# Questions tagged [groebner-bases]

The groebner-bases tag has no usage guidance.

92
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### Extracting implications in polynomial constraint system from Groebner basis

Given a Groebner basis for a system of polynomial constraints over $\mathbb{Q}$, are there any known methods for extracting the low degree factorable polynomials in the ideal generated by that basis?
...

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### Gorenstein property from initial ideal

My question is:
If $I$ is a homogenous ideal of $S=K[x_1,\dots,x_n]$ and $in_{<}(I)$ is the initial ideal of $I$, with respect to a term order $<$ on $S$, then $S/I$ is Gorenstein if and only if ...

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### Ideal membership and change of fields

Let $R=k[x_1,...,x_n]$ be a polynomial ring over a field. Let $f$ be a homogeneous polynomial and $I=(f_1,...,f_m)$ a homogeneous ideal.
With Macaulay2, one can compute the Groebner basis of $I$ when $...

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### Gröbner implicitization with relationships between the variables

I have the following parametric equations, where cost$=\cos t$, cos2t$=\cos 2t$, and $A^2+B^2=1$:
...

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### Resultants and elimination theory

Consider an ideal $I = \langle f_1,\dotsc,f_n\rangle$ in the ring $k[x_1,\dotsc,x_m]$.
Define the $i$-th elimination ideal to be $I_i = I \cap k[x_{i+1},\dotsc,x_m]$.
For any two polynomials $f$ and $...

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157
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### Are zero dimensional ideals radical?

I have a question about Theorem 3.7.25. of Computational commutative algebra I by M. Kreuzer and L. Robbiano.
Let $K$ be a perfect field, $I \subseteq K[x_1, \ldots, x_n]$, be a zero dimensional ...

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### How can Gröbner bases be generalized to differential algebra?

I'm aware of the Rosenfeld-Gröbner algorithm "for computing a regular decomposition of a radical differential ideal generated by a set of polynomial differential equations, ordinary or with ...

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### Compatibility conditions for quadratic equations

In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$:
\begin{eqnarray}
0 &= A_1x^2 + B_1x + C_1 \\
&...

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### How to speed up the process for calculating the Groebner basis?

I am currently trying to get the Groebner basis for 9 equations with 12 variables:
$
a_1^2+b_1^2+c_1^2+d_1^2-48.73=0\\
a_2^2+b_2^2+c_2^2+d_2^2-50.53=0\\
a_3^2+b_3^2+c_3^2+d_3^2-40.69=0\\
a_1a_2+b_1b_2+...

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### Groebner basis with parameters

I need to compute a Groebner basis of a polynomial system with parameters.
The only recent results I found is Groebner cover:
https://www.sciencedirect.com/science/article/pii/S0747717110000970
Are ...

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### 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\...

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### Library/Database of parametric polynomial systems

Could anyone please recommend a known website where I can find a database/library that has systems of polynomial equations with $n$ variables and $m$ parameters?
I need some real examples to test my ...

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### Recommendations for distributed calculations of Groebner Bases

There are many computer algebra systems available which can compute a Groebner basis, including:
Mathematica
Singular
Macaulay2
Magma
Maple
CoCoA
However (please correct me if I've missed something) ...

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### Techniques for estimating dimension of algebraic variety without calculating Groebner basis of polynomial constraints?

I have a system of multilinear polynomial constraints that I believe has no solution. Unfortunately, the number of variables and number of constraints is too large to afford direct computation of a ...

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### 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 ...

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### Buchberger's criterion for Gröbner bases in $k$-algebras with multiplicative basis and admissible order

Let $R$ be an associative $k$-algebra with multiplicative basis $\mathcal B$ with an admissible order on $\mathcal B$.
Let $G \subseteq R$ be a subset.
A multiplicative basis $\mathcal B$ means that $...

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### 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 ...

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### Methods for multivariate polynomial equations over large finite fields

I am trying to get a rough overview of the best methods one might use to find solutions of multivariate polynomial equations over large finite fields. We can suppose for simplicity that the given ...

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148
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### Sufficient syntactic conditions for zero-dimensionality of polynomial systems

Consider a system $S$ of polynomial equations, $p_1=0,...,p_m=0$, for $p_i\in K[x_1,...,x_n]$, for a field $K$: the system $S$ is zero-dimensional if it has finitely many solutions. It is well-known ...

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### The minimum number of polynomial equations the components of linearly dependent vectors must satisfy

Context:
Consider $m<n$ vectors $v_1,\dotsc,v_m\in\mathbb{C}^n$ with complex components. We can study if they are linearly dependent by constructing the following matrices. First the $n\times m$ ...

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### Dependence of the complexity of solving polynomial sytems on the multidegree

Let $f_1,\ldots,f_n\in \mathbb{Q}[X_1,\ldots,X_n]$ be a system of $n$ polynomials in $n$ indeterminant, which only has finitely many solutions. Supose that the each of the variables $X_i$ appears at ...

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### 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 ...

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### How general are Gröbner degenerations?

While working with flat degenerations of flag varieties and Schubert varieties I've noticed that among the numerous known constructions there doesn't seem to be a single one that doesn't turn out to ...

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### grobner basis of an ideal dependent on some parameter

Suppose $I = \langle f_1, ... , f_l \rangle$ is an ideal generated by polynomials $f \in k[x_1,\dots,x_n]$, where $k$ is a field of rational functions in some parameters $s_1,\dots,s_m$.
What are the ...

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### Groebner basis of a toric ideal

I know about toric ideals that it is a sort of binomial ideal i.e. generated by $x^u - x^v$, where $Au = Av $ ( A is the associated matrix). So by finding all integer solutions of $AX = 0$, can we ...

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### Is the Hilbert series of an ideal related to the Hilbert series of its homogenization?

Suppose we have a field $k$ of characteristic 0, let $I$ be an ideal of $R=k[x_1,...,x_n]$, and let $H$ be the homogenization of $I$ in $S=R[z]$. Is there any relationship between the Hilbert series ...

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### Gröbner basis via integer programming

I have studied some papers related to solving integer programs via Gröbner bases. I wonder if the other way is possible or not — i.e., given any ideal, can we find the Gröbner basis by translating ...

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### How to obtain a linear basis from a Groebner basis?

Let $A$ be a finite dimensional algebra generated by $x_1, \ldots, x_n$ subject to certain relations $I_1, \ldots, I_m$. Could we obtained a linear basis $B$ consisting of monomials in $x_1, \ldots, ...

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### Groebner Bases for submodule over polynomial ring with integer coefficients

It is well-known that there exist Groebner bases for ideals in polynomial ring $\mathbb Q[x]$ which can be found algorithmically Moreover, I don't think it is hard to show that there exist Groebner ...

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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 ...

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### Computing the codimension of the variety defined by a system of quadratic forms

Suppose I have an $m \times n$ matrix $L$, where $m \leq n$ and each entry is $L_{i,j}(x_1, ..., x_s)$ which is a linear form over $\mathbb{C}$. Let $\mathbf{y} = (y_1, \ldots, y_n)$. Let us consider ...

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### Can a minimal generating set for an ideal always be made into a Groebner basis?

Let $I\subseteq k[x_0,\ldots,x_n]$ be an ideal, generated by some polynomials $F_1,\ldots,F_r$, all homogeneous and of the same degree. Suppose $r$ is the smallest number of generators that will ...

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### Infinitely many initial ideals for non-Artinian monomial orders?

Consider the polynomial ring $R=\mathbb Z[x_1,\ldots,x_n]$ and an ideal $I\subset R$. Let $<$ be a monomial order, i.e. a total order on the set of monomials in $R$ such that for any monomials $a$, ...

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### System of polynomial equations with a known root

I have 5 polynomial equations for 5 variables and I know that the set of roots is finite. All coefficients are integers. Ultimately I'd like to find all roots but finding the Groebner basis is ...

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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< ...

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### 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 $\...

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### Solving system of multivariable algebraic equations over $\mathbb Q$ by reducing over $\mathbb F_p$

I try to solve the finite system of multivariable algebraic equations with coefficients from $\mathbb Q$. It would be sufficient for me to prove that there is only finite number of solutions over $\...

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### 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 ...

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### 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 $<_{rev}$ ...

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### Gröbner bases of resultants and their monomial ideals

$\newcommand{QQ}{\mathbb{Q}}$
Consider the ring $R = \QQ[x, a_1,\ldots,a_m]$ for a certain integer $m$ and the homogeneous polynomial
$$
f = x^{m+1} + \sum_{i=1}^m a_i^i x^{m+1 - i}
$$
Now let
$$
...

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### Memory usage of Gröbner basis computation

I've been calculating some Gröbner bases in preparation for finding non-commutative Hilbert series (and, once I recreate that, characters of group actions). Specifically, I've been using the ...

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### $\mathbb Z$-torsion for some quadratically presented Lie rings

$\newcommand{\Z}{\mathbb{Z}}$
I asked this question on MSE but no answer so far, so I'm also asking it here.
Let $L$ be a Lie ring (a Lie algebra over $\Z$) with generators $x_1,\dots,x_n$ and ...

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### Geometric significance of Anick's resolution

Given an augmented graded associative $K$ -algebra $A$, we can construct a free resolution of $K$ given by $K_A$ modules which gives nice combinatorial informations about the homology classes of the ...

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### Size of the Groebner basis and change of coordinates

Given an ideal $I$ of $k[x,y]$, the size of the Groebner basis depends on the monomial ordering. For example, if
$$
I = \langle x^3y^4 , x^2 + y^2 \rangle,
$$
then the Groebner basis with the ...

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### Automatic proof in Euclidean Geometry using Theory of Groebner Bases

I've done the same question in math.stackexchange here "https://math.stackexchange.com/questions/1938261/automatic-proof-in-euclidean-geometry-using-theory-of-groebner-bases?noredirect=1#...

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### Integer programming and Groebner basis

I enjoyed reading different papers about using Groebner basis to solve integer programming.
Is there any literature about the complexity and/or comparison with other (more classical) methods like ...

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### Does a polynomial system with precisely e solutions have a Groebner basis of degree bounded by e?

Let $k$ be a field and let $R=k[X_1,...,X_n]$ be a polynomial ring. Let $F \subset R$ be a finite subset generating a radical ideal $I$ with precisely $e$ solutions over an algebraic closure of $k$. ...

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### Weyl algebra acting on a polynomial ring

Let $\mathbb K$ be a characteristic-$0$ field, $R=\mathbb K[x_1,\ldots,
x_n]$ be a polynomial ring, and $W=\mathbb K[x_1,\ldots,x_n,\partial_1,\ldots \partial_n]$ be the Weyl
algebra. As usual $W$ ...

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### Computing intersection of Weyl algebra ideal with certain subring

Let $D=k [x_1,\ldots, x_n, \partial_1,\ldots, \partial_n] $ be the nth Weyl algebra over the characteristic zero field $k $. Set $\theta_i=x_i\partial_i $. Let $I $ be a left ideal in $D $. Is there a ...

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### Groebner bases for differential operators with field coefficients (reference request)

Let $K$ be a field, $\partial_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial_1\ldots \partial_n]$ (it is implicitly assumed that the derivations do not commute with elements ...