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.
575
questions
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49
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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 ...
- 181
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votes
1
answer
76
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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 ...
- 116
1
vote
0
answers
19
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Normal Basis using complex coefficients in SageMath [closed]
The following code does not work with complex coefficients:
...
- 99
0
votes
0
answers
13
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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 ...
- 1,469
2
votes
0
answers
25
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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 ...
- 1,469
2
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0
answers
25
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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 ...
- 1,469
1
vote
1
answer
60
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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 ...
- 102
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0
answers
94
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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) ...
- 890
0
votes
0
answers
46
views
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 ...
- 253
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0
answers
34
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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&...
- 579
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votes
1
answer
69
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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.
...
- 35
0
votes
1
answer
35
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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 ...
- 291
2
votes
1
answer
38
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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:
...
- 83
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45
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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
\begin{equation}
C=\{\sum_{i=1}^mt_i\alpha_i|\ t_i\...
- 321
4
votes
1
answer
200
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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 ...
- 7,283
4
votes
1
answer
112
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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)^{...
- 194
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votes
0
answers
31
views
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 ...
- 1,443
2
votes
1
answer
83
views
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 ...
- 2,334
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votes
0
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20
views
Computationally evaluating messy symbolic sums involving geometric series
Let $t$ be a positive integer, let $p$ be a prime number, and let $q$ be a real number. I need to evaluate the sum
$$
\sum_{\substack{1 \leq c \leq t \\ c \not \equiv 1 \pmod{p}}} q^{-\big((p-2)c + \...
- 3,931
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votes
0
answers
34
views
Numerically extracting generators of polynomial ideal from witness sets of an algebraic variety
I have been using Bertini, a numerical algebraic geometry tool, to explore the (positive dimensional) space of complex solutions to a set of polynomials. While the details of its techniques are ...
- 890
1
vote
2
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101
views
If $R \sin \psi = r \sin \phi$ and $R \cos \psi = -ar \cos \phi$, what are $R$ and $\psi$?
Given $$-ar \cos \phi = R \cos \psi \\
r \sin \phi = R \sin \psi$$
how do I find $R, \psi$ in terms of $r, \phi$?
My work so far below:
$$\tan \psi = \frac 1 {-a} \tan \phi \\
\psi = \arctan(\frac {-...
- 4,470
1
vote
0
answers
45
views
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 ...
- 890
9
votes
2
answers
574
views
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 ...
- 1,039
-1
votes
1
answer
79
views
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, ...
- 2,171
0
votes
1
answer
62
views
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?
- 3,950
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votes
2
answers
79
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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^...
- 10.8k
4
votes
1
answer
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views
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 ...
- 151
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vote
0
answers
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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 ...
- 11.4k
3
votes
1
answer
135
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Which CAS can do non-commutative differential algebra?
I am looking for a CAS (possibly incl. additional packages/libraries) that can compute generic non-commutative differential expressions. Let me illustrate what I mean by two examples.
Let $(R,\partial)...
- 1,184
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0
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84
views
Calculate $\int_{\mathbb{R}^3} x^2\exp(-a\sqrt{x^2+y^2+z^2})d\mathbf{x}$ without spherical coordinates
Which tool can I use to (numerically?) evaluate
$$
\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty x^2\exp\left(-a\sqrt{x^2+y^2+z^2}\right) dx ~ dy ~ dz,~a>0
$$
without having to ...
- 143
1
vote
1
answer
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Variant of Segre embedding
We work over a field $k$. We know that there is the Segre embedding $\def\P{\mathbb{P}} \P^2 \times \P^1 \to \P^5$.
Now I want an embedding of $\P^2 \times \def\A{\mathbb{A}}\A^1$ into some projective ...
- 631
2
votes
1
answer
158
views
Multiplicity of a singular point, Ideals, and Maple/Algorithms
I am teaching myself about algebraic geometry, and the classification of singular points on algebraic curves $f(x,y)=0$, where $x,y\in\mathbb{C}$.
One way to classify these singular points (a set of $...
- 107
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votes
0
answers
35
views
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 ...
- 1,020
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votes
1
answer
43
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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 ...
- 1,072
1
vote
1
answer
78
views
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 ...
3
votes
0
answers
86
views
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 ...
- 890
1
vote
2
answers
101
views
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 ...
- 890
1
vote
0
answers
39
views
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 $\mathrm{S}^kV$ ...
- 163
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votes
1
answer
48
views
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:
...
- 2,334
3
votes
1
answer
127
views
Problem about the Galois group elements in Sage
I am computing a Galois group as below in Sage.
K.<a> = NumberField(x^4-2)
L = K.galois_closure('b')
G=L.galois_group()
Here, $G=\text{Gal}(L/\mathbb{Q)} \...
- 2,817
1
vote
0
answers
41
views
Computing the cohomology of a complex over $F[x,y]$
Is there a method for computing the generators of the cohomology of a chain complex over the ring $R=F[x,y]$?
When $R$ is a PID one can use Smith normal form and there are libraries in Sage that ...
- 2,225
0
votes
1
answer
204
views
Pari GP/Magma: how to calculate the Galois group over arbitrary number fields? [closed]
Let's say I have a polynomial $f(x) = x^5 + 3x + 5$. You can check on Pari that the Galois Group over $\mathbb{Q}$ is $S_5$. $K$ be the intermediate field obtained by adjoining a root of $f(x)$ to $\...
- 63
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votes
1
answer
72
views
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
...
- 2,817
2
votes
1
answer
98
views
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 ...
- 99
4
votes
1
answer
86
views
Index of an explicit subgroup of $\mathrm{GL}_4(\mathbb{Z})$
Let $H$ be the subgroup of $\mathrm{GL}_4(\mathbb{Z})$ generated by the $4!$ permutation matrices together with
$$
\begin{pmatrix}
1 & 0 & 0 & 0 \\
-1 & 0 & 1 & 1 \\
0 & 1 &...
- 9,694
3
votes
0
answers
106
views
Best computer algebra system for non-commutative computations
I'm interested in simplifying expressions like
$$\int_{-\infty}^\infty dq_1\int_{-\infty}^\infty dq_2\int_{-\infty}^\infty dq_3f(q_1,q_2,q_3)\langle\ |b(p_1)b(p_1)a^+(q_1)b(q_2)b(q_3)b^+(r_1)b^+(r_2)a^...
- 161
2
votes
1
answer
83
views
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?
- 3,625
1
vote
0
answers
142
views
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 ...
- 923
1
vote
0
answers
46
views
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 ...
- 6,170
2
votes
3
answers
353
views
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, ...
- 133