55 votes
Accepted

Minimal polynomial of cos(π/n)

The minimal polynomial of $\cos(2\pi/n)$ (by William Watkins and Joel Zeitlin, The American Mathematical Monthly Vol. 100, No. 5 (May, 1993), pp. 471-474) has full clarity on this matter (just take ...
Vladimir Dotsenko's user avatar
45 votes
Accepted

Does there exist a complete implementation of the Risch algorithm?

No computer algebra system implements a complete decision process for the integration of mixed transcendental and algebraic functions. The integral from the excellent paper of Schultz may be solved by ...
Sam Blake's user avatar
  • 566
40 votes
Accepted

How does Mathematica do symbolic integration?

An overview by one of the developers of Mathematica, focusing on definite integrals, is at Symbolic definite integration: methods and open issues. Mathematica knows all the entries in Gradshteyn-...
Carlo Beenakker's user avatar
29 votes
Accepted

What is currently feasible in invariant theory for binary forms?

Let $F$ be a binary form of degree $d$, namely, a homogeneous polynomial of the form $$ F(\mathbb{x})=\sum_{i=0}^{d}\left(\begin{array}{c}d\\ i \end{array}\right)f_i\ x_1^{d-i}x_2^i $$ where $\mathbb{...
Abdelmalek Abdesselam's user avatar
26 votes

Does there exist a complete implementation of the Risch algorithm?

Fricas, an open-source clone of Axiom, implements a considerable chunk of Risch, see http://fricas-wiki.math.uni.wroc.pl/RischImplementationStatus Fricas is also available as a optional package of ...
Dima Pasechnik's user avatar
26 votes
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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)?

You can certainly do better than brute force by considering modular constraints. If the solution is $p(x) = \sum_i a_i x^i$ then $p(x) - \sum_{j=0}^{n-1} a_j x^j$ is divisible by $x^n$ and $$\frac{p(x)...
Peter Taylor's user avatar
  • 6,431
25 votes

How does Mathematica do symbolic integration?

Maple uses the Risch algorithm; see Keith Geddes and George Lebahn, Symbolic and numeric integration in Maple
Ben McKay's user avatar
  • 25.3k
21 votes
Accepted

two's and three's survive in gcd of Lagrange

The claim is certainly true for $n$ sufficiently large, and "sufficiently large" could be specified explicitly with more care. We follow the suggestion of Fedor Petrov, and rely on the results of ...
GH from MO's user avatar
  • 96.9k
20 votes

Web interface for GAP (or other computer algebra system dealing with finite groups)?

You can get at GAP through http://sagemath.org, which has a perfectly fine web notebook interface.
Igor Rivin's user avatar
  • 95.5k
18 votes

What is the minimum size of a partial order containing all partial orders of size 5?

(Edited several times from earlier partial answer, which gave $f(5) \ge 11$.) We have exact results $f(5) = 11$ and $f(6)=16$, and bounds $16 \le f(7) \le 25$. 1. Proving $f(5)=11$ A short proof shows ...
Jukka Kohonen's user avatar
17 votes

Minimal polynomial of cos(π/n)

I guess you mean a polynomial $p(x)$ with rational coefficients. Then, once $\cos {\pi/n}$ is a root of $p(x)$, $\deg p=d$, $e^{i\pi/n}$ is a root of a polynomial $t^dp((t+1/t)/2)$. But $e^{i\pi/n}$ ...
Fedor Petrov's user avatar
17 votes

How does Mathematica do symbolic integration?

People usually mention the Risch algorithm first, as other answers have. Another approach, which is surprisingly successful, is to do what you or I would when solving integrals: look for patterns for ...
Szabolcs Horvát's user avatar
16 votes
Accepted

Representing field elements in a computer

You are looking at a computable field (if your focus is on the field), or a computable presentation of a field (if your focus is on the details of how elements and operations are coded). These objects ...
Arno's user avatar
  • 4,346
16 votes

How can you find an integer coefficient polynomial knowing its values only at a few points (but requiring the coefficients be small)?

Perhaps not the best possible polynomial but a very good polynomial can generally be computed by the LLL algorithm: Let $I$ be an integral polynomial taking the given values at the given points and ...
Roland Bacher's user avatar
14 votes
Accepted

How to compute with the Stark conjectures?

The main reference here is the very useful User's Guide, C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, "User's Guide to PARI / GP" (2003) Particularly the sections about bnrstark (pp. 108)...
Myshkin's user avatar
  • 17.4k
14 votes

Computation of a minimal polynomial

In Sage you can do something like: ...
Moritz Firsching's user avatar
13 votes
Accepted

An efficient isomorphism between finite fields

Google provides an answer to this question. The first deterministic polynomial time algorithm for this is due to H. W. Lenstra, Jr., in his paper "Finding isomorphisms between finite fields" (...
Michael Zieve's user avatar
13 votes
Accepted

Verify that a group is hyperbolic via computer algebra

The KBMAG package can be used to verify hyperbolicity of a group defined by a finite presentation. It does it by verifying that geodesic bigons in the Cayley graph are uniformly thin. Then a result of ...
Derek Holt's user avatar
  • 36.4k
13 votes

Is computer algebra or symbolic computation an active area of research?

[This is certainly a biased view, but too long for a comment, and hopefully these are some helpful starting points. ] I'd say it's active but in the US not huge (more elsewhere in the world). A few ...
12 votes
Accepted

A question on a Macaulay2 computation

Commutative algebra is NOT the same as algebraic geometry, especially projective algebraic geometry. The variety in $\mathbb{P}^9$ defined by $I$ and the variety in $\mathbb{P}^9$ defined by $I_0$ are ...
Alexander Woo's user avatar
12 votes

About the complexity of some operation involving integers

There is a simple algorithm because the minimal path from $A$ to $B$ using these operations must have a very constrained form. First, an optimal sequence of $x+1$ and $x-1$ operations from $a$ to $b$ ...
Emil Jeřábek's user avatar
11 votes
Accepted

A centralised website for computational attempts in graph theory and metric geometry?

There are a few websites with lists and/or databases of graphs, maps and polytopes. House of graphs has a searchable database of interesting graphs and aims to serve as a repository for lists of ...
Katja Berčič's user avatar
11 votes

Algebraization of Bayesian networks?

A bayesian network corresponds to an independence (algebraic) variety and hence to a polynomial ring over $\mathbb{R}$. You can start from Garcia, Luis David, Michael Stillman, and Bernd Sturmfels. ...
Henry.L's user avatar
  • 7,941
11 votes
Accepted

Web interface for GAP (or other computer algebra system dealing with finite groups)?

There is the Magma calculator which can be used to do calculations in finite groups. One problem is that you have to type in all of your input before executing it, but with practice you can do quite ...
Derek Holt's user avatar
  • 36.4k
11 votes
Accepted

Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials

This is false. Let $p$ be an odd prime, let $\ell$ be another prime, and let $m$ be a small prime divisor of $p^{\ell}-1$, that doesn't divide $p-1$. Let $n= 1 + \frac{ p^{\ell}-1}{m}$. Then $n-1$ ...
Will Sawin's user avatar
  • 133k
11 votes

How does Mathematica do symbolic integration?

One approach to symbolic integration is using representations as Holonomic functions (https://en.wikipedia.org/wiki/Holonomic_function): solutions to differential equations of the form: $$p_r f^{(r)} +...
TomKern's user avatar
  • 429
11 votes
Accepted

Finite subgroups of $\mathrm{SO}(n)$ and $\mathrm{O}(n)$

Finite subgroups of $\operatorname{SO}(4)$ and $\operatorname{O}(4)$ are enumerated in Chapter 4 of the book: Conway, John H.; Smith, Derek A., On quaternions and octonions: their geometry, arithmetic,...
Frieder Ladisch's user avatar
10 votes
Accepted

From polynomial ideal over $\mathbb{Q}$ to polynomial ideal over $\mathbb{Z}$

I am aware that I am replying to an old question that has been satisfactorily answered, but I think it might be worth while pointing out, for the completeness of MathOverflow, that the question is ...
Gro-Tsen's user avatar
  • 29.7k
10 votes

Most helpful math resources on the web

Sci-Hub is pretty helpful in accessing articles, even for those researchers who already have access to several journals. The interface is great, the site is pretty fast, and the database is huge. See ...
10 votes

Where to publish a long classification?

Perhaps I can try a suggestion: a journal which accepts long papers on the topics you sketched, aimed towards pure and applied mathematicians (including physicists and engineers), which has also an ...

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