# Questions tagged [computer-algebra-systems]

A computer algebra system (CAS) is a program which is able to carry out various symbolic manipulations with mathematical expressions. Some well-known computer-algebra systems are Mathematica, Maple, Wolfram Alpha, GAP, SAGE. Questions on this site about computer algebra systems should be mathematical in nature, and not just about the syntax or the mechanics of the CAS. Otherwise the question would be better suited for a different Stack Exchange site.

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49 views

### References on a variant of Geometric Calculus

Geometric algebra and (standard) calculus, when synthesized, give rise to geometric calculus, a very powerful formalism. I have read a bit about fractional calculus and time-scale calculus, both very ...
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### How to tell WolfranAlpha that variables given in the equation are positive integers? [closed]

This question is similar to one here, but it's not the same. So, I want to ask WolframAlpha with math input (I don't know natural language) if the following identity holds (of course it does, it's a ...
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1 vote
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### Normal Basis using complex coefficients in SageMath [closed]

The following code does not work with complex coefficients: ...
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### How can I declare a ring in Singular CAS with variable names coming from a list of strings?

The following code constructs a list of strings list l = ("x1", "x2", "x3"); Now I want to declare a ring with variables ...
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### Algorithm to determine closedness of orbits?

Consider a reductive group $G$ acting on an affine variety $X$. It is known that for every $x\in X$, we have $G.x\subseteq\overline{G.x}$ is open dense. Then $\partial({G.x})\subseteq X$ is a closed ...
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### Algorithm computing invariants of $\mathrm{SL}(2)\curvearrowright\mathbb{A}^5$.

Consider a field $\Bbbk$ of characteristic zero. The natural action $\mathrm{SL}(2)\curvearrowright\Bbbk^2$ induces an action on $\Bbbk^5\cong\mathrm{Sym}^4(\Bbbk^2)$. View $\Bbbk^5$ as the affine ...
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### Solve 1 Exponential Equation With 3 Initial Conditions [closed]

I have an exponential equation, \begin{align} y = A - Be^{kx} \end{align} where $(x,y) = (0,3500)$, $(x,y) = (6600,100)$, and $(x,y) = (6000,1000)$ are the initial conditions (IC). I tried ...
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### How to break symmetry of a polynomial ideal to simplify Groebner basis?

I have an ideal $I$ generated by a set of polynomials $\{ p_i \}$. There are some variable permutations to which the ideal is symmetric. By this I mean (apologies if there is a standard term for this) ...
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### Software language specification: null VS empty objects.

I have noticed that in software language specifications, there is pretty much always a NULL element and I am wondering if it is strictly necessary and how it maps to algebraic structures, given that ...
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### Finding the general term of a given P-recursive sequence

I managed to prove that a certain sequence of rational numbers $(c_n)_n$ satisfies the following P-recursive equation: $$3(n-1)(n+2)c_{n+2}=(3n^2-3n+2)c_{n+1}+6nc_n$$ with "initial conditions&...
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### Expand product of sums in sympy by introducing new dummy indices

I want to use sympy to simplify some expressions which contain products of sums, this will require expanding out the products and cancelling equal terms. ...
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### MAGMA: How to efficiently do coercion of element of HomGrp to element of GrpAuto? [closed]

Suppose I have a finite $p$-group $G$ as GrpPC in MAGMA. The computation of the automorphism group $\mathrm{Aut}(G)$ takes a very long time. Suppose that I also ...
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### GAP orthogonal groups: Specifying the invariant bilinear form

If I read the GAP manual correctly, one should be able to specify the underlying invariant bilinear form, when constructing an orthogonal group. However, when I try something like: ...
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### Reference Request: Computer program for calculating conical hull

Suppose we are given several vectors $\alpha_1,\dots,\alpha_m$ in an $n$-dimensional space. The convex cone spanned by them is defined to be the set C=\{\sum_{i=1}^mt_i\alpha_i|\ t_i\...
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### Is There an Open Source Equivalent for Mathematica's SeriesCoefficient Function?

I am exploring options for using computer to extract the coefficient of $x^n$ from the Taylor series expansion of a function of $x$, where $n$ is kept as a symbolic entity. While Mathematica offers a ...
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### Simplifying rational function in Sage

Let $g\geq 1$ be an integer and define three rational functions and their sum by \begin{align*} H(q,t) &= f_1(q,t) + f_2(q,t) + f_3(q,t) \\ &:= \frac{t^{8g-4}q^{2g-1}(1+tq)^{2g-1}(1+q^2t^3)^{...
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### Degree zero part of localized graded ring in computer algebra

Does anyone know how to do the following computation in a standard computer algebra package (MAGMA, SAGE, Macualy2) Let $S$ be a graded ring given by explicit generators (not necessarily all in degree ...
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### Finding orbits of an action on a set of basis vector with GAP

I work with the computer algebra system GAP in this question. Let $K$ be a field (for example the rational numbers). I have a set $W$ consisting of sets of vectors basis of $K^n$ that only have ...
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### How to use CAS to determine if complex solutions to system of polynomial equations are connected or disconnected?

Given a system of polynomial equations in several complex variables, is there a way to determine if the solution space has disconnected components using some computational algebra system like ...
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### Can anyone come up with a method for integrating product of logarithms?

So recently I’ve been studying integrals of products of logarithms $$\int \prod_{n=1}^N \log(x+a_n) dx$$ Can anyone come up with general method for dealing with such integral? Let's start simple, with ...
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### Compute integral closure using a computer

Suppose given an inclusion $A\subset B$ of finitely-presented commutative algebras over a field. Is there a CAS which can decide whether $B$ is a finite $A$-module? What if instead of f.p. k-algebras, ...
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### Iterate through the finite groups in MAGMA

How may I iterate over the finite groups in MAGMA of orders between $1$ and $N$ (where possible), and compute a given property of them?
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### How to use computer algebra system

Consider the two vector-valued functions \begin{align} &f(x,y)=(x+y^2, y+x^3) \\ & g(x,y)=(2x+y^3, 2y+x^4). \end{align} Then \begin{align} f(g(x,y))&=((2x+y^3)+(2y+x^4)^2,(2y+x^4)+(2x+y^...
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### Why do CAS struggle with $\int\frac{dx}{1+\sinh x}$?

$\int\frac{dx}{1+\sinh x}$ is a slightly annoying but still easily solved integral using a weierstrass substitution and PFD. I'm mainly referring to WolframAlpha, but I've seen other computer algebra ...
1 vote
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### Computer Algebra System for Homotopy Types of Finite Simplicial Sets

I hope this is the right place to ask the following question, if not please guide me towards the correct forum... I want to show that two certain finite simplicial sets are not weakly homotopy ...
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### Can you solve $X A X^T = B$ for $X$ over Gaussian rationals in GAP?

I have two symmetric matrices : $A$ is $a \times a$ and $B$ is $b \times b$. I'm trying to find solutions for $X A X^T = B$. $A$ and $B$ are constant and $X$ is the unknown $b \times a$ matrix. All ...
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### Converting a group element to an ordered list of generators in GAP

Consider the free group on {a,b} which is gap is constructed with FreeGroup("a","b"). I want to cyclically reduce words. For example, given word $a^{-1}*b*a$, this would be ...
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1 vote
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### Automatic Differentiation and Product Rule

There is an explanation for the algorithm for Automatic Differentiation on the MathWorks Website. Their example equation is: $$f(x) = x_1 \exp\left(-\frac{1}{2}(x_1^2 + x_2^2)\right)$$ Their ...
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### Groebner basis over rational vs finite field

Some algorithms for calculating a Groebner basis are optimized for calculating with coefficients in a finite field. Having determined the basis over a finite field, I'd like to understand what ...
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### Polynomial constraints for restricting: $a=0$ if and only if $b\neq 0$

For this discussion, I will be considering polynomials over multiple complex variables, and a system of polynomial constraints, where the constraints on the variables can be written as a set of ...
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### Computer algebra system for Tensor Algebra

I am searching for a computer algebra system that allows to tackle the following problem: Let $V = \mathbb{R}^3$. Consider the free commutative algebra $\mathrm{S}^\bullet V$, with $\mathrm{S}^kV$ ...
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### Viewing monoid rings as rings with identity in GAP

I look at monoid algebra of finite monoids with GAP and want to force GAP to view them as algebras with one. But it seems it does not work: ...
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### Question about automorphisms of the cyclotomic field and Sagemath

Let $K = \mathbb{Q}(\zeta_5)$ be the fifth cyclotomic field. I write the following code in sage ...
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### Central Products in SageMath

I would like to compute the Central Product of two groups in SageMath. I cannot find any builtins and I'm not sure what group theory packages for Sage may exist. Is there anything out there or am I ...
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### Algorithm to check whether rational polynomial is irreducible over $\mathbb{C}$?

Let $F\in\mathbb{Q}[x_1,\ldots,x_n]$ a multivariate polynomial. Is there an algorithm for checking whether $F$ is irreducible over $\mathbb{C}$? Is this implemented somewhere?
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### How to calculate Hilbert polynomial

Consider the ideal $I=\langle x_0^2+x_1^2+x_2^2+x_3^2,x_0x_1+x_2x_3\rangle \subseteq k[x_0,x_1,x_2,x_3]$ for $k=\mathbb{F}_{32749}$. I am trying to calculate the dimension of the projective variety ...
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### Unit group structure GAP code. [duplicate]

I want the structure of the unit group of the group algebra $F_{3^k}D_{30}$ using GAP, where $F_{3^k}$ is any finite field of characteristic $3$ and $D_{30}$ is the dihedral group of order $30.$ I ...
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### How to solve a trinomial with a rational exponent $x^{q}-x-c=0$

I require solutions to equations of the following form: $x^{q}-x-c = 0$ where $q$ is a rational number very close to 1 with $q$ such that: $$q=\frac{p^k+1}{p^k}$$ $$q=\frac{p^k}{p^k-1}$$ $p$ prime, ...
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