Questions tagged [groebner-bases]

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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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127 views

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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39 views

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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60 views

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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41 views

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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1answer
124 views

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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90 views

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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1answer
109 views

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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1answer
380 views

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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134 views

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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107 views

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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145 views

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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1answer
131 views

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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2answers
805 views

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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1answer
88 views

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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1answer
148 views

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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78 views

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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30 views

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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76 views

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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1answer
173 views

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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68 views

$\mathbb Z$-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 ...
2
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1answer
164 views

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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90 views

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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2answers
539 views

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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1answer
344 views

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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69 views

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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247 views

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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57 views

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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1answer
210 views

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 ...
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132 views

Computations in Weyl algebra with rational function coefficients

I am looking for a software to perform calculations with modules over the algebra $R_n=\mathbb{C}(x_1\ldots x_n)\langle \partial_1\ldots\partial_n\rangle$ of differential operators with rational ...
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2answers
2k views

Solving a system of equations using Gröbner basis

In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the ...
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1answer
502 views

Solving Non-Linear Equations over a Finite Field of a Large Prime Order

I want to know is there is an efficient way to figure out whether or not a ( underdetermined) system of non-linear equations have a solution over a finite field of large prime order. The equations ...
4
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1answer
321 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 ...
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102 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, ...
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2answers
408 views

Free $k[x_1, \dots, x_n]^{S_n}$-module?

Let $\text{char}\,k = 0$ and $n \ge 2$. What is the easiest way to see that $k[x_1, \dots, x_n]$ is a free $k[x_1, \dots, x_n]^{S_n}$-module with basis$$x_2^{m_2}x_3^{m_3} \dots x_{n-1}^{m_{n-1}} x_n^{...
4
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1answer
195 views

Solving over-determined system of polynomials

I am trying to solve the following over determined system of polynomials \begin{align} & p_1(x_1,x_2,\ldots,x_n)=0, \\ & p_2(x_1,x_2,\ldots,x_n)=0, \\ & \vdots \\ & p_m(x_1,x_2, \...
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3answers
607 views

Polynomials vanishing modulo some integer $n$

It is well-known that a polynomial $q \in \mathbb Z[t]$ vanishes modulo $p$ only if it lies in the ideal $J_p$ generated by $p$ and $t^p-t$. This means that either the degree is large (at least $p$) ...
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2answers
586 views

From polynomial ideal over $\mathbb{Q}$ to polynomial ideal over $\mathbb{Z}$

Is there an algorithm to compute, given a polynomial ideal $I\subset \mathbb{Q}[x_1,\dotsc,x_n]$, the ideal $I\cap \mathbb{Z}[x_1,\dotsc,x_n]$ in $\mathbb{Z}[x_1,\dotsc,x_n]$? The input and output ...
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2answers
178 views

Pseudo-decision procedures for first order arithmetic

I was reading this paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.2911&rep=rep1&type=pdf in which the author describes an algorithm, based on Groebner basis, ...
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1answer
392 views

cup-length of the first Chern class of complex grassmannian

Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian. Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where $f_k=\sum_{i=0}^{[k/2]}(-1)^{k-i}{{...
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3answers
964 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
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0answers
103 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 $$\mathfrak{I}_d=\{ \text{ideals for which there is minimal generating system with $d$ elements} \}\setminus \{\text{ ideals generated by $d$ monomials}\}$$ ...
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1answer
396 views

Is there a Gröbner basis analogue that exists for vector spaces?

Suppose I have a coordinate system $t_1,\ldots t_N$ with a lexicographical ordering. Let LT denote choosing the lowest term of a polynomial with respect to this ordering. e.g. LT$(t_1 + t_2)=t_2$. ...
12
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1answer
1k views

Ways to show a system of polynomial equations has no solution

I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$: $$ \begin{cases} 2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\ X_sX_{m-s}+(-1)^...
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5answers
4k 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 ...
4
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0answers
157 views

Bound for the height of equations defining the singular locus of a variety

Fix positive integers $m, n, d$. In what follows, the height of an algebraic number will mean the absolute multiplicative height. Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...
2
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1answer
73 views

Degenerations and spanning monomials

Let $R = \mathbb{C}[x_1,…,x_n]$, let $J\subset R$ be a graded ideal, and consider the initial monomial ideal $\operatorname{in}(J)$ with respect to some term order. Suppose that we are given a linear ...
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2answers
313 views

Computing toric ideals via saturation and Groebner bases of toric ideals

About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer. Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full ...
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2answers
717 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}...
4
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1answer
137 views

Algorithm to detect if an element of a (commutative) ring is in a subring?

For rings finitely-generated over a field, the theory of Groebner bases gives us quite an efficient algorithm for determining whether an element of the ring is in a given ideal of the ring. Is there ...