# Questions tagged [groebner-bases]

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**8**

votes

**1**answer

742 views

### 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 ...

**2**

votes

**2**answers

507 views

### Nonstandard monomial orders?

Are there any articles/books/examples where a non-standard monomial order is used?
What are the applications of these monomial orders? In particular, uses in groebner bases and variable elimination.
(...

**4**

votes

**2**answers

2k views

### Numerical solution for a system of multivariate polynomial equations

Hi all,
I have a system of 6th-order polynomial equations in 4 variables $q_1, q_2, q_3, q_4$ (i.e. polynomials with all the terms such as $q_1^6, q_2^6, q_2^4 q_3^2$):
$P_k(q_1, q_2, q_3, q_4) = 0$ ...

**3**

votes

**1**answer

363 views

### Is the first part of Eisenbud's Proposition 15.15's proof o.k?

In the chapter on Gröbner bases from Eisenbud's "Commutative Algebra" the following statement appears as Proposition 15.15 (page 344):
Let $F$ be a free $S$ module with basis and monomial order ...

**4**

votes

**4**answers

1k views

### Systems of polynomial equations

Hi all,
I'm an engineer assigned to determine some parameters of a manipulator (ie., calibration). It has a number of parameters, but after some manipulations of its dynamic equations, I can have the ...

**6**

votes

**2**answers

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 ...

**4**

votes

**4**answers

380 views

### Lower bounds on the degrees of representatives of $u^n$ as $n \to \infty$

Let $k$ be an algebraically closed field and $A$ a finitely generated $k$-algebra, together with a specified surjective morphism $\phi \colon k[x_1, \dotsc, x_n] \to A$. For $f \in A$, define $\...

**1**

vote

**1**answer

312 views

### Any implemented algorithm to compute the closure of an affine variety in a product of projective spaces?

Let $I$ be an ideal of $k[x_1, \ldots, x_m, y_1, \ldots, y_n]$, $k$ being a field. Does any of the computer algebra systems implement any algorithm to calculate the generators of the 'bi-...

**26**

votes

**4**answers

2k views

### 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 ...