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.
368
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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 ...
129
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74
answers
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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 ...
65
votes
2
answers
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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 ...
65
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3
answers
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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 ...
50
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5
answers
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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 ...
43
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3
answers
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Is there a systematic method for differentiating under the integral sign?
This MO question by Tim Gowers reminded me of a question I've wondered about for some time. In the delightful book Surely You're Joking, Mr. Feynman!, Feynman praises the technique of differentiating ...
40
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23
answers
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Open source mathematical software
I want some recommendation on which software I should install on my computer. I'm looking for an open source program for general abstract mathematical purposes (as opposed to applied mathematics).
I ...
32
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4
answers
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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 ...
28
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4
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Can Gröbner bases be used to compute solutions to large, real-world problems?
In particular, suppose I have an affine algebraic variety over $\mathbb{R}^n$ described by generators of a radical ideal $I$ and I want to find (perhaps not all of the) points in the variety. Several ...
28
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1
answer
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Algebraic dependency over $\mathbb{F}_{2}$
Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$
such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall i\in[n]:f_{i}(a)=a_{i}$....
26
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5
answers
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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(\...
26
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0
answers
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Where to submit this work with several unusual features?
I appreciate that questions about where to submit are generally considered off-topic, but I hope that the unusual features of the present case may make it acceptable.
I have put a monograph on github ...
24
votes
5
answers
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Fastest algorithm to compute the sum of primes?
Can anyone help me with references to the current fastest algorithms for counting the exact sum of primes less than some number n? I'm specifically curious about the best case running times, of ...
24
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7
answers
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Computer package for representation theory of the symmetric group
Is there a computer algebra package in which I can compute the following for representations of a specific symmetric group (e.g. $S_7$):
(1) $V \otimes W$
(2) $S_\lambda V$, where $S_\lambda$ is a ...
24
votes
1
answer
682
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two's and three's survive in gcd of Lagrange
Lagrange's four square_theorem states that every positive integer $N$ can be written as a sum of four squares of integers. At present, let's focus only on positive integer summands; that is, $N=a_1^2+...
23
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2
answers
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What is currently feasible in invariant theory for binary forms?
When Paul Gordan became a professor in 1875 he could show the binary form in any degree has some finite complete system of (general linear) invariants, but he could not actually give a complete system ...
21
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4
answers
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Is there a way of canonically labelling permutation groups?
When working with large numbers of graphs, a canonical labelling routine is essential as, after the initial cost of canonically labelling each graph, it permits isomorphism checks to be replaced with ...
21
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5
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Fast computation of a Groebner basis. What is possible?
I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or ...
21
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1
answer
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Main open computational problems in quantifier elimination?
A language is said to have quantifier elimination if every first-order-logic sentence in the language can be shown to be equivalent to a quantifier-free sentence, i.e., a sentence without any $\forall$...
20
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5
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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 ...
20
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2
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Where to publish computer computations
In a paper I developed some theory; some of the applications require extensive computations that are not part of the paper. I wrote a Mathematica notebook. I want to publish a PDF and .nb version ...
18
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3
answers
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What is the minimum size of a partial order containing all partial orders of size 5?
This earlier MO question asks to find the minimum size of a partial order that is universal for all partial orders of size $n$, i.e. any partial order of size $n$ embeds into it, preserving the order. ...
17
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4
answers
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An experiment on random matrices
A bit unsure if my use/mention of proprietary software might be inappropriate or even frowned upon here. If this is the case, or if this kind of experimental question is not welcome, please let me ...
17
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6
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"Oldest" bug in computer algebra system?
The goal of this question is to find an error in a computation by a computer algebra system where the 'correct' answer (complete with correct reasoning to justify the answer) can be found in the ...
17
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2
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Compute Lie algebra cohomology
Is there a computer algebra system that is able to compute the Lie algebra cohomology in a given representation? What if the Lie algebra is finite dimensional?
In my case I would like to be able to ...
16
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3
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What to do when your research runs into a computationally challenging problem?
Occasionally, but more frequently lately, I would like to perform some hard computations. As an example, yesterday the following question came up:
What is the projective dimension of the edge ideal ...
16
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2
answers
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From polynomial ideal over $\mathbb{Q}$ to polynomial ideal over $\mathbb{Z}$
Is there an algorithm to compute, given a polynomial ideal $I\subset \mathbb{Q}[x_1,\dotsc,x_n]$, the ideal $I\cap \mathbb{Z}[x_1,\dotsc,x_n]$ in $\mathbb{Z}[x_1,\dotsc,x_n]$?
The input and output ...
16
votes
1
answer
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Computer software for periods
Kontsevich and Zagier define a period as an integral of a rational function (over $\mathbb{Q}$) defined on a $\mathbb{Q}$-semialgebraic set. They conjecture that if two periods are equal, then the ...
15
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4
answers
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Program for computing group cohomology
Is there any computer program with which I can compute the group cohomology H^n(G,V) for a group G acting linearly on a vector space?
I mainly care about infinite groups.
14
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3
answers
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Computing (on a computer) higher ramification groups and/or conductors of representations.
I am supervising an undergraduate for a project in which he's going to talk about the relationship between Galois representations and modular forms. We decided we'd figure out a few examples of weight ...
14
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9
answers
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Is there a non self-referencing non-computable function?
I've seen in college that some functions are not computable.
The proof for that was the case of Halt(x,y) function.
The thing is, the proof used a very artificial (IMHO) case
which is evaluating ...
13
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4
answers
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how to determine whether an ideal is prime or not by an algorithm
Given polynomials $f_{1},\cdots,f_{n}\in \mathbb{C}[x_{1},\cdots,x_{m}]$, do we have an algorithm to determine whether the ideal $I=(f_{1},\cdots,f_{n})$ is prime ideal or not? Of course, we assume ...
13
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3
answers
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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 ...
13
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1
answer
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An efficient isomorphism between finite fields
Let $p$ be a prime number. Let $f$ and $g$ be irreducible polynomials over $\mathbb{F}_p$, both of degree $n$. We know that factor-rings $\mathbb{F}_p[x]/(f)$ and $\mathbb{F}_p[x]/(g)$ are isomorphic ...
13
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1
answer
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Computational Question about finite local rings:
Let $(A,\mathfrak{m})$ be a local Artinian ring with
finite residue field, which I'm happy to assume is $\mathbf{F}_3$.
(In particular, $A$ has finitely many elements.)
I would like to do some ...
13
votes
1
answer
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How can I tell if a variety is normal?
Suppose $R$ is a subalgebra of ${\mathbb C}[x_1,...,x_N]$ generated by polynomials $p_1,...,p_k.$ I know that ${\mathbb C}[x_1,...,x_N]$ is the integral closure of $R$.
Is there an algorithm to ...
12
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1
answer
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Verify that a group is hyperbolic via computer algebra
I would like to know whether there is some computer algebra software that can be used to verify if a group, given by a finite presentation, is hyperbolic (in the sense that it terminates with "yes" if ...
12
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4
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Is Gauss-Seidel guaranteed to converge on *semi* positive definite matrices?
I know that the Gauss-Seidel method is guaranteed to converge given that the matrix you want to solve is positive definite. I've looked at the proofs of convergence, and it appears that one cannot ...
12
votes
1
answer
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Where to publish a long classification?
Suppose that the classification of some mathematical (say algebraic) notions requires (say) 70 pages. Let clarify that (say) 90% of the pages are used to write the result itself, whereas only 10% are ...
12
votes
2
answers
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Ideal Membership without Certificate?
I have a homogeneous ideal $I=\langle f_1,\ldots,f_r\rangle$ of the polynomial ring $\mathbb C[X_1,\ldots,X_n]=:R$ where each of the $f_i$ is actually over $\mathbb Z$. My computations are usually ...
12
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2
answers
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Faster multiplication with a restricted set of multiplicands?
Let $A$ be a set of $k>1$ distinct elements from a semigroup. We wish to compute the product
$$ p=b_1 b_2 \cdots b_n$$
where each $b_i\in A$.
Clearly $n-1$ multiplications suffice to compute $p$; ...
12
votes
1
answer
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Tarski-Seidenberg for strict inequalities and bounded quantification
This theorem says that quantifiers over real variables can be eliminated from classical first order formulae built from equations and inequalities between polynomials with rational coefficients, ie in ...
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5
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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 ...
11
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1
answer
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Representing field elements in a computer
I'm wondering if there is existing terminology to describe fields $F$ with the properties below. I don't have a completely precise description of the concept I have in mind, but hopefully this will be ...
11
votes
1
answer
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Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials
Let $n>1$ and $p$ be an odd prime with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. Notice that, since $p-1 \mid n-1$, we have $T^p - T \mid T^n-T$ in $\mathbb{F}_p[T]...
11
votes
1
answer
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How do computer algebra packages like Sagemath implement rank of a matrix
I am not sure if this is the right place to ask this question, but I believe there will be people here who do computations on computer algebra packages like Sage in their work.
I have been using ...
11
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1
answer
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A 2F1 Hypergeometric identity from a Feynman integral
Using two different approaches to evaluating the dimensionally regularized ($d=4-2\epsilon$ dimensional Euclidean space), equal mass ($x=m^2$), 2-loop vacuum Feynman diagram
$$
\begin{align}
I(x) &...
11
votes
3
answers
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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 ...
11
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3
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Algorithms in Invariant Theory
Let $V$ be a polynomial representation of the general linear group $\Gamma:=\DeclareMathOperator{\Gl}{Gl}\Gl_n(\newcommand{\C}{\mathbb C}\C)$.
In chapter 4.6 of his book "Algorithms in Invariant ...
11
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0
answers
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When is cohomology of a finitely presented dg-algebra computable?
Given a smooth affine variety $X$ defined over $\mathbb{Q}$, its singular cohomology is isomorphic to the algebraic de Rham cohomology, which is the cohomology of the complex $\Omega_X^0\to\Omega_X^1\...