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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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: (...
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0
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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
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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
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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
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0
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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
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3
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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
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1
answer
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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
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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
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1
answer
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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
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0
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122
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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
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0
answers
63
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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
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2
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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
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2
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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
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0
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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
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2
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443
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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
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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
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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
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2
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328
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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
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1
answer
126
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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
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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/...
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1
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131
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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
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580
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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
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0
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175
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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
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148
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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
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0
answers
57
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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
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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
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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
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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
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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
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0
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134
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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0
answers
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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
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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
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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
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3
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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
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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
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4
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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
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1
answer
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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
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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
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1
answer
470
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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 ...