**7**

votes

**2**answers

144 views

### Fast computation of a Groebner basis - What is Possible

I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or ...

**0**

votes

**0**answers

127 views

### Understanding a program for computing Khovanov homology

I would like to understand how a computer program for computing Khovanov homology works. The particular program I have in mind is by John Baldwin: https://web.math.princeton.edu/~baldwinj/Kh.cpp
The ...

**1**

vote

**1**answer

68 views

### Algorithm for Polynomial Reduction in a Quotient Ring

Any reference or suggestion for the following problem would be greatly appreciated.
I am working on the quotient ring $Q=R[X_1,\dots,X_n]/<f_1,\dots,f_k>$. Given polynomials $p$ and $q$ I want ...

**-2**

votes

**1**answer

143 views

### AI / Machine Learning related to high/modern/front mathematics [closed]

I major math and cs. and i'm interested in ai/machine learning/data mining.
so i want to know what math subjects are used in frontier of these technology.
especially, high mathematical tool, like ...

**1**

vote

**0**answers

105 views

### Calogero-Moser eigenfunction

The folllowing function
\begin{equation}
J(t_1,t_2,t_3,m,h)=[(1-e^{t_1-t_2})(1-e^{t_2-t_3})(1-e^{t_1-t_3})]^{-m/h} ...

**4**

votes

**0**answers

84 views

### Computing Tamagawa numbers for jacobians of hyperelliptic curves

Do exist some computational approach to calculation of Tamagawa number for the jacobian of hyperelliptic curve at prime $p$?
As followed from this question one can compute $\Phi(\overline{\mathbb ...

**1**

vote

**2**answers

393 views

### Given a formal power series ,decide whether there exists a polynomial the series satisfies and if it exists,how to write it down?

Given a formal power series $$y(x)=\sum_{i=0}^{\infty} a_i x^i$$ Is there an algorithm that decides whether there exists a polynomial$$ P(x,y)=p_n(x)y^n+p_{n-1}(x)y^{n-1}+\cdots+p_0(x)=0,p_j(x)\in ...

**5**

votes

**0**answers

119 views

### Software for BMW algebra calculations?

Does software exist for computations in the BMW algebra?
For example, I'd like to be able to express elements in a basis of "totally descending tangles" as in a paper of Morton–Wassermann. At ...

**4**

votes

**0**answers

149 views

### Dimension of a commuting nilpotent variety

Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...

**6**

votes

**1**answer

310 views

### Using math software to show that the following groups are infinite?

I would like to show that the following finitely presented group in 3 generators $P, Q, R$ is infinite in certain cases:
$$P^p, Q^q, R^r, (PQ)^2, (QR)^2, (PQR)^2, (QR^{r/2+1})^a (RQR^{r/2})^b$$
For ...

**0**

votes

**1**answer

68 views

### Software for noncommutative Groebner bases over rational function fields

I am wondering whether there is any software package that can compute Groebner bases for noncommutative algebras defined over the field of rational functions $\mathbb{Q}(q)$.
I have tried the GAP ...

**3**

votes

**0**answers

138 views

### What is the function like when its Mobius inversion is $\sum_{w|r, (w,t)=1}\mu(w)q^{r/w}$?

Everyone, I am now reading a paper named The Irreducible Factors of $(cx+d)x^{q^m}-(ax+b)$ over $GF(q)$, http://qjmath.oxfordjournals.org/content/14/1/61.extract. And I’m confused with one of its ...

**1**

vote

**0**answers

90 views

### Benchmark problems for computing rational points on varieties

Are there standard benchmark problem sets used for empirically evaluating algorithms designed for computing rational points on (various classes of) algebraic varieties?
If so, could you please point ...

**11**

votes

**1**answer

411 views

### Faster multiplication with a restricted set of multiplicands?

Let $A$ be a set of $k>1$ distinct elements from a semigroup. We wish to compute the product
$$ p=b_1 b_2 \cdots b_n$$
where each $b_i\in A$.
Clearly $n-1$ multiplications suffice to compute $p$; ...

**3**

votes

**1**answer

133 views

### Is the ideal membership problem solvable for differential ideals? Is there a good notion of a Gröbner basis?

Let $K$ be a field of characteristic zero. Let $\Omega = K[x_1, \dots, x_n, dx_1, \dots, dx_n]$ be the differential ring of algebraic differential forms over $K[X_1, \dots, X_n]$.
Is there an ...

**2**

votes

**1**answer

231 views

### efficient arithmetic with (short) Conway games?

We consider "games" in the sense of ONAG. Conway's definition of a game $G$ as a pair $G = \{L \mid R \}$ of sets of games, together with the definitions of inequality and the arithmetic operations ...

**2**

votes

**0**answers

107 views

### Finding relations between invariant polynomials

Suppose I have an action of a linear reductive group ($GL(2,\mathbb{C})^2$ in this case) on a complex vector space (of dimension $16$) and I want to compute explicitly the ring of invariants of this ...

**2**

votes

**1**answer

124 views

### Finding particular reduced words for Weyl group elements

I am studying cluster algebra structures on the coordinate rings of partial flag varieties, as defined in the paper Partial flag varieties and preprojective algebras by Geiss, Leclerc and Schröer. One ...

**1**

vote

**0**answers

175 views

### Finding a generator of an ideal in an algebraic function field

I have an algebraic function field $\mathbb{Q}(x,y)$, where $y$ satisfies
$$ (y^2-1)^2 = x^2(1+x^2), $$
and I need to find a rational function that has a first order root at $x=0,y=1$ a first order ...

**4**

votes

**1**answer

198 views

### The Representation of $\mathrm{Sp}_{2n}$ of Dimension $2^n$ in characteristic 2

Let $G:=\mathrm{Sp}_{2n}$ be the simple algebraic group of simply connected type with root-system $C_n$.
Is there a way, to explicitly construct the
highest weight representation ...

**1**

vote

**0**answers

76 views

### Testing functional equivalence

We are looking for the most efficient (most recent, or best) techniques to check if two algebraic expressions (elementary, Calculus-type functions) are equivalent (or if an expression is equivalent to ...

**17**

votes

**1**answer

729 views

### Main open computational problems in quantifier elimination?

A language is said to have quantifier elimination if every first-order-logic sentence in the language can be shown to be equivalent to a quantifier-free sentence, i.e., a sentence without any ...

**2**

votes

**0**answers

194 views

### Using the Affine Maxima Package

The Maxima computer algebra system has a package called Affine for doing the calculations implicit in Bergman's diamond lemma for rings. It can be viewed as a kind of noncommutative analogue of ...

**1**

vote

**1**answer

224 views

### Serre's conditions under blow-ups, Blowup and normalization

Suppose $X = \mathbb{Z}[x, y, z]/(f,g)$ is a 2-dimensional Cohen-Macaulay surface. In particular, $X$ satisfies Serre's condition $S_2$. Suppose it is irreducible, reduced but not normal.
...

**4**

votes

**2**answers

368 views

### Algorithm to decide if ideal is principal

Suppose $R = \mathbb{Q}[x_1, ..., x_n]/I$, and $J \subset R$ is a given height one ideal. Is there a quick algorithm one could write to determine if $J$ is a principal ideal or necessarily not ...

**1**

vote

**1**answer

122 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

342 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

346 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

210 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 ...

**4**

votes

**1**answer

426 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 ...

**2**

votes

**1**answer

203 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

213 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 ...

**43**

votes

**2**answers

964 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

357 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 ...

**3**

votes

**1**answer

197 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 ...

**0**

votes

**0**answers

142 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

121 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

185 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

517 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

293 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

454 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 ...

**1**

vote

**0**answers

136 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

331 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

441 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

807 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 ...

**3**

votes

**0**answers

265 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

117 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 $x_1 = \cdots = x_n$ ...

**5**

votes

**1**answer

322 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

99 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

329 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 ...