Questions tagged [groebner-bases]
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12 questions
21
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5
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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
28
votes
4
answers
3k
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Can Gröbner bases be used to compute solutions to large, real-world problems?
In particular, suppose I have an affine algebraic variety over $\mathbb{R}^n$ described by generators of a radical ideal $I$ and I want to find (perhaps not all of the) points in the variety. Several ...
- 3,059
12
votes
2
answers
882
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Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$
This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< ...
- 245
11
votes
1
answer
1k
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PBW theorem over a Q-algebra, without freeness or flatness
Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-Lie algebra, which is not necessarily free as a $k$-module. Let $S\left(L\right)$ denote the symmetric algebra of $L$ (over $k$), constructed ...
- 33.8k
8
votes
1
answer
656
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Given a zero-dimensional ideal $(f_1,...,f_n)$, is the ideal $(f_1-\alpha_1,...,f_n-\alpha_n)$ also zero-dimensional?
Suppose you have a zero-dimensional ideal $I=(f_1,...,f_n)$ in a polynomial ring $R=k[x_1,...,x_n]$ over a field $k$, so that $\dim_k(R/I)<\infty$. Take indeterminates $\alpha_1,...,\alpha_n$ and ...
- 573
7
votes
2
answers
872
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Dimension of a homogeneous polynomial system
Let $m\geq4$ be an even integer, $V\subset\mathbb{C}^{m-1}$ be the solution set of the following polynomial equations:
\begin{cases}
&\sum\limits_{s=1}^{2t-1}z_sz_{2t-s}+\sum\limits_{s=2t+1}^{m-1}...
7
votes
2
answers
815
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Differential ideal membership problem
We know that in the ordinary algebra ideal case, the ideal membership problem can be solved by the Grobner Base theory, then, is there a counterpart theory in the differential ideal case?
To be ...
- 1,528
5
votes
1
answer
3k
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Existence of a real-valued solution to system of multivariate polynomial equations
Given a system of multivariate, polynomial equations, is there a way to determine if it has a solution in a given field (for instance the set of all reals). I don't care what the solution is, I just ...
- 53
4
votes
1
answer
547
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How can I include irreducibility in a Groebner basis calculation?
I'm trying to prove impossibility of certain systems of differential/polynomial equations using Groebner basis techniques.
For example, consider the equation $qn = mf$, where each of the variables ...
- 415
2
votes
1
answer
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Efficiency of Groebner basis for constraints of the form $(a_i x_i+b_i)(a_j x_j+b_j)$
This is based on numerical experiments in sage.
Let $K$ be a ring and define the ideal where each polynomial
is of the form $(a_i x_i+b_i)(a_j x_j+b_j)$ for constant $a_i,b_i,a_j,b_j$.
Q1 Is it true ...
- 25.4k
1
vote
0
answers
126
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Polynomial constraints triggered by irreducibility [closed]
I've come across an interesting connection between irreducible polynomials and polynomial constraints. For example, consider the basic quadratic:
$$af^2 + bf + c = 0$$
If we're working in a ring, ...
- 415
0
votes
0
answers
118
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Maximal elements for ideals and subrings ordered by inclusion with fixed number of minimal generating polynomials
Let $R=\mathbb{R}[X_1,\dots,X_n]$, and
$$\mathfrak{I}_d=\{ \text{ideals for which there is minimal generating system with $d$ elements} \}\setminus \{\text{ ideals generated by $d$ monomials}\}$$
...
- 1,256