Questions tagged [groebner-basis]

A Gröbner basis is a type of a generating set of an ideal in a polynomial ring over a field. It is a multivariate non linear generalization of Gaussian elimination and Euclid's algorithm.

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Two (different ?) definitions of a Gröbner-Basis

I have two slightly different definitions for Gröbner-Bases. 1.Definition from book Let $I$ be an ideal and $G=(g_1,\ldots,g_s)$ a basis for $I$. $G$ is called a Gröbner-Basis if $\langle LT(g_1),\...
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Finding Grobner basis using Buchberger's algorithm by Hand

I could use some help with this question. I have multiplied the first section by z and then subtracted the two polynomials from each other which has given me y=0, or x^2=z^2. Am I on the right track?...
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Solving a system of polynomial equation - can I trust numerical results?

To finish a proof, I need to solve a system of two polynomials with integer coefficients in two variables, $\{F_1(x,y)=0,\,F_2(x,y)=0\}$, and then show that no solutions satisfy $0<x<1$ and $y&...
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Computability theory in relation to other fields

In Geometric Group Theory (or, perhaps more specifically, in Combinatorial Group Theory) we phrase algebraic concepts as operations on words. This gives the subject a more combinatorial flavor (thus ...
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Bounds on the degree and number of polynomials in the reduced Gröbner basis of an ideal and its radical over a field of positive characteristic.

Disclaimer: Throughout, fix a field $K$ with $\text{char}(K) = p >0$, and we assume that all the computations related to Gröbner bases are done with a fixed elimination ordering. I am currently ...
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Are there instances of canonical Grobner bases (i.e. independent of term order)?

I'm trying to define the computation of Grobner bases in some form of logic. In particular, this means that I should come up with a basis which is independent of the chosen order on terms/monomials. ...
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Groebner basis reference request

Someone recommended Eisenbud's Section 15 to me, but I am finding it to be a little too terse. Does anyone know of a fairly self-contained introduction to the topic? I am just now finishing up a ...
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Multidegree of product of polynomials: $\partial(fg)=\partial(f)+\partial(g)$

I'm trying to show that $\partial(mn)=\partial(m)+\partial(n)$, where $\partial$ indicates the multidegree and $m,n$ are monomials in $\mathbb{F}[x_1,\dots,x_n]$, for a field $\mathbb{F}$ and a ...
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F5 algorithm for non-regular sequences

Faugère proves in his first Paper about the F5 algorithm the termination of the algorithm for regular sequences and mentions that some slight changes can be done to adapt this algorithm for non-...
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Decide if certain polynomial is in an Ideal

Let $I$ be the ideal $ I = (x^3y-x^2y^2,x^3z+z^2yx,x^2-xz) \subset \mathbb{Q}[x,y,z]$. I have to decide if $x$ is part of $I$ or $\sqrt I$. My first take was computing the Groebnerbasis $G$ of $I$ by ...
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Finding lcm and gcd of two ideals

I was studying Gröbner bases and I wanted to find $\operatorname{lcm}$ and $\gcd$ of two ideals $\langle x_1^2 + x_2x_3^2 - x_3^2\rangle $ and $\langle x_1x_2+x_2^2-x\rangle $. I know I should find a ...
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Macaulay Matrix Reduction in Macaulay2

I am trying to get the Groebner bases of an ideal in Macaulay 2 through the triangularization of the Macaulay matrix. Indeed, I would like to know how far I should go -how big the matrix should be- ...
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How to find polynomial multipliers that relate a specific element in the Groebner basis to the original generators?

If I have a particular set of generators for an ideal, (eg. $f_1$, $f_2$, $\ldots$, $f_n$), it is obviously straightforward to compute the Groebner basis for this ideal (say, $g_1$, $g_2$, $\ldots$, $...
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Expediting Groebner basis computation

I am trying to solve a bunch of homogenous polynomial equations in several variables, using Groebner basis method. I expect to get a unique rational solution. The problem is that the coefficients of ...
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Is a generalization of Bézier curves to 3 dimensions possible?

Curiosity of the application of groebner bases to problems involving finding the envelope of a family of curves drove me to verify the following: Given a quadratic Bézier curve with control points $(...
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Krull dimension with Grobner bases

Let $R=\mathbb{k}[x,y]$ be a polynomial ring in two variables $x,y$, and consider the ideal $I = \langle x^2+y^2\rangle\subset R$. $G\equiv \{x^2+y^2\}$ is is a Groebner basis for $I$ for any ...
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Why are the number of Groebner bases for a set of polynomials sometimes greater than the number of polynomials

Its easy to see why a set of $n$ polynomials $\{p_1\cdots p_n\}$ may have fewer than $n$ Groebner bases. For example, if $p_n=p_1*p_2+p_3$ then $\{p_1\cdots p_{n-1}\}$ and $\{p_1\cdots p_n\}$ generate ...
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Can I use groebner basis to solve polynomial equation of this kind?

I want to solve a polynomial equation of the following kind. $a (x+y^2-3z) + b (x^2 + 3y^3) = 0$ I want to compute integer coefficients a, b s.t. the above equation is satisfied for all x,y,z. In ...
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Buchberger's algorithm for finding groebner basis

Consider running Buchberger's algorithm on the polynomials $f=x_3-x_1^5,$ $g=x_2-x_1^3$ with lexicographic order and $x_1>x_2>x_3$. We first compute the $S$-polynomial. We have $LT(f)=-x_1^5,\ ...
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Gröbner Basis for a sum of ideals

Suppose that ideals $I$ and $J$ of $k[x_1,\dots,x_n]$ are given with $\{g_1,\dots,g_m\}$ and $\{f_1,\dots,f_n\}$ as their respective Grobner bases. Under what conditions is $\{g_1,\dots,g_m,f_1,\dots,...
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Normal form of the Hironaka decomposition

In "Algorithms in Invariant Theory" (B. Sturmfels) the author gives the following method for testing if a polinomial is part of a Cohen-Macaulay ring $R\in \mathbf{C}[x_1,\dots,x_n]$ that has ...
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Trying to understand Grobner basis

While studying Grobner basis, I realized that creating a basis from a given set of polynomials is not that hard, it is reduced to solving with Gauss Jordan a system of equations. What I don't ...
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Condition for Projective Variety

The definition of projective variety is equivalent to the locus of zero set of homogeneous polynomials that generate a prime ideal in the algebraically closed field they are a subset of (polynomial ...
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Remainder of multivariate division of polynomials

Consider a homogeneous polynomial of several variables $f(x_1,x_2,\ldots,x_n)$ with the leading term (with respect to lex ordering) having the maximum degree of any $x_i$s to be $k$. Take the ideal I ...
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Find a reduced Groebner basis

Problem: Let ideal $I = \langle f_1,f_2,f_3 \rangle \subset \mathbb{Q}[x,y,z]$ in which $f_1 = x-3y-4z, f_2 = -x+y+6z-2, f_3 = x-z+2$. Find a reduced Groebner basis of $I$ with lexicographic ordering....
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Some question in computing Groebner basis

Problem: Let $G = \{z^5 - z^2, -z^3+y^2,x-1-yz^4\}$ and $I$ be an ideal which is generated by $G$ in $\mathbb{C}[x,y,z]$. Show that $G$ is a Groebner basis of $I$ with lexicographic ordering $x>y&...
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Prove that every descending sequence of monomials terminates

Problem: Given a monomial ordering in polynomial ring of n variables. Prove that every descending sequence of monomials terminates. My attempt: Let $>$ be a monomial ordering on $\mathcal{M}$, the ...
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Calculate the dimension of $K$-vector space $V$

Problem: Let monomial ideal $I = \langle x^3,y^3 \rangle \subseteq K[x,y]$, consider quotient ring $V = K[x,y]/I$ as a $K$-vector space. Calculate the dimension of $K$-vector space $V$. Could you ...
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how to solve quadratic homogeneous equations

How to solve quadratic homogeneous equations(may not have quadratic terms) over a polynomial ring of multiple variables in a field
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GCD in multivariate polynomial ring and quotient ring

1. GCD in multivariate polynomial ring I would like to prove the following but couldn't figure out how to. Let $d$ and $h_1, h_2, \cdots, h_k$ be multivariate ...
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Find a finite Gröbner basis for ideal $I \subseteq \mathbb{R}[x, y, z]$

Find a finite Gröbner basis in lexicographic ordering $x \prec y \prec z$ for ideal $I \subseteq \mathbb{R}[x, y, z]$, where $$ I = \{ f \in \mathbb{R}[x, y, z] \space | \space f(a, -a, 2) = 0 \...
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Grobner basis sufficient condition

Assume $f \in K[x_1,\dots ,x_m],x_i\in F$ $$p \in K[x_1,\dots, x_m], p = \sum_{i}\phi_ix_1^{{a}_{i1}}x_2^{a_{i2}}\dots x_n^{a_{in}}, L(p) := \phi _lx^l : x^a \prec x^l \forall a\in A(p) \backslash\{l\}...
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Clarification about a proof regarding a sum of polynomials being expressed as a linear combination of S-polynomials

I'm reading this proof from Ideals, Varieties, and Algorithms by David A. Cox, Donal O'Shea, and John Little. You can find an online version here. This is Lemma 5 of Chapter 2, page 85. From my ...
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If $A^3=0$ show that $A^2=0$ (using Groebner basis)

Let $A$ be a $2\times 2$ matrix such that $A^3=0$. Using Groebner basis show that $A^2=0$. Any ideas on that?
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How to find polynomials satisfying a given ideal of relations

I have come across results/algorithms on how, given a set of polynomials in $k[x_1,\ldots,x_n]$ ($k$ a field) to find the ideal of relations. E.g. $k[a^2,ab,c^2] \cong k[x,y,z]/(xy-z^2)$ I am ...
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Does the vector space spanned by a Gröbner basis depend on the monomial order?

Let $k$ be a field, $n$ a natural number, and $I$ an ideal of $k[x_0,\dots,x_{n-1}]$. Given a monomial order $M$ on $k[x_0,\dots,x_{n-1}]$ let $G_M$ be the Gröbner basis of $I$ with respect to $M$ and ...
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Finite Algebras and Grobner Bases

Background Suppose that $A$ is a finite $\mathbb{R}$-algebra, that is, it is finite-dimensional as an vector space. By a consequence of the Hilbert-basisatz, since $\mathbb{R}$ is Noetherian, then so ...
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Finiteness of an algebraic variety $V(I) \subseteq \mathbb{C}^n$ where $I$ is a zero dimensional ideal of $\mathbb{C}[x_1,\dots,x_n]$

How to demonstrate this?: Let I be an ideal of $\mathbb{C}[x_1,\dots,x_n]$ such that $\frac{\mathbb{C}[x_1,\dots,x_n]}{I}$ has finite dimension ($I$ is a zero dimensional ideal). Then $V(I) \subseteq ...
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Vector space basis for a quotient ring

Given the polynomial ring $\mathbb{F}_2[x,y,z]$ with some ideal $I$, what is the method to find a basis (as a vector space over $\mathbb{F}_2$) for the quotient ring $R/I$? For example, take $$ S = \...
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Can I solve a system of high-degree polynomial equations with the Gröbner basis method?

I have two equations, two unknown (ISP, ch) and one parameter ($n_C$). My aim is to find the values of the unknown according to the parameter. The original rational equations are here : https://...
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Question about the exact definition of a Groebner basis

So the definition I learned is that a set $G = \{g_1,...g_t\}$ is a Groebner basis for an ideal I if $<Lt(g_1),...Lt(g_t)> = <LT(I)>$ However I'm not sure if it also requires the set G ...
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Gröbner bases for sum of ideals

Let $\{g_1,\dots,g_n\}$, $\{f_1,\dots,f_m\}$ be Gröbner bases of polynomial ideals $G,F\subset \mathbb{R}[x_1,\dots,x_k]$, respectively, under some monomial ordering. When is $\{g_1,\dots,g_n,f_1,\...
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How to solve simultaneous linear equations with only two possible values per variable?

I am trying to solve a system of simultaneous linear equations whose unknowns have only two possible values. How do I approach this, or what area of mathematics do I employ inorder to arrive at the ...
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Coefficients such that linear combination lies in an ideal

Let $R$ be a ring, $I$ an ideal, and $\langle g_1, \ldots, g_m \rangle$ a finitely generated ideal. Considering the intersection $I \cap \langle g_1, \ldots, g_m \rangle$, I became interested in the ...
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Skew-symmetric multi-derivations of $k[x_1,\ldots,x_n]/I$

Let $I = \langle f_1, \ldots f_r \rangle$ be an ideal in $R=k[x_1,\ldots,x_n]$ where $k$ is a field, and put $A = R/I$. (If $k$ is algebraically closed and $I$ is radical then $A$ is the coordinate ...
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Groebner base from tdeg order to plex order

I try to solve an equation by using Groebner bases. When I use Maple to find its Groebner basis with plex order, Maple take too long to calculate and the proceed does not terminate. Thus, I try to ...
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How to minimize the following polynomial over a four-dimensional sphere: $(c_0x_0^2+c_1x_1^2+c_2x_2^2+c_3x_3^2)^2+\sum_{nm}c_{nm} x_nx_m $

I need to find the global minimum of the following polynomial over a four-dimensional sphere. $$f(x_0, x_1, x_2, x_3) = \left( \sum\limits_{j=0}^3 c_j x_j^2 \right)^2 + \sum\limits_{n,m=0}^3 c_{nm} ...
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Why doesn't Buchberger's algorithm solve Hilbert's tenth problem?

I've been reading a bit about Buchberger's algorithm and Groebner bases. I'm not really trying to understand the math behind it at this point, just trying to get an idea of how the method could be ...
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1answer
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Choosing different term orders for Groebner basis calculations in GAP

I am using the QPA and GBNP packages in GAP to analyze path algebra quotients. I use the GBNP package for computing Groebner bases for the ideals in the path algebras, and the term orders for these ...
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Quotienting by the Leading Term Ideal

Somewhat related to this question, I'm looking for sources that give a proof of $$S / I \cong S/\mathrm{LT}(I),$$ where $I \lhd S = k[x_1, ..., x_n]$ for $k$ a field, and the isomorphism is an ...