All Questions
Tagged with ac.commutative-algebra computer-algebra
66 questions
2
votes
0
answers
130
views
How to find a single-variable polynomial in a zero-dimensional ideal?
Given finitely many multivariate polynomials with algebraic coefficients that generate a zero-dimensional ideal, is there an easy way to find a nonzero single-variable polynomial in this ideal?
If we ...
- 7,633
4
votes
1
answer
334
views
GCD in $\mathbb{F}_3[T]$ with powers of linear polynomials
This is a continuation of my previous question on $\gcd$s of polynomials of type $f^n - f$.
Let us call $n > 1$ simple at a prime $p$ when $p-1 \mid n-1$ but $p^k - 1 \not\mid n-1$ for all $k > ...
- 63.1k
2
votes
0
answers
188
views
Help with Macaulay2 computation of invariant ring
Consider the algebraic group $G:=\operatorname{SL}_{2}\times\operatorname{SL}_{2}$ acting on $V:=\operatorname{Mat}_{2\times 2}\oplus\operatorname{Mat}_{2\times 2}$ via the action $(A,B)\,\cdot\,(X,Y)=...
- 839
5
votes
0
answers
126
views
Koszul algebras among finite dimensional commutative algebras
Given a local commutative artinian algebra $A$ of the form $K[x_1,...,x_n]/I$ with quadratic ideal $I$ and $K$ a field.
Question 1: Is there a computer algebra system that can check whether such an ...
- 26.5k
5
votes
0
answers
107
views
Size of minimal generating set of ideal over Laurent polynomial ring
Recently in attacking a problem in algebraic topology relating to the construction of stably-free non-free modules over integral group rings I’ve noticed that it is often fairly easy to reduce to ...
- 187
3
votes
0
answers
120
views
Checking the generic rank of a matrix
Suppose that $A,B\in M_{p,q}(\mathbb{Z})$ are two rectangular integer matrices of the same size. Suppose that one has a conjecture stating that the rank of the matrix $A+tB$ for Zariski generic values ...
- 16.9k
0
votes
1
answer
213
views
Grobner basis of a submodule of a free module over polynomial ring
Let $A=\mathbb{Q} [x_1,\dots,x_n]$ be the ring of polynomials with rational coefficients. Let $T$ be an $m\times n$ matrix with entries from $A$. Consider it as a morphism of $A$-modules $T\colon A^n\...
- 21.8k
0
votes
1
answer
127
views
Software to compute generators of a module over polynomial ring
Let $A=\mathbb{R}[x_1,\dots,x_n]$ be the algebra of real polynomials in $n$ variables. Fix polynomials $p_1,\dots,p_k\in A$.
Consider the subset
$$M:=\{(q_1,\dots,q_k)\in A^k|\, p_1q_1+\dots+p_kq_k=0\}...
- 21.8k
1
vote
0
answers
259
views
Computer algebra programs that can solve polynomial systems on algebraically closed fields besides MAGMA
I was wondering which computer algebra programs out there can solve polynomial systems on the algebraic closure of $\mathbb{Q}$ analytically and efficiently.
So far, I only found MAGMA with its ...
- 21
1
vote
1
answer
332
views
Are algebraic power series in positive characteristics D-finite?
We know that in characteristic $0$, all algebraic series are differentiably finite.
Is this true in positive characteristic? I look at the proof, indeed we need to the
characteristic to be $0$ for the ...
- 385
3
votes
0
answers
68
views
Finding generators and relations for special commutative algebras with a computer
Let $K[x_1,...,x_n]$ be the polynomial ring in $n$ variables and $a_1,...,a_m$ elements in the quotient field $K(x_1,...,x_n)$.
Let $A:=K[a_1,...,a_m]$ the ring generated by the $a_i$ in $K(x_1,...,...
- 26.5k
1
vote
1
answer
209
views
Primary decomposition of huge ideals using M2/Singular
I used to ask similar questions in other communities, but so far never received any feedback.
Given four Hermitian $n\times n$ matrices $A_1,A_2,B_1,B_2$ together with the constraints $[A_i,B_j]=0$, I ...
- 71
3
votes
1
answer
238
views
Number of rings with additive group $(\mathbb{Z}_{16})^2$. A341547(16) in OEIS
I would like to know if somewhere the number of non-isomorphic rings with additive group $(\mathbb{Z}_{16})^2$ is mentioned. If not, is someone able to calculate it?
And (easier) the commutative case? ...
2
votes
1
answer
668
views
Efficiently computing Gröbner basis to prove no solution to polynomial constraints
In a similar vein to these now quite old questions on advice for calculating a Gröbner basis:
Fast computation of a Groebner basis. What is possible?
What is the state of art in Groebner bases
I am ...
- 21
2
votes
1
answer
202
views
How to compute cup product of derived limits / presheaf cohomology
I have a finite category $\mathcal{C}$, along with a functor $F \colon \mathcal{C}^{\mathrm{op}} \to \mathsf{GradedCommRings}$. If $F_j$ is $j$-th graded piece of $F$, then I write $H^i(\mathcal{C},...
- 317
2
votes
0
answers
113
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....
- 980
5
votes
0
answers
2k
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)= \{...
- 6,018
7
votes
0
answers
181
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 ...
- 26.5k
0
votes
0
answers
71
views
Low rank approximation
Can we solve low rank approximation problem by using concept of Gröbner basis? I was trying to find it by Macaulay2 but didn't find the answer. I was trying to do by toric ideals as for them Gröbner ...
5
votes
0
answers
220
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 ...
- 51
11
votes
1
answer
475
views
Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials
Let $n>1$ and $p$ be an odd prime with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. Notice that, since $p-1 \mid n-1$, we have $T^p - T \mid T^n-T$ in $\mathbb{F}_p[T]...
- 63.1k
1
vote
2
answers
679
views
Computing Groebner basis for a complicated systems of polynomials
I am trying to solve complicated systems of polynomial equations. The first step is to determine maximal sets of independent variables for the solution manifold (ideal) or the number of isolated ...
2
votes
0
answers
61
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 ...
- 297
4
votes
0
answers
104
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 ...
- 2,788
4
votes
1
answer
273
views
Resultants for compactly represented product form polynomials?
Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the ...
- 13.9k
1
vote
1
answer
108
views
Finding a characteristic for which the zero-locus of an ideal is not empty
I have a set of polynomials $f_1, \dots, f_m \in \mathbb{Z}[x_1, \dots, x_n]$ and I am interested in finding if these polynomials have a common root inside either $\mathbb{C}[x_1, \dots, x_n]$ or $\...
- 113
1
vote
1
answer
394
views
Is this algorithm for primary decomposition correct?
I've written some code for Sage to compute radical ideals and primary decompositions over $\overline{Q}$ (the field of algebraic numbers), and I'm not sure if it's right.
Since Singular (the ...
- 415
4
votes
0
answers
98
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 ...
- 4,254
6
votes
1
answer
148
views
Software for computing equivariants
If $\Gamma$ is a finite group with action on two vector spaces $\mathbb R^n$ and $\mathbb R^m$ denoted by $\gamma_n$ and $\gamma_m$ respectively, the fundamental equivariants are the polynomials $f: \...
- 225
1
vote
0
answers
35
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}$ ...
- 163
3
votes
1
answer
539
views
Generalized Newton Identities
I learnt a lot of new words (Hall-Littlewood, Jack and Macdonald polynomials) but unfortunately everything I dug up is written without a single example and I still don't know the answer to a very ...
- 4,829
1
vote
2
answers
337
views
Finding all submodules
Given a finite dimensional local commutative algebra over a finite field $K$ and a finite dimensional module $M$. What is the fastest/best way to obtain all submodule from $M$ using a Computer algebra ...
- 26.5k
6
votes
1
answer
269
views
Algebraization of Bayesian networks?
The algebraization of classical propositional logic is Boolean algebra.
Bayesian networks are a generalization of classical propositional logic with probability truth-values.
What is the ...
- 558
2
votes
2
answers
411
views
Computer algebra system that test zero divisors in a quotient algebra
I have an algebra $A$ over a Noetherian ring and an ideal $I=(x,y)$, where $x,y \in A$. I need to examine whether a polynomial $h \in A$ is a zero divisor in $A/I$ or not.
Is there a computer algebra ...
- 21
2
votes
1
answer
191
views
programming to compute kernel quotient image of a $\mathbb{Z}$-module endomorphism
Let the integers $n\geq 2$, $k\geq 1$, $v=0$ or $1$ and $n_1,\cdots,n_k\geq 1$ such that
$$
\sum_{i=1}^k n_i+v=n.
$$
Define $P_a^b=0$ if both $a,b$ are odd and $P_a^b={{[a/2]}\choose {[(a+b)/2]}}$ ...
- 1,990
1
vote
0
answers
110
views
Grobner basis for a general algebra
Let $R$ be a quotient of the polynomial ring $\mathbb{C}[x_1,\dots , x_n]$. We fix a $\mathbb{C}^*$ action on $R$ which preserve homogenous components and the multiplication. (The geometric analogue ...
- 2,384
1
vote
0
answers
280
views
Algebraic independence criterion
Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...
- 11
6
votes
0
answers
293
views
Constructing the normal sheaf for the plucker embedding in MAGMA (or a similar programming language)
How would one construct the normal sheaf $N_{G(2,6)/\mathbb P^{14}}$ to the plucker embedding of the grassmannian $G(2,6) \rightarrow \mathbb P^{14}$ as a sheaf in MAGMA (or another programming ...
- 1,308
16
votes
2
answers
721
views
From polynomial ideal over $\mathbb{Q}$ to polynomial ideal over $\mathbb{Z}$
Is there an algorithm to compute, given a polynomial ideal $I\subset \mathbb{Q}[x_1,\dotsc,x_n]$, the ideal $I\cap \mathbb{Z}[x_1,\dotsc,x_n]$ in $\mathbb{Z}[x_1,\dotsc,x_n]$?
The input and output ...
- 7,836
1
vote
0
answers
221
views
How do I check if a sequence of R-modules is exact?
Let R be a ring. For example, take $R=k[x_1,\ldots,x_n]$ or, if possible, $R = \Bbb{Z}[x_1,\ldots,x_n]$.
Consider a sequence of free R-modules
$$R^a \stackrel{f}\to R^b \stackrel{g}\to R^c$$
where $f$...
- 8,866
28
votes
1
answer
1k
views
Algebraic dependency over $\mathbb{F}_{2}$
Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$
such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall i\in[n]:f_{i}(a)=a_{i}$....
- 583
5
votes
1
answer
255
views
computing the nonnegative part of a $\mathbb{Z}$-graded ring
Let $R = \bigoplus_{n \in \mathbb{Z}} R_n$ be a $\mathbb{Z}$-graded commutative ring with nonnegative part $R^+ = \bigoplus_{n \geq 0} R_n$ and nonpositive part $R^- = \bigoplus_{n \geq 0} R_{-n}$. By ...
- 211
21
votes
5
answers
6k
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 ...
- 1,256
1
vote
1
answer
361
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 ...
- 1,256
4
votes
0
answers
312
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 ...
- 768
5
votes
1
answer
557
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 ...
- 2,399
1
vote
1
answer
470
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.
$\bf{...
- 3,555
4
votes
2
answers
1k
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 ...
- 3,555
1
vote
1
answer
160
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 ...
- 13.9k
12
votes
2
answers
588
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 ...
- 4,902