# 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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### 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 ...
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 ...
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) ...
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
293 views

1 vote
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### 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
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
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 ...
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 ...
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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 ...
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 ...
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 ...
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
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
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### 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 ...
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 ...
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 ...
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?
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### 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 ...
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 ...
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} ...
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 ...
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
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
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 ...
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 ...