# 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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### Testing whether the endomorphism ring of an $A$-module is isomorphic to $A$

Let $A=KQ/I$ be finite dimensional quiver algebra and $M$ an $A$-module. Question: Is there a good/fast test whether we have $A \cong End_A(M)$ (or $A^{op} \cong End_A(M)$) using for example the GAP-...
1 vote
31 views

### Is there a more efficient computer algebra system to solve the system of nonlinear equations in N-R method or other numerical methods?

Consider the system of infinite series \begin{align} &F=x+\frac{y^{3^2}}{3}+\frac{x^{3^5}}{3^2}+\frac{y^{3^7}}{3^3}+\frac{x^{3^{10}}}{3^4}+\frac{y^{3^{12}}}{3^5}+\cdots=0 \\ &G=y+\frac{x^{3^3}}...
1 vote
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### Are algebraic power series in positive characteristics D-finite?

We know that in characteristic $0$, all algebraic series are differentiably finite. Is this true in positive characteristic? I look at the proof, indeed we need to the characteristic to be $0$ for the ...
122 views

### Noncommutative algebra question inspired by Young-Jucys-Murphy elements

This is really a followup to Why are Jucys-Murphy elements' eigenvalues whole numbers? , specifically to Igor Makhlin's beautiful answer. I'm trying to make it even more beautiful by getting rid ...
375 views

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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 ...
1 vote
107 views

### 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 ...
130 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 ...
1 vote
59 views

### 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 ...
507 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 ...
396 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 ...
187 views

### 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 ...
70 views

### Linear differential equation hypergeometric series

I am studying the following series $$\sum_{d\geq 1}\left(\frac{x^d}{d \, h^d} \sum_{k=1}^d (-1)^{d-k} \frac{1}{(k-1)! \, (d-k)!}\prod_{i=1}^d G((k-i)h)\right).\tag{*}$$ It's proven that (*) satisfies ...
2k views

### 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 ...
102 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} ...
1k 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 ...
138 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 ...
103 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 ...
1 vote
152 views

### 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 ...
165 views

1 vote
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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 ...
65 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 ...
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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
115 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 ...
1 vote
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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 ...
58 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 ...
130 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 ...
142 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 ...
1 vote