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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2 votes
0 answers
71 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
69 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 ...
3 votes
1 answer
432 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 ...
3 votes
1 answer
324 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 ...
1 vote
0 answers
78 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 ...
43 votes
3 answers
5k views

Is there a systematic method for differentiating under the integral sign?

This MO question by Tim Gowers reminded me of a question I've wondered about for some time. In the delightful book Surely You're Joking, Mr. Feynman!, Feynman praises the technique of differentiating ...
11 votes
1 answer
1k views

A 2F1 Hypergeometric identity from a Feynman integral

Using two different approaches to evaluating the dimensionally regularized ($d=4-2\epsilon$ dimensional Euclidean space), equal mass ($x=m^2$), 2-loop vacuum Feynman diagram $$ \begin{align} I(x) &...
21 votes
5 answers
6k views

Fast computation of a Groebner basis. What is possible?

I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or ...
21 votes
1 answer
1k views

Main open computational problems in quantifier elimination?

A language is said to have quantifier elimination if every first-order-logic sentence in the language can be shown to be equivalent to a quantifier-free sentence, i.e., a sentence without any $\forall$...
5 votes
0 answers
122 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 ...
2 votes
0 answers
63 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 ...
5 votes
2 answers
500 views

What are the most general methods for solving equations in closed form with Lambert W?

What are the most general methods for solving equations with help of Lambert W function or with a generalization of Lambert W function in closed form? I gave a method in MSE here. Which algorithms ...
23 votes
2 answers
1k views

What is currently feasible in invariant theory for binary forms?

When Paul Gordan became a professor in 1875 he could show the binary form in any degree has some finite complete system of (general linear) invariants, but he could not actually give a complete system ...
4 votes
0 answers
85 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 $A$ ...
4 votes
2 answers
443 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
168 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 ...
4 votes
1 answer
316 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
2 answers
328 views

Finding all submodules

Given a finite dimensional local commutative algebra over a finite field $K$ and a finite dimensional module $M$. What is the fastest/best way to obtain all submodule from $M$ using a Computer algebra ...
1 vote
1 answer
126 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},\...
4 votes
1 answer
191 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
131 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 ...
1 vote
2 answers
580 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 ...
1 vote
0 answers
175 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(...
4 votes
0 answers
148 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
57 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 ...
3 votes
1 answer
143 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 ...
8 votes
1 answer
215 views

Is there a good computer program for searching for endomorphisms between finite algebras which make diagrams commute? Is this problem NP-complete?

Let $(X,*),(Y,*),(Z,*)$ be finite algebras. The binary operations $*$ are not required to satisfy any identities though I am interested in the special case where $*$ is associative. Suppose that $f:X\...
3 votes
2 answers
2k views

Computer program to solve a system of polynomial equations over a finite field

I have a set of polynomial equations for which I want to know the solutions (actually really the number of solutions). It would be great if I could get a computer to do it, but I'm not sure exactly ...
2 votes
0 answers
61 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
134 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 ...
6 votes
1 answer
401 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
104 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
237 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]^...
4 votes
1 answer
269 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
442 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) ...
8 votes
1 answer
218 views

Algebraic Shifting Computer Code

Is anyone aware of computer code that will algebraically shift a simplicial complex (as in this Kalai paper)? Ideally, I am looking for an implementation that can run in something like Sage or ...
12 votes
2 answers
560 views

Ideal Membership without Certificate?

I have a homogeneous ideal $I=\langle f_1,\ldots,f_r\rangle$ of the polynomial ring $\mathbb C[X_1,\ldots,X_n]=:R$ where each of the $f_i$ is actually over $\mathbb Z$. My computations are usually ...
2 votes
0 answers
209 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\...
1 vote
1 answer
107 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
332 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
111 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
94 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 ...
0 votes
0 answers
32 views

Limit/Expansion Problems for Benchmarking

I am interested in collections of ‘interesting’ problems involving limits and/or asymptotic expansions of univariate real-valued functions. The purpose is to test a particular algorithm that I ...
3 votes
1 answer
231 views

Finding all selforthogonal indecomposable modules

Given a finite dimensional algebra $A$ with finite global dimension such that there are only finitely many basic tilting modules. Then every selforthogonal indecomposable module $M$ (that is a module ...
7 votes
3 answers
3k views

Using MAGMA for Group Theory

I've just started a PhD in Group Theory and need to use the computer programme MAGMA. I wonder if anyone could help me with a couple of (probably very basic things). I need to produce a Hasse diagram ...
12 votes
4 answers
3k views

Is Gauss-Seidel guaranteed to converge on *semi* positive definite matrices?

I know that the Gauss-Seidel method is guaranteed to converge given that the matrix you want to solve is positive definite. I've looked at the proofs of convergence, and it appears that one cannot ...
28 votes
4 answers
3k views

Can Gröbner bases be used to compute solutions to large, real-world problems?

In particular, suppose I have an affine algebraic variety over $\mathbb{R}^n$ described by generators of a radical ideal $I$ and I want to find (perhaps not all of the) points in the variety. Several ...
6 votes
1 answer
1k views

Computing kernels of maps of modules over a finitely presented algebra

I have the following problem: I have an associative (noncommutative) algebra $A$ defined over a rational function field $k = \mathbb{Q}(\delta, \lambda)$. $A$ is given by a presentation in terms of ...
6 votes
1 answer
141 views

Software for computing equivariants

If $\Gamma$ is a finite group with action on two vector spaces $\mathbb R^n$ and $\mathbb R^m$ denoted by $\gamma_n$ and $\gamma_m$ respectively, the fundamental equivariants are the polynomials $f: \...
11 votes
1 answer
470 views

Representing field elements in a computer

I'm wondering if there is existing terminology to describe fields $F$ with the properties below. I don't have a completely precise description of the concept I have in mind, but hopefully this will be ...

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