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.
368
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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 ...
0
votes
0
answers
86
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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 ...
3
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0
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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,...,...
5
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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 ...
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0
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128
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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 ...
3
votes
0
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176
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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 ...
1
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0
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62
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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 ...
11
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3
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657
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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 ...
3
votes
1
answer
460
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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 ...
2
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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 ...
20
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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 ...
7
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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}
...
13
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3
answers
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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 ...
2
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146
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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 ...
4
votes
1
answer
152
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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 ...
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 ...
3
votes
2
answers
235
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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,...
1
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0
answers
76
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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 ...
0
votes
0
answers
222
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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 ...
3
votes
1
answer
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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? ...
32
votes
4
answers
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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 ...
7
votes
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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 ...
2
votes
0
answers
43
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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 ...
12
votes
1
answer
967
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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
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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 ...
4
votes
1
answer
127
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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 ...
1
vote
1
answer
89
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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 ...
1
vote
1
answer
275
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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, ...
2
votes
1
answer
519
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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 ...
4
votes
0
answers
75
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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 ...
3
votes
0
answers
147
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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?
...
1
vote
1
answer
159
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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 ...
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0
answers
133
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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 ...
2
votes
0
answers
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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 ...
5
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0
answers
145
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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 ...
4
votes
2
answers
159
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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 ...
1
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0
answers
214
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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 ...
2
votes
1
answer
190
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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},...
4
votes
0
answers
212
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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 ...
4
votes
1
answer
205
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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 $F_i$ ...
1
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0
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176
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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 ...
5
votes
2
answers
351
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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:
...
5
votes
0
answers
132
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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 ...
2
votes
1
answer
169
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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 ...
6
votes
1
answer
225
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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 ...
4
votes
0
answers
75
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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?
4
votes
0
answers
119
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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 ...
1
vote
1
answer
61
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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:
\...
2
votes
1
answer
532
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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 $\...
4
votes
1
answer
446
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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 ...