Questions tagged [groebner-bases]
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97 questions
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Example polynomial system where Macaulay bound is tight
I have been solving systems of polynomial equations by forming the Macaulay matrix of different degrees and computing its null space. If the degree is large enough, namely at or above the degree of ...
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Are there algorithms for lowering the degree of the polynomials which generate an ideal by strategically adding new variables?
Suppose I have an ideal $I$ in a polynomial ring $F[x_1,...,x_n]$ generated by a few polynomials of maximum degree $k$.
I want to embed the ideal $I$ into a larger ideal $J$ in a larger ring with ...
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Algorithm to determine closedness of orbits?
Consider a reductive group $G$ acting on an affine variety $X$. It is known that for every $x\in X$, we have $G.x\subseteq\overline{G.x}$ is open dense. Then $\partial({G.x})\subseteq X$ is a closed ...
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Solving sparse bilinear systems with a relatively large number of variables
I'm trying to solve a bilinear system of equations over a finite field. (More specifically: I'm trying to find a single solution, if one exists.) The system consists of equations of the form
$$y^T A_i ...
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The theory of Groebner bases in Jordan case
There are many papers regarding spreading the theory of Groebner-Shirshov bases from Lie algebras to other nonassociative algebras. Also, it has been studied for associative algebras with operators ...
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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 ...
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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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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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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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