# Questions tagged [computer-algebra]

Using computer-aid approach to solve algebraic problems. Questions with this tag should typically include at least one other tag indicating what sort of algebraic problem is involved, such as ac.commutative-algebra or rt.representation-theory or ag.algebraic-geometry.

66
questions with no upvoted or accepted answers

**25**

votes

**0**answers

767 views

### Where to submit this work with several unusual features?

I appreciate that questions about where to submit are generally considered off-topic, but I hope that the unusual features of the present case may make it acceptable.
I have put a monograph on github ...

**9**

votes

**0**answers

172 views

### When is cohomology of a finitely presented dg-algebra computable?

Given a smooth affine variety $X$ defined over $\mathbb{Q}$, its singular cohomology is isomorphic to the algebraic de Rham cohomology, which is the cohomology of the complex $\Omega_X^0\to\Omega_X^1\...

**7**

votes

**0**answers

108 views

### Classification of Frobenius algebras of small dimensions

Despite (commutative) Frobenius algebras over a field $K$ being a very popular class of algebraic objects, it seems no attempt of classification (up to $K$-algebra isomorphism) for them has been ...

**7**

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**0**answers

293 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 ...

**7**

votes

**0**answers

490 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 ...

**6**

votes

**0**answers

198 views

### Constructing the normal sheaf for the plucker embedding in MAGMA (or a similar programming language)

How would one construct the normal sheaf $N_{G(2,6)/\mathbb P^{14}}$ to the plucker embedding of the grassmannian $G(2,6) \rightarrow \mathbb P^{14}$ as a sheaf in MAGMA (or another programming ...

**5**

votes

**0**answers

101 views

### Rank of matrix over UFD polynomial ring

I have a matrix, $M$, size of approximately $20\times 25$, over a polynomial ring $\mathbb{Q}[x_1, \cdots, x_n]$ and a number $r$ such that I would like to test whether or not there exist values for ...

**5**

votes

**0**answers

109 views

### Stable equivalence and stable Auslander algebras

Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules.
Recall that the Auslander algebra of $A$ is $End_A(M)$ and the ...

**5**

votes

**0**answers

64 views

### Computing centralizers of finite sets in right angled Artin groups (RAAGs) / partially commutative groups / graph groups

This question concerns right angled Artin groups (RAAGs), also called partially commutative groups or graph groups.
A student of mine, Adi Ben-Zvi, needs for an algorithm in RAAGs, a subalgorithm ...

**5**

votes

**0**answers

137 views

### Computations in Weyl algebra with rational function coefficients

I am looking for a software to perform calculations with modules over the algebra $R_n=\mathbb{C}(x_1\ldots x_n)\langle \partial_1\ldots\partial_n\rangle$ of differential operators with rational ...

**5**

votes

**0**answers

184 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 ...

**5**

votes

**0**answers

331 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. ...

**5**

votes

**0**answers

252 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 "...

**4**

votes

**0**answers

79 views

### Deciding whether two algebras are derived equivalent

Given two finite dimensional quiver algebras $A$ and $B$ (over a nice field in case that helps, for example a finite field).
Question: Can an there be a finite algorithm that decides whether $A$ ...

**4**

votes

**0**answers

125 views

### Recovering the bimodule from the trivial extension

Given a ring $S$ with a non-zero $S$-bimodule $M$, the trivial extension of $(S,M)$ is defined as the ring $R:=T_M(S)$ with $R= S \oplus M$ with multiplication $(s,m)(s',m')=(s s', sm' +m s')$.
We ...

**4**

votes

**0**answers

98 views

### Compute the closure of graph of function from complement of hypersurface in $\mathbb{A}^n$

I'm hoping someone can give me some tips to help speed up computation on the following problem:
Suppose I have a map $G=(g_1/f,\dots,g_m/f):\mathbb{A}^n\setminus{V(f)}\to \mathbb{A}^m$. I'm ...

**4**

votes

**0**answers

79 views

### Compute equalizer of maps of polynomial rings, perhaps using Gröbner bases

Suppose that $k$ is a field and I have two ring homomorphisms
$$
\phi, \psi :k[x_1,...,x_m] \to k[y_1,...,y_n].
$$
How can I use Gröbner bases (or other computational tools) to compute the subring of ...

**4**

votes

**0**answers

165 views

### Computing the volume of intersection between a ball and a box

$C$ is the set of vectors which are coordinate-wise less than $\overline{c}\in [-1,1]^d$ and greater than $\underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex in $d$ that ...

**4**

votes

**0**answers

159 views

### Finding Rational Curves on a Surface

Let the field of rational numbers be our base field $k$. I hope to find all rational curves on the following surface $S$ defined by $f$. You can find the motivation in the end.
$f= (x^2y^2)z^3 + (5x^...

**4**

votes

**0**answers

167 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 F}...

**4**

votes

**0**answers

279 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 ...

**4**

votes

**0**answers

204 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 ...

**4**

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**0**answers

235 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 ...

**4**

votes

**1**answer

230 views

### Higher roots modulo prime complexity best algorithm

Given integers $a,\ell$ and prime $p$ we need to find the roots of the algebraic equation $x^\ell\equiv a\bmod p$. We know there are at most $\ell$ such $x$.
What is the best method to find all such ...

**3**

votes

**0**answers

62 views

### Frobenius algebras of small dimensions

In Classification of commutative Frobenius algebras , Jeremy Rickard showed that there are infinitely many commutative (local without loss of generality) Frobenius algebras of vector space dimension ...

**3**

votes

**0**answers

91 views

### Finite test for periodicity of a module

Let $A$ be a finite dimensional quiver algebra and $M$ a finite dimensional $A$-module.
Assume we want to test whether $M$ is a periodic module, meaning that $\Omega^n(M) \cong M$ for some $n \geq 1$. ...

**3**

votes

**0**answers

90 views

### Deterministic procedure to find irreducible polynomials

In $\Bbb F_q[x_1,\dots,x_n]$ given $d_1,\dots,d_n\in\Bbb N$ is there a deterministic $O(poly(nd\log q))$ algorithm to find an irreducible polynomial with $d=\max_{i\in\{1,\dots,n\}}d_i$ and $d_i=deg(...

**3**

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**0**answers

85 views

### Can one do better than using general purpose determinant algorithms when using the Fisher-Kasteleyn-Temperley method for perfect matchings?

Questions.
(numerical.generalPfaffian) Is it proved anywhere that in general it is not easier0 to calculate the determinant (over $\mathbb{Q}$) of the skew-symmetric signed adjacency matrix defined ...

**3**

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**0**answers

159 views

### Explicit roots in algebraic extention of Q with roots

Denis Bouhineau in "Solving Geometrical Constraint System Using CLP Based on Linear Constraint Solver" gave a method to find explicit square root in algebraic extention of Q with square roots. For ...

**3**

votes

**0**answers

309 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 ...

**3**

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**0**answers

331 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. ...

**2**

votes

**0**answers

104 views

### Computing whether a set of polynomials cuts out a projective variety

I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a projective variety, i.e. whether the radical of the ideal $I$ that they generate is homogeneous....

**2**

votes

**0**answers

67 views

### Affine projection of polynomials for a given set of points

(Not sure this question fits here, I will remove it in case it doesn't)
Let $F_{\mathrm{ML}}[x_1, \ldots,x_n]$ denote the set of multilinear polynomials over a finite field $F=\mathbb{F}_q$ (i.e. a ...

**2**

votes

**0**answers

48 views

### Determining the symmetry group of a system of ODEs as tensor product decomposition

Suppose you have a linear system of ODEs in implicit form represented by an $n \times n$ matrix $M$ and you are trying to identify a subgroup of $\mathrm{GL}(n)$ that is the symmetry group of this ...

**2**

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**0**answers

68 views

### CAS for finite-dimensional complex representations of $S_n$

Does there exist a computer algebra system that can work with finite-dimensional complex representations of the symmetric groups on finitely many letters? It should have the following functionality: (...

**2**

votes

**0**answers

57 views

### Constructing stable equivalences for finite dimensional algebras

Given a finite dimensional (non-selfinjective) algebra $A$.
Is there a method (for example using QPA) to construct algebras stable equivalent to $A$?
Such a thing is easily possible for derived ...

**2**

votes

**0**answers

80 views

### Obtaining the reduced incidence algebra in QPA

Given a finite poset $P$ (we can assume it is connected), the reduced incidence algebra of $P$ is the subalgebra of the incidence algebra of $P$ consisting of functions constant on isomorphic ...

**2**

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56 views

### Efficient algorithm to prove that a polynomial ideal contains 1

I have the following problem:
Suppose to have an ideal $I\triangleleft k[x_1,...,x_n]$ defined by generators. There exists an efficient algorithm (perhaps more efficient than calculating the Groebner ...

**2**

votes

**0**answers

199 views

### Combinatorial and computational problem related to Weyl groups and the coroot lattice

Let $W$ be a Weyl group with root system $R$ and with set of positive roots $R^+$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. positive roots $\alpha$ which satisfy $\ell(s_\alpha)=2\...

**2**

votes

**0**answers

107 views

### Finding generators of symmetric cones

I have a bunch of vectors $\mathbf v_i$ in $\mathbb R^n$. I would like to consider the cone $C$ spanned by these vectors, together with all the other vectors that can be obtained by permuting the ...

**2**

votes

**0**answers

110 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 ...

**2**

votes

**0**answers

168 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 ...

**2**

votes

**0**answers

251 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

101 views

### Saturated ideals in computational algebra

Let $R$ a commutative ring with one and $I, J \triangleleft R$ two ideals.
The saturated ideal $I^{sat}_J$ with respect $J$ is the ideal
$$(I : J^\infty )= \cup_{n \geq 1} (I:J^n)$$
where $(I:J^n)= \{...

**1**

vote

**0**answers

68 views

### Degree bounds on coordinates of points in a zero-dimensional variety

Let $S = \{f_1, \dots, f_s \in \mathbb{Q}[x_1, \dots, x_n]\}$ have a zero-dimensional nullset $V \subset \mathbb{C}^n$, and suppose that each $f_i$ has total degree at most $d$.
Is there a shared ...

**1**

vote

**0**answers

145 views

### Computing the class-preserving automorphism group of finite $p$-groups

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called a class-preserving if for each $x\in G$, there exists an element $g_x\in G$ such that $\alpha(...

**1**

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**0**answers

30 views

### Grobner basis of the toric ideal $I_{A_P}$ with respect to $<_{rev}$ consists of those binomials $t_αt_β − t_{α\cap β} t_{α\cup β}$

I try to understand the proof of the Theorem. 10.1.3.(page 185) from ''Monomial Ideals'' by Herzog & Hibi.
The reduced Grobner basis of the toric ideal $I_{A_P}$ with respect
to $<_{rev}$ ...

**1**

vote

**0**answers

108 views

### generalized sieve implementations

Suppose I have a multiplicative function $f$ and I want to compute it for all integers from $1$ to $N$ (or maybe just for a long sequence of consecutive integers). It is clear that this is far faster ...

**1**

vote

**0**answers

159 views

### Coefficient perturbations of polynomials with real roots only

Let
\begin{align}
P(x) &= x^n+a_{n-1} x^{n-1} +\ldots+a_0 = \prod_{i=1}^n (x-p_i)\\
Q(x) &= x^n + b_{n-1} x^{n-1} + \ldots +b_0 = \prod_{i=1}^n (x-q_i)\\
p_i, q_i& \in \mathbb{R},\ 0<...

**1**

vote

**0**answers

146 views

### Efficient deterministic algorithms of factorizing

My question is about efficient deterministic algorithms of factorizing polynomials of degree $n$ over $\mathbb{F}_q$.
Are there such algorithms that use poly$(n, \log q)$ bit operations?
I know ...