**4**

votes

**3**answers

1k views

### Differential Geometry/General Relativity Computer Algebra

Hi,
could anybody recommend a CAS suited to DG/GR applications such as computation of connection coefficients or generating (and possibly solving) PDEs for, for example, an unknown metric of given ...

**9**

votes

**5**answers

1k views

### Multipolynomial resultants

We know that the resultant of two polynomials can be computed as the determinant of their Sylvester matrix ( http://en.wikipedia.org/wiki/Sylvester_matrix ). How do we compute the resultant of more ...

**31**

votes

**3**answers

3k views

### 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 ...

**3**

votes

**2**answers

327 views

### Any CAS that deals with the word problem

I'm new to the combinatorial group theory, so maybe my question is a bit naiive.
I know that the word problem is generally "unsolvable". On the other hand there are specific cases, when the problem ...

**1**

vote

**1**answer

248 views

### Computing the connected component without primary decomposition

Given an algebraically closed field $\mathbb{F}$ of characteristic $0$ and a closed subgroup $G$ of $GL_n(\mathbb{F})$. Let $\{g_1,\ldots, g_r\}$ be a Gr"obner basis for the correpsonding ideal ...

**4**

votes

**2**answers

408 views

### Is there a free action on a given variety?

Given a variety $V$, and a prime $p$ I want to decide if there is a free action of $\mathbb{Z}/p\mathbb{Z}$ on $V$, and to find the generator of an action if it exists. Is there a known algorithm to ...

**3**

votes

**3**answers

283 views

### Computational solutions to families of systems of linear equations

Question
Does there exist a computer package that will solve families of systems of linear equations over a field of prime characteristic?
An Example
Suppose I wanted to know when the following ...

**9**

votes

**2**answers

2k views

### 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 ...

**4**

votes

**1**answer

910 views

### Quantum Group Calculations in Mathematica

I'm trying to learn how to do algebraic manipulations in Mathematica but not finding the help very helpful. I'm going to ask about a specific quantum group example related to a previous question of ...

**5**

votes

**2**answers

1k views

### Algorithm for Weierstrass Preparation Theorem for Formal Power Series

The Weierstrass preparation theorem for formal power series rings guarantees that if a given formal series $f(z) = \sum a_k z^k \in R[[z]]$ where $R$ is a complete local ring with maximal ideal $M$ ...

**3**

votes

**2**answers

923 views

### Finding generators of subalgebra of polynomial algebra $K[x_1,\cdots,x_n]$ that are invariant under the action of symmetric group

Let $I =\langle f_1,\cdots,f_m\rangle \subset K[x_1,\cdots,x_n]$be an ideal,
where $f_k\in K[x_1,\cdots,x_n].$
$K[e_1,\cdots,e_n]$ the polynomial algebra generated by the elementary symmetric ...

**-2**

votes

**1**answer

398 views

### Mul + div using only add/sub ? [closed]

In an algorithm book once the first example was how to compute a multiplication in a loop (only that, so I just remembered, and wanted to do it programmatically but with all operations)
...

**13**

votes

**1**answer

759 views

### 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 ...

**4**

votes

**2**answers

651 views

### Bounds on remainder term of power series of elementary functions

This is mainly a question about the remainder term of power series for elementary functions.
I'm very interested in aspects of calculating or computing elementary operations and functions, by which I ...

**4**

votes

**2**answers

593 views

### Indexed tensor manipulation CAS

hello.
I am looking for tensor manipulation software that would allow me:
declare indices
declare results of contraction (or simplification rules)
allow algebraic simplifications and expansion
...

**5**

votes

**3**answers

382 views

### Automatic proving some expression is positive

Is there any automated (i.e., some algorithm) to prove that a certain algebraic expression is always non-negative in some range ? If so, is there any implementation you would suggest? My concrete ...

**8**

votes

**1**answer

536 views

### Mathematical software for computing in integral group rings of discrete groups?

I'm doing computations in the integral group ring of a discrete group,
in particular the discrete Heisenberg group. In this case elements
are integral combinations of monomials $x^k y^m z^n$, where ...

**6**

votes

**3**answers

384 views

### Complexity of high-order differentiation

Let $g(x) = \exp(f(x))$. Assuming numerical (or symbolic) values of $f(x), f'(x), f''(x), \ldots, f^{(n)}(x)$ are known, is there a way to compute $g'(x), g''(x), \ldots g^{(n)}(x)$ (or even the ...

**0**

votes

**0**answers

749 views

### Efficient quadratic residue mod 2^32

I want to determine if a value is a quadratic residue mod $2^{32}$.
I've developed a very fast pre-screening method based on a Bloom Filter that identifies quadratic residues for mod $2^7=128$ in ...

**5**

votes

**1**answer

626 views

### Computer power in plane geometry

I often hear that modern computer programs "may prove any theorem in elementary Eucledian geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for arbitrary or ...

**3**

votes

**1**answer

697 views

### Adjoint/transpose of wavelet transform

I'm using a wavelet transform in Matlab, so I think of it as a black-box. I'll represent it here as $W(x)$. There's a reconstruction function as well, which I'll write as $W^\dagger(y)$. I can ...

**6**

votes

**6**answers

1k views

### What are you using for symbolic computation?

What are the pluses and minuses of different software packages? Anything new worth checking out?
I'm especially interested in open source packages.

**13**

votes

**4**answers

1k views

### 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 ...

**25**

votes

**5**answers

7k views

### The unification of Mathematics via Topos Theory

When the paper The unification of Mathematics via Topos Theory by Olivia Caramello, says "one can generate a huge number of new results in any mathematical field without any creative effort." is this ...

**6**

votes

**3**answers

2k views

### Computing only the order of Galois group (not the group itself).

My question is related to this one: Computing the Galois group of a polynomial.
I was wondering if there is a faster algorithm just to compute the order of the group rather than the group itself.
...

**5**

votes

**1**answer

378 views

### software for computations on flag varieties in arbitrary characteristic

Is there any software that will compute cohomology of vector bundles (or just line bundles) on flag manifolds?
The only one I know of is Macaulay2, via the Schubert2 package, but it works with what ...

**7**

votes

**1**answer

510 views

### Are there any sofware packages for computing Picard numbers?

Are there any computer algebra systems (e.g. Macaulay2 og singular) that allows one to compute the Picard number (i.e. the rank of the Neron-Severi group) of a given variety?

**2**

votes

**1**answer

418 views

### Efficient derivation of null space of large symbolic matrices?

Hi all,
I'm wondering if anyone is aware of an efficient mechanism by which to derive the null space of a "large" symbolic matrix. Here, large means on the order of 10^2 rows, not necessarily ...

**7**

votes

**4**answers

2k views

### Basis for modular forms of half-integral weight

Given a character $\chi$ and $k$ odd how can one compute a basis for the space of modular forms $M_\frac{k}{2}(\Gamma_0(4),\chi)$. By compute a basis I mean, finding the beginning of the Fourier ...

**5**

votes

**2**answers

663 views

### Software for computing multi-graded Hilbert series

The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights (1,1,-1,-1)
is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series
$\frac{1 - ...

**36**

votes

**23**answers

9k views

### 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 ...

**16**

votes

**2**answers

727 views

### 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 ...

**12**

votes

**3**answers

2k views

### 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 ...

**13**

votes

**6**answers

2k views

### “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 ...

**6**

votes

**3**answers

2k views

### Rational exponential expressions

Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by:
The leaves 1 and $x$ for $x$ drawn from a class of variables; and
Closed under the binary ...

**68**

votes

**33**answers

16k views

### 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 ...

**6**

votes

**1**answer

616 views

### Choosing a fast computer algebra system that works in characteristic p?

Hi all,
I want to compute in $\mathbb{F}_q (x)((y))$ i.e. a Laurent series ring over the rational functions over $\mathbb{F}_q$. The computations are fairly basic, but they involve raising to the qth ...

**3**

votes

**1**answer

386 views

### Decomposition of modules using computer packages

I am interested in computing direct sum decomposition of modules over some quotients of polynomial rings over a field (do not care much about the field at this point). Does any one know a package ...

**10**

votes

**2**answers

361 views

### Counting points on varieties of low codimension

The graduate students here at MIT have been thinking about questions like the following: Over $\mathbb{F}\_q$, how many symmetric matrices are there with nonzero determinant and $0$'s on the diagonal? ...

**0**

votes

**2**answers

107 views

### Properties of adjacent submatrixes [closed]

Hi!
I've encountered a matrix problem when designing an algorithm, which I cannot seem to figure out. I have a (square) matrix with the following properties:
j<k → aij<aik, aji<aki
...

**23**

votes

**4**answers

2k views

### 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 ...

**2**

votes

**4**answers

2k views

### Symmetrical Presentation of 4-Dimensional Rotation Matrix

This question is not urgent; just a matter of curiosity...
It is relatively easy to generate an arbitrary 3D or even 4D rotation matrix using conjugation (i.e. YXY−1) of orthogonal rotations. I ...

**7**

votes

**5**answers

3k views

### 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 ...

**9**

votes

**9**answers

2k views

### 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 ...

**1**

vote

**4**answers

387 views

### CAS for finding closed form solutions to PDEs and SDEs?

Are there any specialized Computer Algebra Systems (or packages for these) for finding closed form solutions to
a) partial differential equations,
b) stochastic differential equations?
If yes, what ...

**9**

votes

**3**answers

1k views

### Is there a stable algorithm for polynomial division (in several variables)?

Suppose you have a homogeneous ideal $I$ inside the algebra $\mathbb{C}[x_1,...,x_d]$ of complex polynomials in $d$-variables. Can one find a basis for $I$, say $\{f_1,...,f_k\}$, such that every $h ...

**5**

votes

**4**answers

439 views

### Sections of a divisor on elliptic curve

I'm interested in producing explicit bases for the sections of a line bundle on an embedded genus 1 curve. Let me restrict to the first case that I don't know how to do, so that I can be as concrete ...

**9**

votes

**8**answers

3k views

### Which computer algebra system should I be using to solve large systems of sparse linear equations over a number field?

This is related to Noah's recent question about solving quadratics in a number field, but about an even earlier and easier step.
Suppose I have a huge system of linear equations, say ~10^6 equations ...

**91**

votes

**69**answers

15k views

### 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 ...

**5**

votes

**4**answers

395 views

### Software for rigorous optimization of real polynomials

I am looking for software that can find a global minimum of a polynomial function in R^n over a polyhedral domain (given by some linear inequalities say). The number of variables n is not more than a ...