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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.

3
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1answer
106 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
0answers
80 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
1answer
189 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
2answers
159 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
0answers
180 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\...
3
votes
1answer
134 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
1answer
74 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
1answer
84 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
0answers
45 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
0answers
50 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 ...
4
votes
0answers
69 views

Computing the volume of intersection between a hyper-rectangle and a ball

$C$ is a region bounded above, below coordinate-wise by $\overline{c},\ \underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex with $d$ that computes the volume of $C\cap B(0,1)$ ...
0
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0answers
23 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
1answer
157 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 ...
6
votes
1answer
99 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: \...
10
votes
1answer
428 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 ...
3
votes
0answers
71 views

Deterministic procedure to find irreducible polynomials

In $\Bbb F_q[x_1,\dots,x_n]$ given $d_1,\dots,d_n\in\Bbb N$ is there a deterministic $O(poly(nd\log q))$ algorithm to find an irreducible polynomial with $d=\max_{i\in\{1,\dots,n\}}d_i$ and $d_i=deg(...
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0answers
25 views

Grobner basis of the toric ideal $I_{A_P}$ with respect to $<_{rev}$ consists of those binomials $t_αt_β − t_{α\cap β} t_{α\cup β}$

I try to understand the proof of the Theorem. 10.1.3.(page 185) from ''Monomial Ideals'' by Herzog & Hibi. The reduced Grobner basis of the toric ideal $I_{A_P}$ with respect to $<_{rev}$ ...
0
votes
0answers
73 views

Shape Basis Generator of an Ideal

Let $R=K[x_1,...,x_n]$, for a field $K$, and $I$ is radical zero dimensional ideal of $R$. We define $s=\Sigma_{i=1}^{n}a_ix_i$, with $c_i\in K$, a shape basis generator of $I$, if $(s+I)$ generates ...
5
votes
1answer
108 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 in closed form with help of Lambert W function or with a generalization of Lambert W function? I gave a method in MSE here. Which algorithms ...
2
votes
1answer
280 views

Generalized Newton Identities

I learnt a lot of new words (Hall-Littlewood, Jack and Macdonald polynomials) but unfortunately everything I dug up is written without a single example and I still don't know the answer to a very ...
10
votes
1answer
319 views

How do computer algebra packages like Sagemath implement rank of a matrix

I am not sure if this is the right place to ask this question, but I believe there will be people here who do computations on computer algebra packages like Sage in their work. I have been using ...
21
votes
3answers
3k views

Minimal polynomial of cos(π/n)

I know that $\cos(\pi/n)$ is a root of the Chebyshev polynomial $(T_n + 1)$, in fact it is the largest root of that polynomial, but often that polynomial factors. For example, if $n = 2 k$ then $\cos(\...
17
votes
1answer
447 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 ...
6
votes
1answer
281 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 ...
2
votes
1answer
139 views

Memory usage of Gröbner basis computation

I've been calculating some Gröbner bases in preparation for finding non-commutative Hilbert series (and, once I recreate that, characters of group actions). Specifically, I've been using the ...
6
votes
6answers
906 views

Computation of a minimal polynomial

It is relatively easy (but sometimes quite cumbersome) to compute the minimal polynomial of an algebraic number $\alpha$ when $\alpha$ is expressible in radicals. For example, the simple query "...
3
votes
0answers
58 views

Can one do better than using general purpose determinant algorithms when using the Fisher-Kasteleyn-Temperley method for perfect matchings?

Questions. (numerical.generalPfaffian) Is it proved anywhere that in general it is not easier0 to calculate the determinant (over $\mathbb{Q}$) of the skew-symmetric signed adjacency matrix defined ...
3
votes
1answer
173 views

Explicit endomorphisms of Jacobians of genus $2$ and the Theta divisor

I hope this question is good to be here. Let $J$ be the Jacobian of a hyperelliptic curve $H$ of genus $2$ given by $y^2 = x^5 + h$. I was calculating an explicit formula in Mumford coordinates of $[...
3
votes
0answers
155 views

Explicit roots in algebraic extention of Q with roots

Denis Bouhineau in "Solving Geometrical Constraint System Using CLP Based on Linear Constraint Solver" gave a method to find explicit square root in algebraic extention of Q with square roots. For ...
2
votes
2answers
201 views

Computer algebra for calculating curvature when the tensor metric is very big

Is there a computer algebra method to compute the curvature of a Riemannian metric on the plane when the metric tensor has long entries $E,F,G$ The computation by hand is very ...
0
votes
1answer
116 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 ...
2
votes
0answers
137 views

Finding Rational Curves on a Surface

Let the field of rational numbers be our base field $k$. I hope to find all rational curves on the following surface $S$ defined by $f$. You can find the motivation in the end. $f= (x^2y^2)z^3 + (5x^...
1
vote
0answers
105 views

generalized sieve implementations

Suppose I have a multiplicative function $f$ and I want to compute it for all integers from $1$ to $N$ (or maybe just for a long sequence of consecutive integers). It is clear that this is far faster ...
1
vote
1answer
120 views

Omitting constraints of polynomial system

Let $n_1, n_2 \geq 1$ be known integer constants. Suppose that we have the following system of $n$ polynomial inequalities for which we know that there exists a feasible solution $(p_1, p_2) \in (0,1)...
9
votes
3answers
586 views

Web interface for GAP (or other computer algebra system dealing with finite groups)?

GAP is computer algebra system which allows to make calculations with finite groups. (See wikipedia link for an example). Is there web interface for it ? (I cannot google it.) Or may be some other ...
4
votes
2answers
302 views

(Efficient) computation of symmetric powers of square matrices

I'm looking for software that can compute symmetric powers of medium-size square (say rational, 100 by 100) matrices, and ideally can do so efficiently if the matrix is sparse enough. I haven't found ...
8
votes
1answer
636 views

How to compute with the Stark conjectures?

I would like a convenient basis for the elements of a fixed abelian extension $E$ of a real quadratic field $\mathbb{Q}(\sqrt{d})$. The accepted answer to this MO question suggests that the Stark ...
6
votes
1answer
244 views

Check irreducibility of an explicit polynomial, without computer

I have a polynomial of degree 8 in 6 variables given explicitly by $$ (\sqrt{1+(x_1+x_2+x_3)^2+(y_1+y_2+y_3)^2}+\sqrt{1+x_1^2+y_1^2}+\sqrt{1+x_2^2+y_2^2}+\sqrt{1+x_3^2+y_3^2})\times\text{the other ...
6
votes
0answers
106 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
1answer
135 views

speeding up Gosper and WZ algorithms

In our ongoing work to speed up symbolic summation and other similar algorithms in Sagemath, we notice that naive implementations of Gosper and Wilf-Zeilberger (a.k.a. WZ) algorithms are usually quite ...
1
vote
0answers
117 views

Coefficient perturbations of polynomials with real roots only

Let \begin{align} P(x) &= x^n+a_{n-1} x^{n-1} +\ldots+a_0 = \prod_{i=1}^n (x-p_i)\\ Q(x) &= x^n + b_{n-1} x^{n-1} + \ldots +b_0 = \prod_{i=1}^n (x-q_i)\\ p_i, q_i& \in \mathbb{R},\ 0<...
22
votes
1answer
570 views

two's and three's survive in gcd of Lagrange

Lagrange's four square_theorem states that every positive integer $N$ can be written as a sum of four squares of integers. At present, let's focus only on positive integer summands; that is, $N=a_1^2+...
6
votes
1answer
225 views

Problem on triangles

Let $T\subset \mathbb{R}^2$ be any triangle and $T^t$ a deformation of $T$. Call $l_1,l_2,l_3$ the squares of the lengths of the sides of $T$ and $l_1^t,l_2^t,l_3^t$ the squares of the lengths of the ...
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vote
0answers
39 views

How to compute the set of minimal elements of Σ(L)?

Let $L=(f)\subseteq\mathbb{Z}[x]$ be the ideal generated by $f$. For $g\in L$ such that $\textrm{lc}(g)>0$, write $g=g^+-g^-$, where $g^+$ and $g^-$ are the positive part and the negative part ...
8
votes
1answer
143 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 ...
23
votes
0answers
654 views

Where to submit this work with several unusual features?

I appreciate that questions about where to submit are generally considered off-topic, but I hope that the unusual features of the present case may make it acceptable. I have put a monograph on github ...
1
vote
0answers
139 views

Efficient deterministic algorithms of factorizing

My question is about efficient deterministic algorithms of factorizing polynomials of degree $n$ over $\mathbb{F}_q$. Are there such algorithms that use poly$(n, \log q)$ bit operations? I know ...
12
votes
1answer
792 views

An efficient isomorphism between finite fields

Let $p$ be a prime number. Let $f$ and $g$ be irreducible polynomials over $\mathbb{F}_p$, both of degree $n$. We know that factor-rings $\mathbb{F}_p[x]/(f)$ and $\mathbb{F}_p[x]/(g)$ are isomorphic ...
11
votes
1answer
447 views

Verify that a group is hyperbolic via computer algebra

I would like to know whether there is some computer algebra software that can be used to verify if a group, given by a finite presentation, is hyperbolic (in the sense that it terminates with "yes" if ...
5
votes
1answer
146 views

Groebner bases for differential operators with field coefficients (reference request)

Let $K$ be a field, $\partial_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial_1\ldots \partial_n]$ (it is implicitly assumed that the derivations do not commute with elements ...