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.

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Finite global dimension via the Cartan determinant

Let $A=T(KQ)$ be the trivial extension algebra of a path algebra of Dynkin type $KQ$. The indecomposable module of $A$ correspond to the roots of $Q$ (and not just the positive roots as for $KQ$). Let ...
Mare's user avatar
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Computational tool for checking the existence of non-trivial rational zero of a cubic form

Suppose we consider a arbitrary cubic homogeneous form $f$ in in four or five variables over the rational field. Is there any computational tool or algorithm to check whether this cubic homogeneous ...
Sky's user avatar
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Finding generators and relations for special commutative algebras with a computer

Let $K[x_1,...,x_n]$ be the polynomial ring in $n$ variables and $a_1,...,a_m$ elements in the quotient field $K(x_1,...,x_n)$. Let $A:=K[a_1,...,a_m]$ the ring generated by the $a_i$ in $K(x_1,...,...
Mare's user avatar
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Conjugacy classes in normalized unit group of a group ring

Let $V(FA_4)$ be the normalized unit group of the group ring $FA_4$, where $F$ is the field containing 4 elements and $A_4$ is the alternating group on 4 symbols. How can I find conjugacy classes of ...
HIMANSHU's user avatar
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Can PARI compute class numbers without factoring the discriminant?

When calculating properties of algebraic number fields, one of the hardest steps is factorizing the discriminant of a defining polynomial. This is necessary in the Pohst-Zassenhaus algorithm for ...
wandersam's user avatar
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3 votes
0 answers
176 views

Enumeration of stable graphs of genus $g$

Let $G=(V,E)$ be a connected undirected finite graph, let us call $G$ stable if each vertex has degree at least $3$. Is there a computer algorithm to efficiently enumerate (repetition allowed) all ...
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1 vote
0 answers
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Finding multivariate binomials with a common zero [closed]

I have a problem for which I have to find binomials over a multivariate polynomial Ring which all have a common zero. Let $\mathbb{F}[x_1,\dots,x_n]$ be some multivariate polynomial ring over some ...
Christoph Strobl's user avatar
11 votes
3 answers
657 views

An open triangle problem in plane geometry

Some years ago, I asked some 'famous' people in an advanced Plane Geometry forum about the following: Let $ABC$ be arbitrary triangle, how can one construct a point $P$ in the plane such that $P$ is ...
Đào Thanh Oai's user avatar
3 votes
1 answer
460 views

Computing conjugacy between two elements of $\mathrm{SL}_2(\mathbb{Z})$

The conjugacy classes of $\mathrm{SL}_2(\mathbb{Z})$ are well characterized (see, e.g., this question). Assuming two matrices $A, B \in \mathrm{SL}_2(\mathbb{Z})$ are conjugate, is there a way to ...
Alex's user avatar
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Integer points on genus 1 curves using CAS

How can I practically find integer points on genus 1 curves with small coefficients using computer algebra systems (CAS), like Mathematica, Maple, SageMath, Magma, etc.? As a specific example, do ...
Bogdan Grechuk's user avatar
20 votes
5 answers
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How can you find an integer coefficient polynomial knowing its values only at a few points (but requiring the coefficients be small)?

Example: How can you guess a polynomial $p$ if you know that $p(2) = 11$? It is simple: just write 11 in binary format: 1011 and it gives the coefficients: $p(x) = x^3+x+1$. Well, of course, this ...
Alexander Chervov's user avatar
7 votes
0 answers
119 views

Softwares to determine semi-simple types of Lie algebras generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of matrices

I wish to determine the type of a Lie algebra generated over $\mathbb{R}$ or $\mathbb{C}$ by a set of square matrices with irrational elements. For example, \begin{align} n^+ = \begin{pmatrix} ...
WunderNatur's user avatar
13 votes
3 answers
2k views

Is computer algebra or symbolic computation an active area of research?

I'm interested in doing PhD in computer algebra or symbolic computation, and was wondering if this is an active area of research? Would this area of research also help me in the transition to ...
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146 views

How to decide if an algebraic number is a root of a given polynomial?

Let $p$ be a polynomial with rational coefficients and $\alpha = \sqrt[n]{q}e^{i2k\pi/m}$ for some natural numbers $n,m,k$ and a rational number $q > 0$. Is there an effective algorithm for ...
Jára Cimrman's user avatar
4 votes
1 answer
152 views

Link invariants from modular categories (strictification and computation)

By the theory of Reshetikhin and Turaev, a modular tensor category $C$ gives rise to a link invariant. While $C$ is strict as a monoidal category (e.g. $\mathbb{Fib}$), calculating the link can be ...
Student's user avatar
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1 vote
1 answer
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Primary decomposition of huge ideals using M2/Singular

I used to ask similar questions in other communities, but so far never received any feedback. Given four Hermitian $n\times n$ matrices $A_1,A_2,B_1,B_2$ together with the constraints $[A_i,B_j]=0$, I ...
Timo59's user avatar
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3 votes
2 answers
235 views

Finding the "Q-span" of vectors in Q(q)

Apologies if this question is quite basic. Consider the $\mathbb{Q}(q)$-vector space $V = \mathbb{Q}(q)^n$ with standard ordered basis $\{e_1,\ldots,e_n\}$. Suppose someone hands you some vectors $v_1,...
Sam Hopkins's user avatar
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1 vote
0 answers
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Fitting point on a Quadric curve [closed]

I am working on research project. I am currently using CloudCompare for my project, which calculates the Gaussian curvature and mean curvature to extract the geometric features of the points. I have a ...
Visal Prabhakaran's user avatar
0 votes
0 answers
222 views

Algorithm to compute automorphism group of a finite group

Is there an algorithm to compute automorphism group of a finite group? GAP has a function to do this, but while perusing their GitHub repo, I could not find an implementation. I'm struggling to find ...
Jerry Halisberry's user avatar
3 votes
1 answer
237 views

Number of rings with additive group $(\mathbb{Z}_{16})^2$. A341547(16) in OEIS

I would like to know if somewhere the number of non-isomorphic rings with additive group $(\mathbb{Z}_{16})^2$ is mentioned. If not, is someone able to calculate it? And (easier) the commutative case? ...
José María Grau Ribas's user avatar
32 votes
4 answers
6k views

How does Mathematica do symbolic integration?

I suppose there was at least once in our lifetime the point where we resorted to mathematica for help with an integral.-Unless you chose not to have the pleasure of using the continuum in your ...
Sascha's user avatar
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0 answers
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Is it easy to certify whether a given set of solutions to a polynomial system is complete?

Given a system of complex polynomial equations, we seek the solution set. If we have more equations than variables, then we might expect a finite solution set. One may obtain the solution set by ...
Dustin G. Mixon's user avatar
2 votes
0 answers
43 views

Transforming a symmetric matrix into pentadiagonal form

Given a symmetric matrix $A$, which has complex values in the diagonal, but whose all other entries are real, I am interested in finding an orthonormal transformation $Q$ such that $Q^tAQ$ is a ...
Qwertuy's user avatar
  • 251
12 votes
1 answer
967 views

Where to publish a long classification?

Suppose that the classification of some mathematical (say algebraic) notions requires (say) 70 pages. Let clarify that (say) 90% of the pages are used to write the result itself, whereas only 10% are ...
2 votes
0 answers
68 views

Checking for norm-Euclidean with a computer

Let $K$ be a finite Galois extension of $\mathbb{Q}$ and $O$ the ring of integers in $K$. Question: Is there an algorithm to test whether $O$ is norm-euclidean? In case the question has a positive ...
Mare's user avatar
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4 votes
1 answer
127 views

String compression algorithms for simplifying an expression by introducing variables

I have a very long algebraic expression computed with Maple, and when I inspect it visually, it is clear that it consists of a set of terms that appear over and over again. For purposes of human ...
Tom Solberg's user avatar
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1 vote
1 answer
89 views

Algorithms for Polynomials Over a Real Algebraic Number Field, a reference

I need to find "Algorithms for Polynomials Over a Real Algebraic Number Field Ph.D. thesis, University of Wisconsin, Madison (1974) by Rubald". However I cannot find it online nor in my ...
Lucio Tanzini's user avatar
1 vote
1 answer
275 views

Find basis for the set of torsion points E[m]

In paper "On the Cost of Computing Isogenies Between Supersingular Elliptic Curves" (source) reads Let ${P, Q}$ be a basis for $E[2^{e/2}]$. Let $R_0 = [2^{e/2}−1]P , R_1 = [2^{e/2}−1]Q, ...
nam_ngn's user avatar
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2 votes
1 answer
519 views

Efficiently computing Gröbner basis to prove no solution to polynomial constraints

In a similar vein to these now quite old questions on advice for calculating a Gröbner basis: Fast computation of a Groebner basis. What is possible? What is the state of art in Groebner bases I am ...
RedCurry's user avatar
4 votes
0 answers
75 views

Finding all nice ideals for quiver algebras

Let $Q$ be a finite, connected and acyclic quiver which is simply-laced. Let $k$ be a field and $kQ$ the path algebra of $Q$ over $k$. Recall that an ideal $I$ of $kQ$ is called admissible if it is ...
Mare's user avatar
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3 votes
0 answers
147 views

For which $n$ is this ring an euclidean domain?

Let $f_n(x)=x^n-\sum\limits_{i=0}^{n-1}{x^i}$ and $A_n$ the number field corresponding to $f_n$. Question: Is $A_n$ for all $n$ an euclidean domain? Is there a good choice for an euclidean function? ...
Mare's user avatar
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1 vote
1 answer
159 views

Sufficient syntactic conditions for zero-dimensionality of polynomial systems

Consider a system $S$ of polynomial equations, $p_1=0,...,p_m=0$, for $p_i\in K[x_1,...,x_n]$, for a field $K$: the system $S$ is zero-dimensional if it has finitely many solutions. It is well-known ...
Michele's user avatar
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1 vote
0 answers
133 views

Polynomial systems and algebraic functions

An algebraic function $y(x)$ is defined as the solution of a polynomial equation of the form $p(x,y)=0$, that is one making the identity $p(x,y(x))=0$ true --- in either analytical or formal power ...
Michele's user avatar
  • 313
2 votes
0 answers
104 views

Computer program which computes the automorphism group of Gram Matrix of lattice?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Fixed $n \geq 2$, given $K \in \GL(n,Z)$. One can view $K$ is a Gram matrix of Lattice. I also imposed that $K$ is symmetric i.e $K^{T}=K$. We ...
en kuo's user avatar
  • 145
5 votes
0 answers
145 views

Quiver and relations of $F\mathrm{SL}(2,q)$

$\DeclareMathOperator\SL{SL}$Let $q=p^n$ be a prime power and $F$ a field of characteristic two. Let $G=SL(2,q)$ the group of $2 \times 2$ special linear matrices over the field with $q$ elements with ...
Mare's user avatar
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4 votes
2 answers
159 views

Using computer algebra to check if a family of algebras are pair-wise non-isomorphic

Given an infinite field $k$, consider a quiver $\Gamma$ with one vertex and two arrows $x,y$ and define $R=k\Gamma/(x,y)^2.$ This is a three-dimensional $k$-algebra. Now consider the additive group of ...
Sergey Guminov's user avatar
1 vote
0 answers
214 views

How to solve special Diophantine equation systems (which one can solve by hand) with the computer?

I have a quadratic Diophantine equation system which is possibly not homogeneous and has some mixed and some linear terms. But I know that there are only finitely many solutions over the integers. One ...
Bernhard Boehmler's user avatar
2 votes
1 answer
190 views

How to compute cup product of derived limits / presheaf cohomology

I have a finite category $\mathcal{C}$, along with a functor $F \colon \mathcal{C}^{\mathrm{op}} \to \mathsf{GradedCommRings}$. If $F_j$ is $j$-th graded piece of $F$, then I write $H^i(\mathcal{C},...
Mr. Palomar's user avatar
4 votes
0 answers
212 views

Computing homology class of curve in product of elliptic curves

I have a smooth, projective curve $X/\mathbb{C}$ of genus $g$, embedded in a product of elliptic curves $A = \prod_{i=1}^g E_i$. Since $H_*(A; \mathbb{Z})$ with the Pontryagin product is isomorphic to ...
Daniel Hast's user avatar
  • 1,806
4 votes
1 answer
205 views

Software computing dimension and degree

Assume a projective scheme $X_{k_1,\dots,k_r}\subset\mathbb{P}^n$ is given as the set of common solutions of homogeneous polynomials $F_1(x_0,\dots,x_n),\dots,F_s(x_0,\dots,x_n)$, where the $F_i$ ...
user avatar
1 vote
0 answers
176 views

Using Bertini software to determine whether or not a variety is empty

I have a system of polynomials $f_1,\dots, f_n \in \mathbb{C}[x_1,\dots, x_m]$, and I would like to determine whether the set of solutions to the system $f_1(x)=\dots=f_n(x)=0$ is empty or not. Since ...
Ben's user avatar
  • 1,010
5 votes
2 answers
351 views

Checking if Hochschild cohomology $\mathit{HH}^2(A)=0$

I am trying to compute the Hochschild cohomology of a particular bound quiver path algebra. The quiver $Q$ consists of one vertex and four loops $x,y, h_1,h_2$, and the relations $I$ are generated by: ...
Sergey Guminov's user avatar
5 votes
0 answers
132 views

A practical way to check whether a module is periodic

A module $M$ over a finite dimensional selfinjective algebra $A$ over a field $K$ is called periodic if $M \cong \Omega^n(M)$ for some $n \geq 1$. We assume here that $M$ is simple and that A is a ...
Mare's user avatar
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2 votes
1 answer
169 views

Software for $S$-unit equation

Is there any implementation available of an algorithm which solves in full generality the $S$-unit equation $x+y=1$ in a number field? It seems that Magma solves $ax+by=c$ but only in the algebraic ...
Ferra's user avatar
  • 509
6 votes
1 answer
225 views

Constructing M-curves à la Hilbert

I have been reading some text about Harnack's theorem. The theorem basically says that for degree $d$, the maximal number of connected components in the real (projective) plane of a plane curve with ...
Jose Capco's user avatar
  • 2,175
4 votes
0 answers
75 views

Parabolic Bruhat graphs for exceptional types

I am looking for some computer software or a reference for some parabolic Bruhat graphs. In particular, what I really need $E_8 \setminus E_7$. Does anyone know where or how I'd find this?
Chris Bowman's user avatar
  • 1,191
4 votes
0 answers
119 views

Linear relation between polynomial roots

Consider an irreducible polynomial $P\in\mathbb{Q}[x]$ of degree $n$ whose second leading coefficient is $0$ and $\alpha_1,\dots,\alpha_n$ its $n$ distincts roots. I am interested on the problem of ...
T. Combot's user avatar
  • 231
1 vote
1 answer
61 views

How to verify that an element in the root lattice is an imaginary root of a non-hyperbolic root system?

In my research I encounter some elements in a root lattice and I would like to verify that these elements are imaginary roots. Consider the root system $J_{6, 11}$ with the following Dynkin diagram: \...
Jianrong Li's user avatar
  • 6,101
2 votes
1 answer
532 views

A question on a Macaulay2 computation

I have an ideal $I$ generated by quadratic and cubic homogeneous polynomials in $10$ variables. Macaulay2 tells me that $I$ defines an irreducible variety $X$ of dimension $5$ and degree $10$ in $\...
user avatar
4 votes
1 answer
446 views

All rational periodic points

I am trying to find all rational periodic points of a polynomial. To specify: a periodic point is the point that satisfy $f^n(x)=x$. It is related to dynamical systems in fact. So the current codes ...
nomadd's user avatar
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