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Questions tagged [groebner-bases]

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12
votes
4answers
3k 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 ...
6
votes
2answers
698 views

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 ...
3
votes
1answer
2k views

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 ...
7
votes
2answers
648 views

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}...
2
votes
0answers
84 views

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, ...
0
votes
0answers
101 views

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 $$\mathfrak{I}_d=\{ \text{ideals for which there is minimal generating system with $d$ elements} \}\setminus \{\text{ ideals generated by $d$ monomials}\}$$ ...
3
votes
1answer
251 views

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 ...