# 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.

260
questions

**6**

votes

**1**answer

80 views

### Is the matrix positive definite given the Gauss-Seidel method converges?

I know that the Gauss-Seidel method converges given that the matrix you want to solve is symmetric positive definite. However, I'm wondering if the "converse" of the statement is true. Namely, if $A$ ...

**2**

votes

**0**answers

66 views

### Affine projection of polynomials for a given set of points

(Not sure this question fits here, I will remove it in case it doesn't)
Let $F_{\mathrm{ML}}[x_1, \ldots,x_n]$ denote the set of multilinear polynomials over a finite field $F=\mathbb{F}_q$ (i.e. a ...

**3**

votes

**0**answers

56 views

### Frobenius algebras of small dimensions

In Classification of commutative Frobenius algebras , Jeremy Rickard showed that there are infinitely many commutative (local without loss of generality) Frobenius algebras of vector space dimension ...

**7**

votes

**0**answers

107 views

### Classification of Frobenius algebras of small dimensions

Despite (commutative) Frobenius algebras over a field $K$ being a very popular class of algebraic objects, it seems no attempt of classification (up to $K$-algebra isomorphism) for them has been ...

**1**

vote

**0**answers

296 views

### Finding elliptic curve with $P=[m]R$

To avoid X-Y problem I am going to write my problem down in detail, so plz bear with me.
The elliptic curve over $Q$ given by a Weierstrass equation is -
$E := y^2 +a_1 xy +a_3 y = x^3 + a_2 x^2+...

**0**

votes

**0**answers

51 views

### Low rank approximation

Can we solve low rank approximation problem by using concept of Gröbner basis? I was trying to find it by Macaulay2 but didn't find the answer. I was trying to do by toric ideals as for them Gröbner ...

**4**

votes

**0**answers

91 views

### Rank of matrix over UFD polynomial ring

I have a matrix, $M$, size of approximately $20\times 25$, over a polynomial ring $\mathbb{Q}[x_1, \cdots, x_n]$ and a number $r$ such that I would like to test whether or not there exist values for ...

**11**

votes

**1**answer

381 views

### Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials

Let $n>1$ and $p$ be an odd prime with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. Notice that, since $p-1 \mid n-1$, we have $T^p - T \mid T^n-T$ in $\mathbb{F}_p[T]...

**0**

votes

**1**answer

91 views

### IntersectInP bug of Macaulay2 [closed]

I am trying to use the intersectInP command in Macaulay2, inside package ReesAlgebra. However, I tried to follow the exact code in the user-guide, but it doesn't run in my Ubuntu app (of win 10). Can ...

**2**

votes

**0**answers

45 views

### Determining the symmetry group of a system of ODEs as tensor product decomposition

Suppose you have a linear system of ODEs in implicit form represented by an $n \times n$ matrix $M$ and you are trying to identify a subgroup of $\mathrm{GL}(n)$ that is the symmetry group of this ...

**2**

votes

**1**answer

119 views

### Intersection of a $\mathbb{Q}$-affine space with $\mathbb{Z}^n$

Let $E$, a $\mathbb{Q}$-affine space of arbitrary dimension included in $\mathbb{Q}^n$. Is it possible to check efficiently if $E \cap \mathbb{Z}^n$ is empty or not?
If is an hard problem could give ...

**4**

votes

**1**answer

136 views

### On nearly Frobenius algebras

Let $A$ be a quiver algebra over a field $k$ with multiplication $m$. By https://arxiv.org/pdf/1705.10222.pdf definition 6, $A$ is called nearly Frobenius in case there exists a (non-zero? Was this ...

**2**

votes

**0**answers

67 views

### CAS for finite-dimensional complex representations of $S_n$

Does there exist a computer algebra system that can work with finite-dimensional complex representations of the symmetric groups on finitely many letters? It should have the following functionality: (...

**1**

vote

**0**answers

61 views

### Is there a workable numerical method for determining the center of a circle through three points? [closed]

I'm a 73-year-old engineer struggling with numerically implementing a math problem.
I am working on a kinematic linkage project that generates motion paths (as long sequences of x,y coordinates) of ...

**0**

votes

**0**answers

136 views

### Quantifier elimination with no free variables and real polyhedral inequalities

In this introductory blog post https://cstheory.blogoverflow.com/2011/11/something-you-should-know-about-quantifier-elimination-part-i/ it is mentioned in the very last line that "I do not know if a ...

**3**

votes

**1**answer

305 views

### Lower bound for polyhedral real quantifier elimination

All known examples for double exponential lower bounds for real quantifier elimination involves polynomial inequalities with degree $>1$.
Is there an example of double exponentiality with ...

**3**

votes

**1**answer

305 views

### Special linear Diophantine system - is it solvable in general?

Background: An equivalent question was asked on MSE almost two years before this post now. It was never fully resolved. - Here, we are asking if further progress can be made.
Motivation
Solving this ...

**1**

vote

**0**answers

66 views

### Degree bounds on coordinates of points in a zero-dimensional variety

Let $S = \{f_1, \dots, f_s \in \mathbb{Q}[x_1, \dots, x_n]\}$ have a zero-dimensional nullset $V \subset \mathbb{C}^n$, and suppose that each $f_i$ has total degree at most $d$.
Is there a shared ...

**2**

votes

**0**answers

56 views

### Constructing stable equivalences for finite dimensional algebras

Given a finite dimensional (non-selfinjective) algebra $A$.
Is there a method (for example using QPA) to construct algebras stable equivalent to $A$?
Such a thing is easily possible for derived ...

**4**

votes

**0**answers

77 views

### Deciding whether two algebras are derived equivalent

Given two finite dimensional quiver algebras $A$ and $B$ (over a nice field in case that helps, for example a finite field).
Question: Can an there be a finite algorithm that decides whether $A$ ...

**5**

votes

**0**answers

108 views

### Stable equivalence and stable Auslander algebras

Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules.
Recall that the Auslander algebra of $A$ is $End_A(M)$ and the ...

**3**

votes

**2**answers

194 views

### On Auslander algebras

Given a connected quiver algebra $A$ that is representation finite, the Auslander algebra $B_A$ of $A$ is defined as the endomorphism ring of the direct sum of each indecomposable $A$-module.
It is ...

**5**

votes

**1**answer

136 views

### Ideals of commutative Frobenius algebras

Given a finite dimensional commutative (connected=local) Frobenius algebra $A$ over a field $K$.
Question 1: Does $A$ have only finitely many ideals? (the answer should be no in the non-commutative ...

**0**

votes

**1**answer

95 views

### Quantifier elimination and where is this quantified convex program in the polynomial hierarchy?

I have a quantified convex program of the form that I need to solve
$$\exists(x_{1,1},\dots,x_{1,n})\in\mathbb R^n\quad\forall(x_{2,1},\dots,x_{2,n})\in\mathbb R^n$$
$$\vdots$$
$$\exists(x_{2t-1,1},\...

**3**

votes

**1**answer

143 views

### Branching to Levi subgroups in SAGE and the circle action

In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup:
http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/...

**-1**

votes

**1**answer

79 views

### Perform a univariate integral, involving a Gauss hypergeometric function

This is a follow-up question to the one posed in Compute the two-fold partial integral, where the three-fold full integral is known . (I hope that doing so is viewed as a legitimate step. If not so, I ...

**3**

votes

**4**answers

525 views

### Compute the two-fold partial integral, where the three-fold full integral is known

I have the following trivariate ($\rho_{11}, \rho_{22}, \mu$) function
\begin{equation}
4 \mu ^{3 \beta +1} \rho_{11}^{3 \beta +1} \left(-\rho_{11}-\rho_{22}+1\right){}^{3 \beta +1}
\rho_{22}^{3 \...

**1**

vote

**2**answers

283 views

### Computing Groebner basis for a complicated systems of polynomials

I am trying to solve complicated systems of polynomial equations. The first step is to determine maximal sets of independent variables for the solution manifold (ideal) or the number of isolated ...

**5**

votes

**2**answers

173 views

### Obtaining quiver and relations for finite p-groups

Given a finite field $K$ with $p$ elements and a finite $p$-group $G$, is there a way to obtain the quiver and relations of $KG$ with GAP (and its package QPA)?
Since $KG$ is local, the quiver should ...

**4**

votes

**0**answers

118 views

### Recovering the bimodule from the trivial extension

Given a ring $S$ with a non-zero $S$-bimodule $M$, the trivial extension of $(S,M)$ is defined as the ring $R:=T_M(S)$ with $R= S \oplus M$ with multiplication $(s,m)(s',m')=(s s', sm' +m s')$.
We ...

**0**

votes

**0**answers

54 views

### Quadrics over the univariate function field with discriminant of minimal degree

Consider a non-degenerate quadric $Q(x,y,z) \subset \mathrm{P}^2$ over the univariate function field $\mathbb{F}_p(t)$, where $\mathbb{F}_p$ is a prime finite field, $p > 2$. For simplicity assume ...

**1**

vote

**0**answers

139 views

### Computing the class-preserving automorphism group of finite $p$-groups

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called a class-preserving if for each $x\in G$, there exists an element $g_x\in G$ such that $\alpha(...

**3**

votes

**1**answer

105 views

### Computing double coset operators in a computer algebra system

I want to do double coset operators computations on modular forms of half integer weight and with character such as the trace operators that map modular forms of congruence subgroups $\Gamma_0(N)$ to ...

**2**

votes

**0**answers

77 views

### Obtaining the reduced incidence algebra in QPA

Given a finite poset $P$ (we can assume it is connected), the reduced incidence algebra of $P$ is the subalgebra of the incidence algebra of $P$ consisting of functions constant on isomorphic ...

**2**

votes

**0**answers

55 views

### Efficient algorithm to prove that a polynomial ideal contains 1

I have the following problem:
Suppose to have an ideal $I\triangleleft k[x_1,...,x_n]$ defined by generators. There exists an efficient algorithm (perhaps more efficient than calculating the Groebner ...

**0**

votes

**0**answers

117 views

### lcalc and the Analytic Rank of $y^2 = x^3 + 432764797 x^2 + 332896 x$

I'm looking at elliptic curves associated with $a/(b+c) + b/(a+c) + c/(a+b) = N$. For the case $N=10400$, Michael Rubinstein's lcalc gives the analytic rank of the associated elliptic curve $y^2 = x^3 ...

**5**

votes

**1**answer

283 views

### Calculating the Ext-algebra with a computer

Given a finite dimensional quiver algebra $A$ over an arbitrary field and a module $M$ of finite injective dimension or finite projective dimension.
Let $B$ be the Ext algebra of $M$, that is $B:=\...

**4**

votes

**0**answers

97 views

### Compute the closure of graph of function from complement of hypersurface in $\mathbb{A}^n$

I'm hoping someone can give me some tips to help speed up computation on the following problem:
Suppose I have a map $G=(g_1/f,\dots,g_m/f):\mathbb{A}^n\setminus{V(f)}\to \mathbb{A}^m$. I'm ...

**3**

votes

**1**answer

145 views

### Solving polynomial inequalities — efficient Positivstellensatz on a computer

I have about twenty five (multilinear) polynomials $f_1(\mathbf{x}), f_2(\mathbf{x}), \dots, f_{25}(\mathbf{x})$ all in fifteen variables and I would like to decide if there is a $\mathbf{y} \in [0,1]^...

**3**

votes

**0**answers

91 views

### Finite test for periodicity of a module

Let $A$ be a finite dimensional quiver algebra and $M$ a finite dimensional $A$-module.
Assume we want to test whether $M$ is a periodic module, meaning that $\Omega^n(M) \cong M$ for some $n \geq 1$. ...

**4**

votes

**1**answer

227 views

### Resultants for compactly represented product form polynomials?

Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the ...

**3**

votes

**2**answers

223 views

### Find parameter values for which a 3x3 matrix has a triple eigenvalue

An Exceptional point generally occurs in eigenvalue problems in which the matrix is dependent on some parameter(s). The particular point in which the eigenvalues become degenerate for the parameter(s) ...

**2**

votes

**0**answers

197 views

### Combinatorial and computational problem related to Weyl groups and the coroot lattice

Let $W$ be a Weyl group with root system $R$ and with set of positive roots $R^+$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. positive roots $\alpha$ which satisfy $\ell(s_\alpha)=2\...

**4**

votes

**1**answer

228 views

### Higher roots modulo prime complexity best algorithm

Given integers $a,\ell$ and prime $p$ we need to find the roots of the algebraic equation $x^\ell\equiv a\bmod p$. We know there are at most $\ell$ such $x$.
What is the best method to find all such ...

**1**

vote

**1**answer

87 views

### Finding a characteristic for which the zero-locus of an ideal is not empty

I have a set of polynomials $f_1, \dots, f_m \in \mathbb{Z}[x_1, \dots, x_n]$ and I am interested in finding if these polynomials have a common root inside either $\mathbb{C}[x_1, \dots, x_n]$ or $\...

**1**

vote

**1**answer

134 views

### Is this algorithm for primary decomposition correct?

I've written some code for Sage to compute radical ideals and primary decompositions over $\overline{Q}$ (the field of algebraic numbers), and I'm not sure if it's right.
Since Singular (the ...

**5**

votes

**0**answers

62 views

### Computing centralizers of finite sets in right angled Artin groups (RAAGs) / partially commutative groups / graph groups

This question concerns right angled Artin groups (RAAGs), also called partially commutative groups or graph groups.
A student of mine, Adi Ben-Zvi, needs for an algorithm in RAAGs, a subalgorithm ...

**4**

votes

**0**answers

78 views

### Compute equalizer of maps of polynomial rings, perhaps using Gröbner bases

Suppose that $k$ is a field and I have two ring homomorphisms
$$
\phi, \psi :k[x_1,...,x_m] \to k[y_1,...,y_n].
$$
How can I use Gröbner bases (or other computational tools) to compute the subring of ...

**3**

votes

**2**answers

285 views

### Alternate descriptions of finite fields

The finite field of order $p^n$ is isomorphic to $(\mathbb Z/p \mathbb Z)[X]/(P)$, where $P$ is an irreducible polynomial in $(\mathbb Z/p \mathbb Z)[X]$ of degree $n$. This describes every finite ...

**4**

votes

**0**answers

150 views

### Computing the volume of intersection between a ball and a box

$C$ is the set of vectors which are coordinate-wise less than $\overline{c}\in [-1,1]^d$ and greater than $\underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex in $d$ that ...