x_n]$ (i.e. $G$ is a Gröbner basis for the ideal it generates).
By Adams, Loustaunau - An Introduction to ...
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Is there a simple way mathematically to convert $X$ and $Y$ into $1$ where $X=Y$, and $0$ where $X \neq Y$
Given two integers, $X$ and $Y$, is there a simple mathematical expression that can be performed on them that resolves to $1$ when $X=Y$ and to $0$ when $X \neq Y$?
Something that can be expressed ...
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How many non-isomorphic groups of order $5832 = 2^3 \cdot 3^6$ are there?
I'm afraid I can't provide much motivation other than personal interest. I have found David Burrell's very recent Ph.D. thesis, which identified a transcription error that resulted in an incorrect ...
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Is there a way to computationally verify that the sporadic groups are simple?
I'm trying to understand the "easy" direction of the CFSG: namely, the proofs that the 18 infinite families and 26/27 sporadic groups are indeed simple. I'm working through Simple Groups of ...
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Multiple differentiation of chain rule $\frac{d^n}{dx^n} f(g(x))$ computationally
The question was to find a simplified expression for
$$\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(...
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1
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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,...
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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}[...
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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_{...
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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 ...
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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 ...
- 5,932
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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, ...
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Calculating the Galois Group for large degree $f(x)$
Let $f(x)\in \mathbb{Q}[x]$ with Galois group $G$ over $\mathbb{Q}$. Are we always able to calculate $G$ although it might be computationally expensive? I know if $f(x)$ has a special form we can ...
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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 ...
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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 $G=GL(\mathbb Q,n)$...
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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}(...
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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 (...
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3
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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 ...
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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
...
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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 ...
- 153
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1
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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 ...
- 45
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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 ...
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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 ...
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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 ...
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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....
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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 ...
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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_{...
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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 ...
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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 ...
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2
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What is the PDA for this language with three 0's per one 1?
I am trying to find a PDA for $L = \{0^{3i} 1^i \mid i \geq 0\}$ but I am struggling. I was trying to find a DFA and then convert it to PDA but now know that DFA's can't keep count. Any ideas?
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More convenient GAP code to verify Additional property of d-maximal groups
Let $G$ be a finite $p$-group and $d(G)$ be its minimal number of generators. We say that $G$ is $d$-maximal if $d(H) < d(G)$ for all $H < G$.
The following code determines weather $G$ is $d$-...
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Irreducible real representations of $D_{2k}$ and $(C_{i}\times C_{j})\rtimes D_{2k}$
I am considering finite groups of types $D_{2k}$ or $(C_{i}\times C_{j})\rtimes D_{2k}$. I would like to find the irreps of these groups over $\mathbb{R}$ on vector spaces of dimensions $N \lesssim 20$...
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Determining the defining polynomial of a parametrized variety
This was a question I came up with when working on twisted cubic curve.
The twisted cubic curve in $\mathbb{A}^3$ is given by $Y=\{(t,t^2,t^3)|,t\in k\}$, we can immediately tell the defining ...
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What is an algorithm to test whether a number field element has a square root in that number field? [duplicate]
Let $K$ be a number field, such as $\mathbb{Q}(\sqrt{d})$ for $d$ a square-free integer. I am looking for an algorithm that outputs whether some $\alpha \in K$ has a square root in $K$, i.e. when ...
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Computational complexity of the word & conjugacy problems for a (classical) reversible circuit
By word problem I mean the decision problem of whether a given reversible circuit is functionally equivalent to the empty one. It is in co-NP. The broader conjugacy problem asks whether, given $f$ ...
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How to explain these ideal factorization results from `gp`?
Here are the two different factorization results of $\mathfrak{p} = 5$ in two different number fields $F = \mathbb{Q}[x]/(x^3 + 10x - 12)$ and $G = \mathbb{Q}[x]/(x^3 - 6x + 41)$.
...
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Solving a Solvable Polynomial by Radicals (Effectively)
I'm trying to actually write some code (in sage) to take a polynomial $f$ with solvable galois group and compute its roots as nested radicals. Right now I'm just trying to get cyclic extensions to ...
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GAP on Jupyter Notebook
I wanted to learn GAP as it is certain that it will be helpful in the future for my research.
I wanted to try GAP in Jupyter notebook in GitHub. I have tried following the step i.e. launch binder and ...
- 11
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How do I proof Groebner basis existence in $R[x_1, \dots, x_n]$?
In An Introduction To Groebner Bases from Loustaunau, it says that:
We further note that the Noetherian property of the ring R, and hence of the ring $R[x_1, \dots, x_n]$, yields to the following ...
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Polynomials which are invariant to the cyclic permutation of variables
I'm trying to solve the following problem from this book. I can find the Gröbner basis of $J$ using Buchberger’s algorithm, and so I don't have any problem with the first part of this problem. But my ...
- 413
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Algorithm for expressing Gröbner basis in terms of ideal generators
Given a polynomial ring and an ideal
$$A \supset I = (f_1, ..., f_m)$$
there are plenty of implementations of an algorithm (e.g. Buchberger's) that produces a Gröbner basis
$$G = (g_1, ..., g_n)$$
and ...
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166
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Existence of reduced Groebner basis [closed]
In my class notes there is a Theorem saying: Every non zero ideal of the ring $F[x_1,\dots,x_n]$ has a reduced Groebner Basis. Unfortunately there is no proof. Can someone give the proof or refer to ...
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Learn Algebra with computational applications [Book Recommendation] [closed]
I am studying Linear and abstract algebra and find it a bit too, again "abstract", could someone recommend me a good book so I can learn it through computational applications? I think it ...
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775
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How to find minimal generating sets efficiently?
Let $G$ be a group. A minimal generating set of $G$ is a subset of $R$ of $G$ that generates $G$ such that no proper subset of $R$ also generates $G$. For arbitrary groups is its not true (unlike ...
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What is the minimal set of comparisons that determines a monomial order?
A monomial order in $k[x_1, x_2, \ldots, x_n]$ for a field $k$ is a relation $\prec$ on the monomials such that:
$\prec$ is a total order;
if $m_1 \prec m_2$ then $m_3m_1 \prec m_3m_2$ for any three ...
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Using a lemma to calculate syzygy.
I want to find the syzygies of the following monomial ideal $I = (x_1^4, x_1^3x_2, x_1^2x_2^2, x_1x_2^3, x_2^4)$ in $S = k[x_1, x_2]$.
To do this I will use Lemma 15.1 on pg. 322 in Eisenbud "...
user889267
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1
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Normal form and remainder - Groebner Basis
I'm quite confused about the concept of normal form of a polynomial $f$ relative to the ideal $I$. It should be the remainder of the division of $f$ with a Groebner Basis of $I$. But does that mean ...
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