**18**

votes

**5**answers

6k views

### 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 ...

**1**

vote

**1**answer

233 views

### Reference for complexity of primitive polynomials

What is the fastest known way to check if a given polynomial of degree $n$ in $F_{2}[X]$ is primitive?
In response to Greg Kuperberg's answer. If we known factorization of $2^{n} - 1$, then what is ...

**9**

votes

**4**answers

1k views

### 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 ...

**5**

votes

**2**answers

615 views

### Is it possible to check two curves on birational equivalence by some computer algebra system?

I have two curves, for example hyperelliptic:
\begin{align}
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 18, \\\\
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 5x + 1
\end{align}
Is it possible to check them ...

**6**

votes

**5**answers

1k views

### Computer algebra system for calculation of characteristic polynomial of sparse matrix

I have a $n \times n$ matrix, for which i need to calculate the characteristic polynomial. The matrix is over $GF(2)$, and $n \approx 10^4$. However the matrix is very sparse, with around $ n $ non ...

**0**

votes

**1**answer

444 views

### Polynomial of degree N with integer coefficient for a given root.

Is it possible to construct a polynomial of degree N, with all of them as integer coefficient have a root as the given value. ...

**7**

votes

**1**answer

2k views

### Constructing a unitary matrix

Setting:
Given a set of $n\times n$ matrices $A_i$, I would like to find a linear combination of these matrices $Q = \sum_i A_i x_i$ with $x_i$ a set of complex numbers, such that $Q$ is unitary: ...

**5**

votes

**1**answer

165 views

### Expressing a element of a Matrix subgroup in terms of subgroup generators

I'm no (computational) algebraist, and my searches have been pretty unyielding (probably due to the vast amounts written on the key words), but perhaps someone may know if this is possible, and if so, ...

**8**

votes

**2**answers

765 views

### Homological computations

Suppose I have a group acting on some Hadamard manifold, and I want to understand as much as possible about the (co)homology of the quotient. In my case I can find a fundamental domain for the action ...

**4**

votes

**0**answers

200 views

### Algorithm/denominators of elements of a rational affine space

I hope it's not a trivial question... Suppose I have a finite dimensional vector space $V$ over $\mathbb{Q}$ with a distinguished basis (in my case it's the $k$th graded piece of the free associative ...

**6**

votes

**3**answers

776 views

### Is there a MAGMA function to calculate the absolutely irreducible components of an algebraic curve defined over the rationals?

Given a curve defined over the rationals, is it computationaly possible to find all its absolutely irreducible components?
Is there an implementation of this in the MAGMA program?

**16**

votes

**3**answers

1k views

### 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 ...

**2**

votes

**0**answers

230 views

### Efficient computing critical points of algebraic function involved radical expression

I am interested in finding local optima of an algebraic function $f(X,Y)$. Suppose, that this expression involves radicals, for example $f(X,Y)= \frac{1}{2}(X+Y)-\sqrt{XY}$. The approach in which i am ...

**1**

vote

**0**answers

214 views

### How to ask Magma to compute the induced morphisim on divisor group

Suppose Magma has computed homomorphism $h$ between function fields $F1 \to F2$. Then we have an induced homomorphism $h$ on the divisor group. Now my question is that if there's a better way to ...

**6**

votes

**0**answers

433 views

### Where can I find tables of dual canonical basis vectors?

Leclerc (arXiv:math/0209133) has given us an algorithm for computing the dual canonical basis of the upper part of a quantised enveloping algebra.
Now presumably this algorithm has been implemented ...

**5**

votes

**1**answer

418 views

### Does a variety contain a cartesian product of two curves?

We are given an affine variety $V\subset \mathbb{A}^n\times\mathbb{A}^n$, and wish to know if it contains a product of the form $C_1\times C_2$, where $C_1$ and $C_2$ are two curves in $\mathbb{A}^n$. ...

**6**

votes

**1**answer

921 views

### 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) ...

**0**

votes

**1**answer

170 views

### the maximal length of a special dicksonian sequence

First, we define a sequence $t_{1},t_{2},\cdots,t_{k}$ of n-tuples dicksonian, if $\forall 1\leq i < j\leq k,$ there does not exist a non-negative n-tuple t such that
$t_{i}+t=t_{j}.$ For example, ...

**9**

votes

**4**answers

2k views

### 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 ...

**3**

votes

**0**answers

305 views

### Rank of Subgroup of Elliptic Curve

I'm currently looking at two rational points $ p, q $ on an elliptic curve $E$ over $ \mathbb{Q} $. SAGE tells me that $E$ has rank 5 and no torsion, and that $p$ and $q$ both have infinite order. ...

**6**

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 ...

**33**

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 ...

**4**

votes

**2**answers

360 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 ...

**4**

votes

**4**answers

3k views

### Computational Algebra - Where?

I'm on my last semester of a math B.Sc. and about to start studying for a math M.Sc in the same institute.
It now seems like a good time to start thinking of a PhD.
I'm interested in both algebra and ...

**1**

vote

**1**answer

253 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

411 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

288 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 ...

**10**

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

1k 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

963 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

411 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

780 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 ...

**6**

votes

**2**answers

682 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

628 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

393 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

562 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

388 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 ...

**5**

votes

**1**answer

635 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

790 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

8k 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 ...

**7**

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

388 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

534 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

460 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

697 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 - ...