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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Computing certain polynomials related to symmetric functions and $\lambda$-rings in Sage

The definition of a $\lambda$-ring (https://en.wikipedia.org/wiki/%CE%9B-ring) makes use of certain "universal" polynomials $P_n$ and $P_{n,m}$, which basically give you the formulas for ...
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Solving the trigonometric equation $\sin(x) + \sin(2x) + \sin(3x) - \sqrt 3 = 0$

Anybody can solve this equation for me? $$\sin(x) + \sin(2x) + \sin(3x) - \sqrt 3 = 0.$$ I imported it to my cas calculator and this was the output: $$\left\{x = 2k π + (1/3) π, x = 2k π + 0....
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Maxima CAS: Is there a way to drop multiplicative constants? [closed]

Is there a way in Maxima CAS to drop multiplicative constants? For example, $f(x):=a\cdot g(x)\cdot \sqrt{b\cdot x}$ would result in $f(x)\propto g(x)\cdot \sqrt{x}$.
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Maxima CAS: How to collect summands? [closed]

Is there a way to collect summands of a Maxima CAS expression into a list? For example a+b*x+sqrt(y+c*x^2) would result in list of 3 summands ...
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Euclidean division algorithm for (quadratic) number rings

I implemented in the computer Euclidean division for Gaussian integers. I’d like to implement it for other Euclidean number rings, or at least for the few norm-Euclidean quadratic rings. Any reference?...
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How to tell Maxima to "join like radicals"?

Maxima presents a solution to a problem in the form of a function formula that contains lots of bits looking like this: ...
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Maxima CAS: How to check if a function is used in expression?

Is there a way to check if, say, sum() function is used in expression in Maxima CAS ? For example, is there a function to return ...
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Finding all small monoids with the help of GAP

In GAP there is the command AllSmallGroups(n) to construct all finite groups of order $n$ up to isomorphism. Question: Is there a similar method (or package) in ...
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Maxima CAS: How to force log() into a product?

Is there a simplification way to turn log of a product into a sum of logs in Maxima CAS? I tried radcan(), but it does not change the expression. See the toy ...
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Maxima CAS: how to simplify a sum algebra to a sigma notation?

Is there a way in Maxima to simplify expression $f$ to the (sigma notation) form of $g$, i.e. to use summation of indexed variables? The summands ((X[i]-1)^2) are ...
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Maxima CAS: How to differentiate w.r.t. an expression?

Is there a way to differentiate with respect to an expression, instead of a single variable in Maxima CAS? Here is a toy example, which should give $\frac{\partial}{\partial x^2}x^2=1$: ...
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Maxima CAS: How to set domain of function's arguments and parameters?

How can I specify that $a>0$ and $f$ is defined on $(0,1)$ for a toy function $f(x|a)=ax$ in Maxima CAS? f(x):=a*x; That is $f:(0,1)\to(0,а)$
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Computer Algebra: How to find out when symbolic equality is decideable?

Richardson's theorem states that for a specific class of expressions $R$, $E \in R$, $E = 0$ is undecideable. This implies (To my understanding) that it is impossible to write a CAS that can solve all ...
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Macaulay2 how to change grading of ReesAlgebra for computing HilbertSeries

I am trying to compute the Hilbert-Samuel polynomials of some examples in Macaualy2. In my toy example, $R = \mathbb{Q}[x,y]/(x^2 - y^3)$ and I am considering the ideal $I = (x,y)$. Then I am looking ...
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Any exemple of a semi-regular sequence to compute a Gröbner basis?

According to the definition of a semi-regular sequence in this paper Hybrid approach for solving multivariate systems over finite fields page 5: Let {$p_{1},...,p_{m}$} $\subset \mathbb{K}[x_{1},...,...
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Computing rank of finitely-generated but large Z-module

I have a collection of $\mathbb{Z}$-modules, each defined by a set of formal generators $x_1, \ldots, x_n$ module a respective set of relations of the form $\sum_j a_i x_i = 0$. I would like to ...
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More convenient GAP code to verify Additional property of d-maximal groups

Let $G$ be a finite $p$-group and $d(G)$ be its minimal number of generators. We say that $G$ is $d$-maximal if $d(H) < d(G)$ for all $H < G$. The following code determines weather $G$ is $d$-...
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Coordinates of the largest small decagon

I've been trying to compute the abcissa $x_1$ of the first point of the largest small decagon in this slide show (p.13) using Maple. I let $x_1=\frac{2p}{1+p^2}, x_2=x_1+\frac{2q}{1+q^2}, x_3=x_2+\...
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Range reduction for the exponential integral?

I would like to compute $\operatorname{Ei}(x)$ to a reasonable precision. Fortunately, I have a computer algebra system (PARI/GP) which includes this function as a built-in, but unfortunately, $\...
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How to solve the following system of $4$ nonlinear equations

How to solve this equation system? $$\frac{-1}{x_1^{2} x_2 x_3} +y(x_2+2x_3) = 0, $$ $$x_3- \frac{1}{x_1 x_2^{2} x_3} + y x_1 = 0, $$ $$x_2 - \frac{1}{x_1 x_2 x_3^{2}} + 2yx_1 = 0, $$ $$x_1 x_2 + x_1 ...
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How do I operate very small numbers [0,1] in Logarithmic Number System

I found no many papers describing this topic, but one of them is this https://web.ece.ucsb.edu/~parhami/pubs_folder/parh20-cee-comp-w-lns-arithmetic-final.pdf Here the formula of representation of ...
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Minimal polynomials of "simple" algebraic numbers

This should be a fairly trivial question, to which I have nevertheless found no satisfactory answer. I am interested in effective algorithmic computation of minimal polynomials of some particularly ...
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Polynomials subrings vs. polynomial ideals

Let us consider $K[x_1,\ldots,x_n]$, the ring of multivariate polynomials in variables $x_1,\ldots,x_n$ with coefficients in some field $K$. With this, we consider the subring $R$ generated by a ...
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How to solve a system of equations with symbolic dimension computationally?

Suppose I have a vector of unknowns $x_i$, with $i=1,...,N$, where $N$ is symbolic, and I have some nonlinear system of $M$ equations, $$f_m(x_1,...,x_N) = 0\quad \forall m,$$ with $f_m: \mathbb{R}^N \...
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Why does "Turn! Turn! Turn!" equal 241217.524881?

If you search for "Turn! Turn! Turn!" on Google, then the second result is this YouTube video of The Byrds performing the Pete Seeger song of that name. But the first result is Google's ...
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Frequency of items in a list in MAGMA

I'm working in MAGMA and have ended up with a sequence with repeated entries. e.g. something somewhat (although much larger in my application) like this: ...
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Factorize with unknown in MAGMA

Say I have a polynomial $f(x)$ over a finite field $\mathbb{F}_p$ such as $$f(x) = x^2 + \alpha x + \alpha^2$$ for some $\alpha \in \mathbb{F}_p$. Then we can express the zeroes $x_0, x_1$ in terms of ...
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Calculations in a group ring with computer algebra system

I am currently trying to read the paper Constructions of difference sets by Applebaum et al. Unfortunately, there are some very elaborate calculations. For example, in example 1.13 in the group ring $\...
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Ideal polynomial equations in Macaulay2

Given a known ring $R$ and an ideal $I\subseteq R$, I would like to be able to solve an "ideal polynomial equation" of the form $$\sum_i I_i^{\alpha_i}=I,$$ i.e to find ideals $I_i$ so that ...
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Find minimal condition number from a non-square matrix

I have an optimization problem for a computational problem and I would like to know how I can solve it. I have a matrix $\left[A\right]_{n \times m}$ with $n < m$ and I would like to get a matrix $\...
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Closed form for definite integral involving Bessel function, $K_1$

Wolfram Alpha knows that, and it can be calculated that: $$ \int_0^1 \exp\bigg(\frac{1}{\log x}\bigg)~dx=2K_1(2).$$ Where $K_1$ is a modified Bessel function of the second kind. I wanted to find out ...
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Using Gröbner bases and Cylindrical Algebraic Decomposition to solve real polynomial systems

I'm working on a project that involves solving systems of multivariate polynomial equations over the reals (and find their real solutions). Assuming that a system has a finite number of complex ...
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Algorithm for finding an isotropy group of a matrix

I'm trying to implement the Cartan-Karlhede algorithm (details) on a computer algebra system (SageMath, specifically), and I'm having trouble with a step that involves determining the isotropy group ...
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1answer
86 views

Desmos plot only for integral $x$ [closed]

I'm trying to plot $2\operatorname{floor}\left(\log_{2}x\right)+1$ but only if $x$ is a natural number. Desmos plots correctly $2\operatorname{floor}\left(\log_{2}x\right)+1$ I would like this ...
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Can I get a polynomial out of a polynomial division? [closed]

I have a to divide a thrid degree polynomial by another one, is it possible for me to get another polynomial out of it? By the way, I am working with decimal numbers (floats in 32bits). If it is ...
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1answer
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Do Groebner bases over rationals lose information about complex solutions?

Given a system of polynomial equations, if I calculate the Groebner basis over the rationals and get {1} then: My understanding is that I can then conclude there are no rational solutions. Is this ...
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587 views

When does a system of polynomial equations have infinitely many solutions?

It seems intuitively correct to say that a system of polynomial equations has finitely many solutions if there are as many equations as there are variables in the system. However, how would you prove ...
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1answer
115 views

Is it possible to give a time restriction to a GAP program? [closed]

I have written a GAP program that returns "true" or "false" as output. I would like to modify the program in the following way: If the computation is not finished after 20 seconds, ...
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1answer
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Name of matrix operation of [ A[0] dot B[0], A[1] dot B[1] ] from 2x2 matrices A, B

Please advise how this matrix operation is called, and what is the numpy operation for it. np.inner, np.dot does not create dot ...
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Computing Groebner Basis with given generators and relations of a module

Suppose I have a module $M$ over a polynomial ring $R=k[X_1,\ldots,X_n]$, viewed as a submodule of $R^m$. I know the generators of $M$ $$ v_1=[a_{11},a_{12},\ldots,a_{1m}],\\ v_2=[a_{21},a_{22},\ldots,...
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Testing whether a module is a two-sided ideal

Let $A$ be a finite dimensional algebra and $T$ a (tilting, if that helps) module of $A$. Question: What is the best way to test whether $T$ is (isomorphic to) a two-sided ideal of $A$ using the GAP-...
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Help understanding the result of a Cylindrical Algebraic Decomposition of $V\left( {x,y,z} \right) = {x^2} + {y^2} + {z^2} < 1$?

From https://mathworld.wolfram.com/CylindricalAlgebraicDecomposition.html it is said that: Define a cell in ${\mathbb{R}^1}$ as an open interval or a point. A cell in ${\mathbb{R}^{k + 1}}$ then has ...
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Computing the number of conjugacy-classes in $GL_{n}(\mathbb{F}_{p})$ of elementary abelian p-subgroups by GAP and Magma

I'm trying to compute the number of conjugacy-classes of elementary abelian p-subgroups of rank $2$ in $GL_{n}(\mathbb{F}_{p})$ by GAP and Magma. So I consider the following GAP function: ...
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1answer
95 views

Computing the rank of a specific group

The rank of a group is the minimum number of elements needed to generate the group. In general, the rank of a group is not algorithmically computable for finitely presented groups. I have a specific ...
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53 views

What's inherently wrong with these Fourier transforms?

I was trying to write a formula for antidifference operator $\Delta^{-1}=(e^D-1)^{-1}$ using Fourier transforms. I obtained the formal formula: $$\Delta^n[f](x)=\frac1{2 \pi }\int_{-\infty }^{+\infty }...
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1answer
89 views

Testing for isomorphism between normal subgroup lattices

The question is whether we can use GAP to construct the normal subgroup lattice of a finite group $G$, call it $\mathcal{N}(G)$, and to then test whether $\mathcal{N}(G) \cong \mathcal{N}(H)$, where $...
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79 views

How to do the given Line Integral problem in Maple ? Find the work done by the force F along C.

How to do the given Line Integral problem in Maple ? Find the work done by the force F along C. F(x, y, z) = (y, −x, −z), C consists of the line segments from (0, 1, 0) to (3, 2, −1), (3, 2, −1) to (−...
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1answer
73 views

Producing a large list of irreducible polynomials of given degree over a specific finite field using a CAS?

I'm curious if there are computer algebra systems that allows one to easily produce a list of irreducible polynomials of a given degree $n$ in $\mathbb{F}_p[X]$, where $p$ is a specific prime. Does ...
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1answer
25 views

Complexity of composing a multivariate polynomial with a vector univariate polynomial.

Let $f(x) = (f_1(x),...,f_n(x)) \in \mathbb{Q}[x]^n$, and $g \in \mathbb{Q}[x_1,...,x_n]$. Assume $\deg(f_i) \leq b$ for all $i$, and $\deg(g) \leq c$. Is there an efficient algorithm for computing $g ...
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170 views

$1/(x^6 + 1)$ partial fraction decomposition, with computer?

This is a re-post from StackOverflow, I was advised to post it here. https://stackoverflow.com/questions/64101194/partial-fraction-decomposition How do I find the constants A,B,C,D,K,S such that $$ \...

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