Questions tagged [computational-algebra]

Question 1

I have some confusion about the definition of birth time, and I hope someone can clarify a few things. I am reading "Computational Topology: An Introduction" by Edelsbrunner and Harer. They ...

Question 2

Let’s fix some natural $0 < m < n$ and consider matrices $m \times n$ with rational coefficients. Let’s call such matrices $A$ and $B$ equivalent iff there are an invertible $m \times m$ matrix $...

Question 3

I am working with some homogeneous polynomials $f\in \mathbb C[x_1, \ldots, x_n]$ belonging to families $\mathcal F$ over $\mathbb C^k$, i.e. I am considering $$f=\sum a_Ix^I, \qquad\qquad a_I\in \...

Question 4

Let $R=\mathbb Q[x_1, \ldots, x_n]<R'=\mathbb C[x_1, \ldots, x_n]$. I often can prove results for $R$ using computational algebra packages (e.g. Macaulay2) but I am never too sure if they extend to ...

Question 5

Let us consider two monic polynomials $f(X), g(X) \in \dfrac{\mathbb{Z}}{p^k\mathbb{Z}}[X]$. Now, we call $h(X)$ is a divisor of $f(X)$, if there exists a $l(X) \in \dfrac{\mathbb{Z}}{p^k\mathbb{Z}}[X]...

Question 6

This is a follow-up question to this question. In that question, we learned that if, $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ , then $T(g(z))$ is divisible by $f(z)$. Now, my question is: If $T(y) = ...

Question 7

I have a doubt concerning monomial orderings. I'm reading Cox, Little, O'Shea book Ideals, Varieties and Algorithms. Well, they say that a monomial $x^\alpha$ divides a monomial $x^\beta$ if there ...

Question 8

I am working on some problem on toric arrangements at the crossroad between topology, combinatorics and algebraic geometry. $\textbf{Setting}$ Let $m,n\geq1$ and let \begin{equation*}\mathcal{S}=\left\...

Question 9

I'm trying to compute the torsion subgroup of the first integral homology group $𝐻_1(\Gamma_1(193),\Bbb{Z})$ This group arises from modular symbols associated to the congruence subgroup $\Gamma_1(193)...

Question 10

In the group of isometries of the plane, let $r$ be a rotation by $\frac{2\pi}5$ radians and let $s$ be a translation. I'd like to find a finite presentation of the subgroup $G=\langle r,s\rangle$. ...

Question 11

I wrote a proof for the following statement: Let $\mathbb{F}$ be a finite field and $V = \mathbb{F}^n$ for $n \in \mathbb{N}$. Furthermore let $A \in \mathrm{GL}(n,\mathbb{F})$ with $\mu_A = \chi_A = ...

Question 12

There are algorithms for computing explicitly a basis of holomorphic differentials for algebraic curves given as the vanishing locus of a polynomial $f(x, y)$. Take for example a smooth hyperelliptic ...

Question 13

Let $P$ be the nonnegative orthant in ${\mathbb R}^4$, $P={{\mathbb R}_+}^4$. For a finite set of vectors $S \subseteq P$, the cone $Cone(S)$ generated by $S$ is the set of all linear combinations of ...

Question 14

Gröbner bases computation works on several computer systems. I have this question. Is it possible to compute (in any of the systems, eventually in which?) Gröbner basis for integer polynomials, in ...

Question 15

Let $G$ be a finite group, and assume you have some computational representation of $G$ that allows efficient iteration through its elements, constant-time checks for equality between elements, and ...

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Questions tagged [computational-algebra]

Birth definition in persistent homology theory

How to algorithmically tell if two matrices are equivalent up to an invertible matrix on the left and a permutation matrix on the right?

Statements about dimension of singularity loci in families of polynomials

Extending proofs via computational algebra from $\mathbb Q$ to $\mathbb C$

Is there any algorithm which can find a common divisor of two polynomials modulo $p^k$?

If resultant $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ have a nontrivial factors then can $f(x)$ also have a nontrivial factors?

Problems with monomial orders and monomial divisibility

Toric arrangements and system of polynomial equations

What is the order of the torsion subgroup of $𝐻_1(\Gamma_1(193),\Bbb{Z})$?

Finding a presentation of the group generated by a translation and a rotation

A-invariant subspaces of cyclic generalised linear eigenspaces

What can be computed algorithmically on cohomology of higher-dimensional varieties?

Cone of rank three between two cones of rank four?

Gröbner bases computation for not fully specified polynomials

How fast is taking the quotient of a group by a subset?

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