56
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 ...
- 16.9k
47
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 ...
- 586
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-...
- 188k
31
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{...
27
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 ...
- 14k
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
- 26.3k
25
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)...
- 7,226
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 ...
- 105k
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.
- 96.4k
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 ...
- 4,219
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}$ ...
- 109k
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 ...
- 911
16
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)...
- 17.6k
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 ...
- 4,667
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 ...
- 17.9k
14
votes
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 ...
- 3,093
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$ ...
- 47.1k
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. ...
- 8,071
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 ...
- 37.4k
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$ ...
- 148k
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)} +...
- 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,...
- 6,919
11
votes
Accepted
Checking for a normal p-complement with a computer
For Question 2, you are asking whether a Sylow $3$-subgroup is normal, and I would expect the fastest and easiest way to do that would be
$\mathtt{G:=SmallGroup(768,k);\ IsNormal(G,SylowSubgroup(G,3))}...
- 37.4k
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 ...
- 32.4k
10
votes
Accepted
Computation of a minimal polynomial
To compute the minimal polynomial of integer multiple of an algebraic integer is easy, so the only thing you need for linear combinations is the minimal polynomials of sums. Now, note that if $A$ is ...
- 96.4k
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 ...
10
votes
How can you find an integer coefficient polynomial knowing its values only at a few points (but requiring the coefficients be small)?
Given $p(x_i)=y_i$ for $i\in\{1,2,\dots,n\}$, finding a polynomial of fixed degree $d$ can be posed as finding a closest vector to $(cy_1,\dots,cy_n,\underbrace{0,\dots,0}_{d+1})^T$ in the lattice ...
- 34.3k
9
votes
Computer algebra errors
I found a mistake made by WolframAlpha, which I also posted on a related thread on Mathematics Educators SE.
Ask WolframAlpha to ...
Only top scored, non community-wiki answers of a minimum length are eligible