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.
405 questions
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Testing whether a given quiver algebra is noetherian
Let $A=KQ/I$ with $Q$ a finite connected quiver and $I \subset J^2$ where $J$ is the ideal generated by the arrows of $Q$.
Question 1: Is there a good theory (or even a finite test) to test whether $...
- 28.2k
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0
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137
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Help for a conjecture on Cayley Hamilton algebras
For $n\in N$ an $n$ Cayley Hamilton algebra is an associative algebra $R$ over a commutative algebra $A$ with a norm $N:R\to A$ that is a multiplicative polynomial map of degree $n$ so that each $r ...
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Is there any algorithm which can find a common divisor of two polynomials modulo $p^k$?
Let us consider two monic polynomials $f(X), g(X) \in \dfrac{\mathbb{Z}}{p^k\mathbb{Z}}[X]$. Now, we call $h(X)$ is a divisor of $f(X)$, if there exists a $l(X) \in \dfrac{\mathbb{Z}}{p^k\mathbb{Z}}[X]...
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2
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On a quotient of the quantum enveloping algebra of $sl_2$
Let $K= \mathbb{C}$ and $q$ a root of unity with $q^2 \neq 1$ and of smallest order $d$ and set $e=d$ if $d$ is odd and $e=d/2$ if d is even. Let $U_q$ be the quantum enveloping algebra of $sl_2$ ...
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How big can a multiprojective variety be for which Macaulay2 can calculate irreducible components and check their smoothness?
I have a multiprojective variety $X$ in a product of projective spaces given by a multigraded ideal $I$. Suppose that the multiprojective variety is embedded into a product of projective spaces the ...
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If resultant $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ have a nontrivial factors then can $f(x)$ also have a nontrivial factors?
This is a follow-up question to this question. In that question, we learned that if, $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ , then $T(g(z))$ is divisible by $f(z)$.
Now, my question is:
If $T(y) = ...
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0
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Automating the resolution of IMO 2025 Problem 1
The problem 1 of the 2025 IMO is the following:
A line in the plane is called sunny if it is not parallel to any of the x-axis, the y-axis, and the line $x + y = 0$.
Let $n ⩾ 3$ be a given integer.
...
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2
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Checking whether an algebra is finite dimensional using a computer
Let $A$ be the $K$-algebra defined as the quotient of the non-commutative polynomial ring in variables a,b,c,d,e,f,g,h,z modulo the relations
...
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Algebras as centralizer algebras
Let $A$ be a finite dimensional $K$-algebra for $K$ a field.
Define the centralizer dimension of $A$ to be the smallest $n$ such that $A$ is the centralizer of $n$ matrices. Any algebra $A$ can be ...
- 28.2k
1
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1
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Efficient computing of alternating multinomial products for elimination of radicals in polynomial equations
I have a problem that involves elimination of the radicals from equations involving a number of radicals of multivariate polynomials to then finally form multivariate polynomial equations (over Z). ...
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0
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160
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Regarding exceptional primes
I am reading Swinnerton Dyer's paper on "On $\ell$-adic representations and congruences for coefficients of modular forms". It defines a prime $\ell$ to be exceptional for an eigenform $f \...
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MAGMA question (kernel of ring hom)
I apologize for posting here, but I don't see much else on stackoverflow where people who know MAGMA might be able to answer. The MAGMA documentation at
https://magma.maths.usyd.edu.au/magma/...
- 3,596
2
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1
answer
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Gröbner Basis That do not Introduce New Indeterminates
Let $\vec{X}$ be a finite set of indeterminates, and $\sqsubseteq$ be a monomial ordering.
A Gröbner Basis $B$ of an ideal $I$ of polynomials can be characterized as a finite set of polynomials ...
1
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1
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Efficient algorithms for inductive enumeration of Lyndon trees below a certain nilpotency / depth
I am working on a computational project for a supervisor working in classical and modern deformation theory; my base framework is the SageMath project. This question is inspired by a serious ...
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2
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Computer algebra systems implementing Schubert polynomials
I've had a python package out for multiplying Schubert polynomials, double Schubert polynomial, quantum Schubert polynomials, and double quantum Schubert polynomials for a little over a year. Recently ...
- 2,448
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Determining if the two different dimension integer matrices are congruent
Given a matrix $A\in M_{6\times 6}(\mathbb{Z})$ that is symmetric and has determinant zero. I want to deterministically figure out if there exists a matrix $T\in M_{6\times 5}(\mathbb{Z})$ such that $...
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1
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Finding nice relations for an explicit matrix group and showing that it is isomorphic to the symmetric group
Let $A$ be the $n \times n$ matrix with all diagonal entries equal to 1 except in the last column and all entries in the last column equal to -1 and all other entries equal to 0.
Let $B$ be the $n \...
- 28.2k
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Finding relations for a finite matrix group
I have a problem where I have groups generated by two $n \times n$ integer matrices and I know that one of those matrices is a permutation matrix.
Question: What is the best way to find minimal nice ...
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3
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2
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Algorithms (or packages) to find recurrence relations for given sequence of q-polynomials?
Assume we have sequence of polynomials : $P_i(q)$ - each term is polynomial in $q$. (With integer coefficients, but hopefully it is not important).
We expect that there exists recurrence relation a ...
- 26.4k
4
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1
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Compute generators for group of totally positive units of a number field?
Given a number field $K$, I would like to compute (in Sage) generators for the group of totally positive units of $K$.
Update: I've tried some code (details below), which I've received some help on in ...
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Computer program for free restricted Lie polynomial
I am conducting numerical experiments involving the Gröbner–Shirshov Basis for restricted Lie algebras. At each step of the computation, I need to work with restricted Lie polynomials. Specifically, I ...
- 1,143
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160
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Free, easy-to-use program for noncommutative algebra over finite fields
I am looking for a computer program that can handle computations in noncommutative algebra over a finite field of prime order $p$.
My requirements are:
The program should be free, as I do not have ...
- 1,143
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2
answers
327
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Compute corestriction map on group cohomology in Magma
I am trying to use Magma to compute the corestriction of a second cohomology class, but I’m not sure how to interpret the output. The details and code are given below.
Consider the group $G = \mathrm{...
- 1,068
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58
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Computing all roots of a function with square-root terms
Given $3n$ positive numbers $a_1, \ldots, a_n$, $b_1, \ldots, b_n$, and $x_1, \ldots, x_n$, we are given a function
$$f(x) = \sum_{i = 1}^n \frac{a_i}{\sqrt{(x - x_i)^2 + b_i}}.$$
Can we find all the ...
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0
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Example polynomial system where Macaulay bound is tight
I have been solving systems of polynomial equations by forming the Macaulay matrix of different degrees and computing its null space. If the degree is large enough, namely at or above the degree of ...
- 11
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vote
1
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160
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MAGMA (pseudo)basis for maximal order in quaternion algebra
I wanted to compute any maximal order in the non-split quaternion algebra $\left(\frac{21, -7}{\mathbf{Q}(\sqrt{-3})}\right)$, so I did the following in MAGMA:
...
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Can computer algebra system compute Galois group/splitting field of a polynomial over $p$-adic number field of higher degree?
I am looking for a computer algebra system that checks if the splitting fields of two polynomials over a $p$-number field are the same or not. At least, knowing their splitting fields are isomorphic ...
- 450
1
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0
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147
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Primitive element theorem for algebraic functions
Given a function $f(x) : \mathbb{R}^n \to \mathbb{R}$, we call it algebraic if it satisfies a polynomial equality $g(y, x) = 0$.
This is analogous to an algebraic number being the root of a univariate ...
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0
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115
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Simplifying sums with CAS or theorem prover
To verify that a certain map is a chain homotopy I could reduce it to an evaluation of
$S = S_1 + S_2$ where
$$
S_1 = \sum_{a=0}^p \sum_{b=0}^{p+1} (-1)^{a+b} n_b \cdot e_a
$$
$$
S_2 = \sum_{b=0}^p \...
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152
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Rewriting systems for finite groups [closed]
This is a question about rewriting systems & languages for finite groups. I'm sure everything must be in the literature somewhere, but I find it hard to navigate the references I have (for example ...
- 2,359
3
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1
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About Shor's quantum algorithm
I know very little about quantum computing, and I've been trying to understand Shor's algorithm for the factorization of an integer $N$. I'm following Computational Complexity — a modern approach by ...
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0
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Computer computation of the first Galois cohomology of a $p$-adic torus?
Let $T\subset {\rm GL}(N,{\mathbb Q})$ be an $n$-dimensional ${\mathbb Q}$-torus
given by its Lie algebra $\mathfrak{t}\subset \mathfrak{gl}(N,{\mathbb Q})$.
I want to compute, in some sense ...
- 15.4k
7
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1
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Which CAS can do basic non-commutative differential algebra?
This is a repost of my question at MSE from 7 months ago, to which I haven't been able to find an answer yet.
I am looking for a CAS (possibly incl. additional packages/libraries) that can compute ...
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10
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0
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What axiomatic system does AlphaGeometry use?
In January 2024, researchers from DeepMind announced AlphaGeometry, a software able to solve geometry problems from the International Mathematical Olympiad using a combination of AI techniques and a ...
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Describing the primes with each cyclic decomposition group in a given finite Galois extension of $\mathbb Q$
$\newcommand{\Q}{{\mathbb Q}}
$Let $f\in \Q[x]$ be a polynomial,
and let $L/\Q$ be the finite Galois extension
obtaining by adjoining to $\Q$ all roots of $f$.
Magma knows how to compute $\Gamma:={\...
- 15.4k
6
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0
answers
231
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Computer programs for decomposition groups?
There is quite a lot of work on computing Galois groups of splitting fields of polynomials over $\Bbb Q$. Magma is quite good at it.
In this answer to Decomposition groups for the Galois module $\mu_8$...
- 15.4k
1
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0
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Is there a better/newer list of Kazhdan-Lusztig polynomials?
I am essentially just re-asking this question, as it's now over a decade old, and I'm hoping that more extensive lists exist. I've started looking at the papers cited in the previous question, and ...
5
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1
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Can one compute the automorphism group of a curve of genus >1?
Given a sufficiently nice perfect field $k$ and a smooth projective curve $C$ of genus $g_C>1$ over $k$, can one compute the automorphism group ${\rm Aut}(C)$? It is known that ${\rm Aut}(C)$ is ...
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Is it decidable whether a statement about reals (in the language of ordered rings) is constructively provable?
The language of ordered rings is a first-order language with operators for $+$, $-$, and $\cdot$, constants for $0$ and $1$, and relations for $<$, $=$ and $>$.
To decide whether such a ...
- 7,158
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How to find a single-variable polynomial in a zero-dimensional ideal?
Given finitely many multivariate polynomials with algebraic coefficients that generate a zero-dimensional ideal, is there an easy way to find a nonzero single-variable polynomial in this ideal?
If we ...
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46
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Finite right-triple convex sets in planes
Let $\mathcal{S}$ be a set of points in $\mathbb{R}^2$. We say that $\mathcal{S}$ is right-angle convex, if for any two distinct points $P,Q\in \mathcal{S}$ there always exists another point $R\in \...
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Relations between non-negativity of multivariate polynomials and SOS over gradient ideal
We know there is a necessary condition for the non-negativity of multivariate polynomials in the paper "Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational ...
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Factorial series $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$ and hypergeometric functions
For positive integer $D$, define $j(D)=\sum_{n=1}^\infty \frac{1}{(n^D)!}$.
For $D \le 6$, sage finds closed form in terms of hypergeometric functions
at algrebraic arguments and fails to find closed ...
- 25.8k
2
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1
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Efficiency of Groebner basis for constraints of the form $(a_i x_i+b_i)(a_j x_j+b_j)$
This is based on numerical experiments in sage.
Let $K$ be a ring and define the ideal where each polynomial
is of the form $(a_i x_i+b_i)(a_j x_j+b_j)$ for constant $a_i,b_i,a_j,b_j$.
Q1 Is it true ...
- 25.8k
11
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Computing homology groups with GAP
I’m studying the homology groups of arithmetic groups such as $SL(5,\mathbb{Z})$. I saw in the answer to this post that we can use GAP to compute some of the homology groups for $SL(3,\mathbb{Z})$. Is ...
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Application of Resultant in Computer Algebra [closed]
Can you guys give me some application of resultant in Computer Algebra, it will be amazing if you guys can give me some paper or book to read more. Thanks so much
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0
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Macaulay2 seems to have divergent behavior on rings with differently ordered variables
I noticed the following strange behavior which I cannot explain.
I wanted to compute the integral closure of the following ring,
$$ A = \mathbb{F}_5[x,t]/(t^2 (1 - x^4) - x^5) $$
Call the integral ...
- 4,480
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1
answer
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How to construct such a real algebraic curve
Suppose $L$ is a real line in $\mathbb{RP}^2$, my question is: for a given postive integer $n$, is it possible to find a real projective algebraic curve $C$ of degree $n$ with maximum connected ...
- 85
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0
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Hall-Littlewood polynomials with sage
I need to make some computations with Sage involving Hall-Littlewood polynomials. I couldn't find any satisfying information in the Sage manuals/tutorials that I found on the internet. What I found is ...
- 6,459
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0
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132
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Isomorphism and counting for tree quivers
Let $Q$ be a quiver which is a connected tree and let $A=KQ/I$ be a quiver algebra with $I$ an admissible ideal, meaning that $I$ is generated by paths of length $\geq 2$. Let $n$ be the number of ...
- 28.2k