# Tagged Questions

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**1**answer

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### 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 ...

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**0**answers

147 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 ...

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**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 ...

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**1**answer

223 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**

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**2**answers

367 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 ...

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**1**answer

121 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 ...

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**2**answers

205 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 ...

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**1**answer

356 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 ...

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**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 : ...

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**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 ...

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**1**answer

321 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, ...

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**0**answers

221 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 ...

**5**

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**1**answer

395 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 $\mathbb{A}^n$. ...

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**4**answers

1k 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 ...

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**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 $M$ ...

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votes

**2**answers

871 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 ...

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votes

**1**answer

708 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 ...

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votes

**1**answer

352 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 ...

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**2**answers

590 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 - ...

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votes

**1**answer

371 views

### Decomposition of modules using computer packages

I am interested in computing direct sum decomposition of modules over some quotients of polynomial rings over a field (do not care much about the field at this point). Does any one know a package ...

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votes

**3**answers

542 views

### Is there a software package that does Schubert Calculus computations?

Is there a good software package for doing computations in the cohomology ring of Grassmannians? Things like, I can write down a polynomial in, in fact, special Schubert classes, but it's one where ...

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votes

**5**answers

576 views

### is there a good computer package for working with bicomplexes?

I'm interested in working with bicomplexes of modules over polynomial rings, specifically tensoring them together, and the operation of taking cohomology in one direction, and then the other. Is ...