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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7 votes
1 answer
808 views

Computer power in plane geometry

I often hear that modern computer programs "may prove any theorem in elementary Euclidean geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for ...
0 votes
0 answers
70 views

Runtime order of Groebner basis computations over prime field

While there seems to be few resources detailing explicit complexity bounds for computing Groebner Bases over the integers, I'm finding it even harder to search for such bounds for computations over ...
3 votes
0 answers
53 views

Recommendations for distributed calculations of Groebner Bases

There are many computer algebra systems available which can compute a Groebner basis, including: Mathematica Singular Macaulay2 Magma Maple CoCoA However (please correct me if I've missed something) ...
7 votes
4 answers
6k views

Computational algebra: where?

I'm on my last semester of a math B.Sc. and about to start studying for a math M.Sc in the same institute. It now seems like a good time to start thinking of a PhD. I'm interested in both algebra and ...
1 vote
2 answers
293 views

Fastest way to solve non-negative linear diophantine equations

Let $A$ be a matrix in $M_{n \times m}(\mathbb{Z}_{\ge 0})$ without zero column. Let $V$ be a vector in $\mathbb{Z}_{> 0}^m$. Question: What is the fastest way to find all the solutions $X \in \...
5 votes
1 answer
177 views

Classification of multiplicative lattices

Question 1:Is there a classification of finite lattices which admit a multiplication making them into a finite multiplicative lattices? (see https://encyclopediaofmath.org/wiki/Multiplicative_lattice ...
3 votes
4 answers
721 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 \...
6 votes
0 answers
162 views

Proving the spectrum of the Young-Jucys-Murphy elements by formal computation in the degenerate affine Hecke algebra

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 ...
1 vote
0 answers
118 views

Computer algebra programs that can solve polynomial systems on algebraically closed fields besides MAGMA

I was wondering which computer algebra programs out there can solve polynomial systems on the algebraic closure of $\mathbb{Q}$ analytically and efficiently. So far, I only found MAGMA with its ...
0 votes
0 answers
66 views

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
0 answers
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}}...
4 votes
1 answer
203 views

Reliability of ILP approach to number-theoretic optimization

This question is inspired by the recent answer, where @RobPratt proposed to use integer linear programming (ILP) for solving a number-theoretic optimization problem. I will consider a very similar ...
1 vote
1 answer
244 views

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 ...
2 votes
3 answers
392 views

Useful software for variable elimination

I have three non-homogeneous trivariate polynomials in $\mathbb Z[x,y,z]$ and I want to eliminate the variables $y$ and $z$ to get a polynomial in $x$. The monomials of the polynomials are $\{1,x^4,x^...
1 vote
0 answers
252 views

Algebraic independence criterion

Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...
1 vote
1 answer
154 views

Find $x$ that solves $x\left(e^{\frac{a}{x}}-1\right)-y=0$

When trying to solve the equation in the title with WA, it produced the following as the solution: now, if you divide the numerator and denominator by $y$ and set $z:=-\frac{a}{y}$ the solution ...
1 vote
0 answers
34 views

Computer verification for hyperbolic trigonometry

I am currently writing a paper that requires some lengthy computations using basic hyperbolic trigonometry. So, several hyperbolic figures appear, and we apply the law of sines and so on in order to ...
50 votes
5 answers
13k views

The unification of Mathematics via Topos Theory

In her paper The unification of Mathematics via Topos Theory, Olivia Caramello says "one can generate a huge number of new results in any mathematical field without any creative effort". Is ...
25 votes
4 answers
5k 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(\...
0 votes
0 answers
55 views

Solving a large system of equations when a large subset factors

I have a large (~1000 equations) overdetermined system of degree 3 polynomials with ~140 variables. I would like to find a complete solution set for this system, however, the Gröbner basis seems to be ...
127 votes
74 answers
19k views

Most helpful math resources on the web

What are really helpful math resources out there on the web? Please don't only post a link but a short description of what it does and why it is helpful. Please only one resource per answer and let ...
13 votes
3 answers
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 ...
3 votes
0 answers
91 views

Finite global dimension via the Cartan determinant

Let $A=T(KQ)$ be the trivial extension algebra of a path algebra of Dynkin type $KQ$. The indecomposable module of $A$ correspond to the roots of $Q$ (and not just the positive roots as for $KQ$). Let ...
0 votes
0 answers
83 views

Computational tool for checking the existence of non-trivial rational zero of a cubic form

Suppose we consider a arbitrary cubic homogeneous form $f$ in in four or five variables over the rational field. Is there any computational tool or algorithm to check whether this cubic homogeneous ...
3 votes
1 answer
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 ...
3 votes
0 answers
57 views

Finding generators and relations for special commutative algebras with a computer

Let $K[x_1,...,x_n]$ be the polynomial ring in $n$ variables and $a_1,...,a_m$ elements in the quotient field $K(x_1,...,x_n)$. Let $A:=K[a_1,...,a_m]$ the ring generated by the $a_i$ in $K(x_1,...,...
136 votes
34 answers
33k views

Computer algebra errors

In the course of doing mathematics, I make extensive use of computer-based calculations. There's one CAS that I use mostly, even though I occasionally come across out-and-out wrong answers. After ...
5 votes
0 answers
67 views

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 ...
3 votes
0 answers
132 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 ...
11 votes
3 answers
527 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 ...
1 vote
0 answers
108 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 ...
1 vote
0 answers
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 ...
2 votes
0 answers
190 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 ...
49 votes
2 answers
6k views

Does there exist a complete implementation of the Risch algorithm?

Is there a generally available (commercial or not) complete implementation of the Risch algorithm for determining whether an elementary function has an elementary antiderivative? The Wikipedia article ...
8 votes
3 answers
2k views

Is there a MAGMA function to calculate the absolutely irreducible components of an algebraic curve defined over the rationals?

Given a curve defined over the rationals, is it computationaly possible to find all its absolutely irreducible components? Is there an implementation of this in the MAGMA program?
0 votes
0 answers
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 ...
20 votes
5 answers
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 ...
6 votes
0 answers
103 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} ...
4 votes
1 answer
106 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 ...
2 votes
0 answers
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 ...
1 vote
1 answer
284 views

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 ...
1 vote
1 answer
155 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 ...
10 votes
5 answers
6k views

Finding minimal or canonical expressions for Boolean truth tables

This is not an urgent question, but something I've been curious about for quite some time. Consider a Boolean function in n inputs: the truth table for this function has 2n rows. There are uses of ...
3 votes
2 answers
168 views

Finding the "Q-span" of vectors in Q(q)

Apologies if this question is quite basic. Consider the $\mathbb{Q}(q)$-vector space $V = \mathbb{Q}(q)^n$ with standard ordered basis $\{e_1,\ldots,e_n\}$. Suppose someone hands you some vectors $v_1,...
1 vote
0 answers
63 views

Fitting point on a Quadric curve [closed]

I am working on research project. I am currently using CloudCompare for my project, which calculates the Gaussian curvature and mean curvature to extract the geometric features of the points. I have a ...
1 vote
1 answer
116 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 ...
65 votes
3 answers
4k views

Reasons to prefer one large prime over another to approximate characteristic zero

Background: In running algebraic geometry computations using software such as Macaulay2, it is often easier and faster to work over $\mathbb F_p = \mathbb Z / p\mathbb Z$ for a large prime $p$, rather ...
0 votes
0 answers
152 views

Algorithm to compute automorphism group of a finite group

Is there an algorithm to compute automorphism group of a finite group? GAP has a function to do this, but while perusing their GitHub repo, I could not find an implementation. I'm struggling to find ...
3 votes
1 answer
224 views

Number of rings with additive group $(\mathbb{Z}_{16})^2$. A341547(16) in OEIS

I would like to know if somewhere the number of non-isomorphic rings with additive group $(\mathbb{Z}_{16})^2$ is mentioned. If not, is someone able to calculate it? And (easier) the commutative case? ...
30 votes
4 answers
5k views

How does Mathematica do symbolic integration?

I suppose there was at least once in our lifetime the point where we resorted to mathematica for help with an integral.-Unless you chose not to have the pleasure of using the continuum in your ...

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