**1**

vote

**1**answer

133 views

### Finding reducible polynomials with restricted factors

Given $f(x),g(x) \in \mathbb{Z}[x]$, two irreducible polynomials, is there a polynomial $h(x) \in \mathbb{Z}[x]$ coprime to $f(x)$ such that $f(x) + g(x)h(x)$ is reducible over $\mathbb{Z}[x]$ with ...

**1**

vote

**2**answers

362 views

### Numerical Determination of Generating Functions from Recursion Relations

Are there computer packages which calculate coefficients of generating functions, such as
$$D_n(q)=\sum_m d_{m,n}q^m= \frac{1}{\prod_{i=1}^n (1-q^i)^2} \text{ or}$$
$$S_d(q)=\sum_m s_{m,d}q^m = ...

**0**

votes

**1**answer

535 views

### What program/language can I use to handle large numbers (over 10^300) [closed]

I'm trying to multiply together a set of numbers.
However, the programs I have tried (Excel, R, FreeMat) do not allow numbers above around 300 digits. This limits me to sets of a few hundred numbers ...

**8**

votes

**2**answers

264 views

### Ideal Membership without Certificate?

I have a homogeneous ideal $I=\langle f_1,\ldots,f_r\rangle$ of the polynomial ring $\mathbb C[X_1,\ldots,X_n]=:R$ where each of the $f_i$ is actually over $\mathbb Z$. My computations are usually ...

**5**

votes

**1**answer

728 views

### Algorithm to find exponential map of differential operators acting on function

I am trying to write a computer program which computes the action of the exponential of a differential operator on a function, for any given differential operator.
Examples:
$\exp(\varepsilon ...

**3**

votes

**1**answer

528 views

### How to maximize the determinant of a matrix of the form VDV^H

Hi,
I have a matrix of the form $A=VDV^H$,
where $V$ is a $M \times 2M$ complex matrix, $D$ is a $2M \times 2M$ diagonal real matrix, thus the dimension of $A$ is $M \times M$.
My problem is how ...

**1**

vote

**1**answer

254 views

### Sage or Magma Implementation of Nilpotent Orbit Varieties

For a given partition $[n_{1},...,n_{k}]$ of $N \in \mathbb{N}$ there exists a corresponding nilpotent orbit variety $O_{[n_{1},...,n_{k}]}$ in $\mathfrak{gl}(N)$ which can be represented by a set of ...

**48**

votes

**2**answers

1k views

### Reasons to prefer one large prime over another to approximate characteristic zero

Background:
In running algebraic geometry computations using software such as Macaulay2, it is often easier and faster to work over $\mathbb F_p = \mathbb Z / p\mathbb Z$ for a large prime $p$, rather ...

**2**

votes

**1**answer

405 views

### Is there an algorithm to decide if an ideal contains monomials?

Let $I\subset k[x_1,\dots,x_n]$ be an ideal in a polynomial ring in commuting variables. Is there a procedure to decide if $I$ contains a monomial and possibly to find one?
Gröbner bases come to ...

**4**

votes

**1**answer

260 views

### The existential theory of the reals

Some definitions of the existential theory of the reals (ETR) allow a real closed field and some definitions allow only rational numbers as coefficients of polynomials. Which one is correct? Will the ...

**1**

vote

**0**answers

173 views

### Computer package to compute HOMFLY polynomial?

I apologize of already asked by someone else, but what (in your opinion) is the best package for computing HOMFLY polynomials?

**1**

vote

**1**answer

148 views

### Recommendations for binomial system solver

I am interested in solving binomial systems of the form
$$
\begin{cases}
a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} +
b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\\\
...

**1**

vote

**2**answers

196 views

### My output of a group and inverse-closed subset in MAGMA is no longer inverse-closed when entered as input to GAP.

In MAGMA, I input the following:
G:=SmallGroup(20,3);
G;
E:=[xx:xx in G];
S:=[E[6],E[7],E[13],E[20]];
S;
S[1]^2;
S[2]^2;
S[3]*S[4];
This gives the output:
GrpPC : ...

**3**

votes

**3**answers

667 views

### Good Computer Package for Calculating Inverse of a Formal Power Series?

This might be a question people already asked or is obvious to experts, or is not appropriate for this forum, if so, I apologize. I am trying to calculate things like $z/(e^z-1)$, or find the inverse ...

**2**

votes

**1**answer

311 views

### Computer algebra system (CAS) with good re-presenting or transformation support

Such heavy-weight transformations as expanding or factoring are provided by most of CAS-es, but what about light-weight, but a useful transformations, like "reorder some terms to make expression more ...

**10**

votes

**3**answers

549 views

### Algorithms in Invariant Theory

Let $V$ be a polynomial representation of the general linear group $\Gamma:=\DeclareMathOperator{\Gl}{Gl}\Gl_n(\newcommand{\C}{\mathbb C}\C)$.
In chapter 4.6 of his book "Algorithms in Invariant ...

**2**

votes

**0**answers

142 views

### Randomized alternative to Buchberger's algorithm

Richard Lipton's blog describes a A New Way To Solve Linear Equations by Prasad Raghavendra.
Can the ideas in this algorithm be generalized to systems of polynomial equations to provide a randomized ...

**7**

votes

**2**answers

358 views

### Computing determinants of matrices of linear forms

Suppose we have three $n \times n$ matrices $A$, $B$, $C$ with floating point entries. We would like to compute the polynomial $\det (xA+yB+zC)$. At least in Mathematica, and I think in all computer ...

**12**

votes

**1**answer

457 views

### How can I tell if a variety is normal?

Suppose $R$ is a subalgebra of ${\mathbb C}[x_1,...,x_N]$ generated by polynomials $p_1,...,p_k.$ I know that ${\mathbb C}[x_1,...,x_N]$ is the integral closure of $R$.
Is there an algorithm to ...

**5**

votes

**4**answers

1k views

### computer algebra system for polynomial algebras over finite fields

Is there a computer algebra system that can do arithmetic over polynomial algebras over finite fields where I can specify the extension?
Exempli gratia, if $f(x), g(x) \in ...

**4**

votes

**0**answers

302 views

### On finding A-polynomials

I have two questions to obtain the explicit forms of A-polynomials.
Takata used the mathematica pacage qMultisum.m to obtain the recursion relation of the colored Jones polynomials for twist knots. ...

**1**

vote

**2**answers

120 views

### Non-Uniform Root of Polynomial in Open Cube

I'm looking to find a root $(x_1,\dots,x_n)$ of a polynomial $p \in {\mathbb R}[x_1,\dots,x_n]$ such that $0 \leq x_i < 1$ for all $i$. Further, I know in advance that setting $x_1 = \cdots = x_n$ ...

**6**

votes

**1**answer

394 views

### Reasonable implementation of finding Gröbner bases over non-field coefficient rings

Gröbner bases are usually considered in the ring of polynomials over a field. However, there are useful definitions and algorithms for Gröbner bases over other coefficient rings; see, for instance, ...

**0**

votes

**0**answers

105 views

### What is the largest computed summatory liouville interval ?

I am interested to know the largest computed summatory liouville interval, an implementation of which is detailed in Section 4.1 of [1].
The wikipedia page [2] for the function charts summatory ...

**1**

vote

**3**answers

382 views

### Finding maximum value of degree-3 homogeneous polynomials when variables sum to 1

I would like to be able to find maximum values of degree-3 homogeneous polynomials, when the variables are non-negative real numbers that sum to 1. For example,
For example, the maximum value of ...

**7**

votes

**0**answers

246 views

### Computer Algebra solution for simplicial resolutions for André-Quillen cohomology

Hello,
I would like to experiment with André-Quillen (co)homology. Especially for singular rings.
A key problem is that the construction of a simplicial resolution of a ring seems to require a rather ...

**20**

votes

**7**answers

2k views

### Computer package for representation theory of the symmetric group

Is there a computer algebra package in which I can compute the following for representations of a specific symmetric group (e.g. $S_7$):
(1) $V \otimes W$
(2) $S_\lambda V$, where $S_\lambda$ is a ...

**1**

vote

**3**answers

439 views

### the Richardson theorem and the base identities problem

In the fields related to school mathematics there is some acitivity on proving (or disproving) deducibility/decidability for some classes of school identities. In particular,
1) In logic they ...

**0**

votes

**1**answer

178 views

### Changing the Type of a Module in MAGMA

I am currently working with irreducible $k[G]$-modules in MAGMA for finite fields $k$ and finite groups $G$. To construct these modules, I am using the commands IrreducibleModules(G,k) This results in ...

**1**

vote

**2**answers

334 views

### Working with group cosets in MAGMA

When working with group cosets in MAGMA is there a way of treating the cosets as subsets of the overlying group. Specifically I have a group $G$ and subgroups $H$ and $K$ . I wish ...

**8**

votes

**4**answers

1k views

### Program for computing group cohomology

Is there any computer program with which I can compute the group cohomology H^n(G,V) for a group G acting linearly on a vector space?
I mainly care about infinite groups.

**1**

vote

**3**answers

369 views

### Checking for invertibility of large matrices in MAGMA

If you have a number of large matrices, and you wish to determine whether each matrix has determinant zero or not, what is the most efficient way to do this in MAGMA
(it appears that calculating the ...

**20**

votes

**3**answers

656 views

### Is there a way of canonically labelling permutation groups?

When working with large numbers of graphs, a canonical labelling routine is essential as, after the initial cost of canonically labelling each graph, it permits isomorphism checks to be replaced with ...

**5**

votes

**0**answers

233 views

### What became of PoSSo and FRISCO

I know PoSSo and FRISCO were pretty big projects involving many European universities.
Interestingly, I couldn't find much information about these projects
(the the top of the PoSSo homepage says ...

**8**

votes

**2**answers

308 views

### Monomial orderings in noncommutative setting

An ordering of monomials in the free associative algebra $k\langle x_1,\ldots,x_n\rangle$ is called a monomial ordering (EDIT: it seems that an equally common term used in this context is "term ...

**4**

votes

**1**answer

387 views

### Basis of a Finite Dimensional Algebra with a Finitely Generated Relation Set By Computer

Let $A$ be a noncommutative finitely generated algebra with a finitely generated set of relations. Moreover, assume that $A$ is finite dimensional as a vector space.
What I want to know is, can ...

**2**

votes

**2**answers

1k views

### Solving a system of equations/inequalities that have trigonometric functions on the left-hand side

Is there any known (symbolic) method that solves a system of equations/inequalities that have trigonometric functions on the left-hand side of the system?
Ex) Find $x,y,\theta \in \mathbb{R}$ that ...

**4**

votes

**2**answers

1k views

### Computing the fixed field of an automorphism of a function field

Let say we have a function field $k(x,y)$ defined by $f(x,y)$ over $k$, with $\sigma \in Aut(k(x,y)/k)$ and. Suppose, I'm not that out of luck, so that either of $\prod \sigma^i(x)$ or $\sum ...

**0**

votes

**1**answer

356 views

### Homomorphisms and their restrictions in MAGMA

I am trying to look at a representation (so a homomorphism) of a group G, and see what the restriction of the representation to a subgroup of G will be. Is there an easy way (or any way!) to do this ...

**4**

votes

**1**answer

472 views

### Finding Presentations of Groups with GAP

Group $A_5$ has presentation $〈 a, b | a^2 = b^3 = (ab)^5 = 1 〉$. Items equal to 1 are relators, so a presentation of $A_5$ as a set of relators could be
$(a^2, b^3, (ab)^5)$
$Q_{16}$ is ...

**6**

votes

**2**answers

2k views

### How to do integrals involving two Bessel functions and another function?

I often encounter the integrals in the following form:
$\int_0^\infty{\rm Bessel}(ax)\cdot{\rm Bessel}(bx)\cdot f(cx)dx$,
where Bessel can be $J$, $N$, $H^{(1)}$, $H^{(2)}$, $I$, or $K$; and $f(x)$ ...

**5**

votes

**3**answers

2k views

### Using MAGMA for Group Theory

I've just started a PhD in Group Theory and need to use the computer programme MAGMA. I wonder if anyone could help me with a couple of (probably very basic things).
I need to produce a Hasse ...

**18**

votes

**5**answers

5k 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

230 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

596 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

443 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

164 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, ...