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 ...

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 ...

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-...

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{...

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 ...

26
votes

Accepted

### 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)...

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

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 ...

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.

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 ...

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 $e^{i\pi/n}$ ...

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 ...

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 ...

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 ...

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)...

14
votes

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" (...

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 ...

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 ...

Community wiki

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 ...

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 $b$ ...

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 ...

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. ...

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 ...

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 $n-1$ ...

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)} +...

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,...

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 ...

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 ...

Community wiki

Only top scored, non community-wiki answers of a minimum length are eligible

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