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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Characterization of Groebner Bases in terms of unicity of remainders

Let $I$ be an ideal of a polynomial ring $k[x_1,\ldots,x_n]$ over a field $k$. A Groebner basis of $I$ is a finite generating set $\{g_1,\ldots,g_m\}$ such that every leading monomial (according to a ...
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Testing if a submodule is free

This is hopefully a very simple question. In "Groebner Bases in commutative algebra" by Ene and Herzog, I find the Problem 4.11, which says ($S$ here is a polynomial ring over a field $K$, ...
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Groebner basis over rings

Let $I$ be an ideal in $A[x_1, \ldots, x_n]$, where $A$ is a Noetherian commutative ring, such that w.r.t some monomial order it has a Groebner basis $G = \{g_1, \ldots, g_t\}$ with all the leading ...
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Showing the polynomials form a Gröbner basis

Let $A$ be an $m \times n$ real matrix in row echelon form and $I \subset \mathbb{R}[x_1,\dots,x_n]$ is an ideal generated by polynomials $p_i = \sum_{j = 1}^na_{ij}x_j$ with $1 \leq i \leq m$. ...
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Computing “sparse” basis

If I have matrix A, and have performed RREF and been able to compute the regular basis for the row space of A. How do i compute “sparse” basis for the row space of A?
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Algorithmic computing kernel of a graded homomorpism

For computing kernel of a module homomorphism we can use module-Grobner basis such as described in notes talking about computing SyZyGies. How can we compute kernel of a homomorphism between a graded ...
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If $I,J$ are ideals in a polynomial ring over a field, how do I prove that $I = J$ if $\operatorname{in}_<(I)=\operatorname{in}_<(J)$?

If $I\subseteq J$ are ideals in a polynomial ring of $n$ variables, how do I prove that $I = J$ if $\operatorname{in}_{\lt}(I)=\operatorname{in}_{\lt}(J)$, where $\lt$ is any monomial ordering? ...
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radical membership and ideal membership [closed]

Consider the ideal $I=(x^3y-x^2y^2,x^3z+z^2yx,x^2-xz)\subset \Bbb Q[x,y,z].$ Is $x\in I?$ Is $x\in \sqrt I?$ I'm assuming a question like this is quite simple and that there is just a method, if ...
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Is a function in an ideal? Verification by hand and Macaulay 2

Suppose $$f_1=-4x^4y^2z^2+y^6+3z^5,$$ $$f_2=-4x^2y^2z^2+y^6+3z^5,$$ $$f_3=4x^4y^2z^2+y^6+3z^5,$$ $$f_4=4x^2y^2z^2+y^6+3z^5$$ and $$I=\langle xz-y^2,x^3-z^2\rangle\subset\mathbb C[x,y,z].$$ Is ...
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About Gröbner Bases

Recently I have come across a book of Gröbner Bases written by Adams & Loustaunau. The book is excellent and I have become interested in Gröbner bases after reading the book. I want to read more ...
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Use Gröbner bases to count the $3$-edge colorings of planar cubic graphs…

I found a nice introduction on how to Use Gröbner bases to construct the colorings of a finite graph. Now my graphs $G=(V,E)$ are the line graphs planar of cubic graphs, so they are $4$-regular. The ...
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Why is $|V(I)| \leq d_1\cdots d_n$?

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). Then if $G$ is a ...
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Groebner basis and projective closures

New to algebraic geometry and Groebner basis, so I just wanted to bounce my argument off of somebody. I have a zero set defined by one polynomial, $Y=Z(S) = \{p(x)\}$ in affine space, I am interested ...
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How to show Buchbergers algorithm terminates in noncommutative cases

I formulated buchberger's Algorithm over certain types of Ore-Algebras, meaning, i have the case $A:=R[y_1,...,y_n]$ with $R$ being a commutative Ring and $y_1,...,y_n$ being noncommutative variables ...
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The fastest Gröbner basis algorithm available?

for my undergraduate thesis I'm (pseudo) replicating algebraic attack on certain cryptosystem using gröbner basis approach. The heart of original attack was F5/2 algorithm (since the cryptosystem is ...
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Explanation of the Groebner Basis of a pair of polynomials?

I was working a problem in which I needed to reduce two polynomials in variables c,p,t into a single polynomial in c,t. Note that t appears only once, at the end of the $q_2$ definition. $$q_1\equiv ...
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A variation of Buchberger algorithm

Let $I$ be an ideal of a polynomial ring $R$. Fix a monomial order. Denote the $S$-polynomial of $f, g\in R$ by $S(f, g)$ and denote the gcd of their leading terms by $T(f, g)$. Consider the ...
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How to get the variety of a 3-dimensional ideal?

I have managed to calculate a Groebner Basis for the problem described here with respect to degree inverse lexicographic term order with help of SINGULAR. Please open the linked page in another tab! ...
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Is the following a valid argument? Finding the Groebner basis of an elimination ideal

I am asked to find the basis for $I \cap k[x]$ and $I \cap k[y]$ where $I=<x^2+2y^2-3,x^2+xy+y^2-3>$. I will omit the calculation here since it is very long and tedious, but I found one Groebner ...
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Can I use the elimination theorem in the following case? Elimination Ideals and Groebner basis

I understand the statement of the elimination theorem tells me that, $I \subset k[x_1,...,x_n]$ be an ideal and $G$ a Groebner basis of $I$ with respect to lex ordering. Then, $G_l=G \cap ...
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A short question on how the following statement is induced: Groebner Basis Lemma

I have a very short proof here for the following lemma, and there's one small bit I am not sure why it is true. Lemma: Let $G$ be a Groebner basis for the polynomial ideal $I$. Let $p\in G$ be a ...
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Polynomial constraint which ensures that a variable can never be zero?

In the answer by Jacques Carette to this question, Jacques Carette suggests a possibility of adding extra variables and introducing extra equations to an already given set of equations in order to ...
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I want a famous application of comprehensive grobner basis w.r.t. lex order and w.r.t. some conditions on parametric coefficients.

Let $I$ be an parametric polynomial ideal in $K[a_1,\cdots,a_m][x_1,\cdots,x_n]$ where $a_1,\cdots,a_m$ is a sequence of parameters and $x_1,\cdots,x_n$ is a sequence of variables. Is there any famous ...
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Groebner basis and syzygy

I am not familiar with polynomial ring. Suppose we have an ideal $\langle f_1, f_2, \ldots, f_n \rangle$ of the polynomial ring $k[x_1, x_2, \ldots, x_n]$. Suppose there is some syzygy relation ...
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Basis for row space of A

Assume that A is not reduced form, and R is the REF of A. I have understand that the set of nonzero rows in R is the basis for the row space of A. My question is why can't the corresponding rows of A ...
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Groebner Basis Question: Where are the other two zeros hiding?

Consider the system of polynomial equations: $$f_1=x^2+y^2-1=0$$ $$f_2=(x+y)(x-y)(2x-y)=0$$ Obviously there are 6 real solutions, which are the intersecting points of the lines $y=x, y=-x,$ and $y=2x$ ...
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Monomial ordering clarification

Reading this paper I stumbled upon a Groebner Basis calculation with a custom monomial order prescription. The algorithm 2) which you can find on page 11 of the above paper deals with polynomials in ...
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How to know when a polynomial belongs to a certain ideal in $\mathbb{C}[x_1,x_2,x_3]$?

I am trying to compute manually a Gröbner basis for $I=\langle f=x_3-x_1^5,g=x_2-x_1^3\rangle$ with the lexicographic order. After the third iteration I get, $$h_1=x_1^2x_2-x_3$$ $$h_2=x_1x_3-x_2^2$$ ...
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Buchberger's algorithm

I am trying to calculate a Gröbner basis for $I=\langle \mathcal{B}\rangle$, where $\mathcal{B}=\{f=x_3-x_1^5, g=x_2-x_1^3\}$, with respect both lexicographic and graded reverse lexicographic orders. ...
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Is every basis a Gröbner Basis with respect to some monomial order?

Given a polynomial ring $R=k[x_1,\ldots, x_n]$, and an ideal $I=\langle f_1,\ldots, f_m\rangle\subseteq R$, does there exist a monomial order $<$ on $\mathbb N^n$ such that $\{f_1,\ldots, f_m\}$ is ...
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How is Buchberger algorithm a generalization of the Euclid GCD algorithm?

It is said in many places (for example, on the Wikipedia article for Buchberger's algorithm) that Buchberger's algorithm to find Groebner basis is a generalization of Euclid's GCD algorithm. This is ...
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Tricks for a Specific System of Polynomial Equations

I'm looking for all the complex solutions to the following 3 equations (and for this consider $a$ to be some given constant, so that there are really just 3 unknowns in solving): ...
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Decide if a given set of monomials is a basis of a polynomial ring quotient

Let $R = \mathbf{k}[x_1,\ldots,x_n]$ be a polynomial ring over some field $\mathbf{k}$ (which can be $\mathbb{C}$ if that makes a difference) and $I$ some ideal of $R$ such that $R/I$ is ...
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Expressing polynomial as linear combinaion

I found these questions in Adams Introduction to Groebner bases Let $f=x^6-1$ and $g=x^4+2x^3+2x^2-2x-3$. Let $I=\langle f,g\rangle$. Calculate the polynomial that generates $I$ alone. After a ...
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Reduce multivariate polynomials by known roots?

Consider three multivariate polynomials $p_1(x,y,z)$, $p_2(x,y,z)$ and $p_3(x,y,z)$ with $x,y,z\in\mathbb{C}$. Imagine that the set of polynomials above is constructed such that they have exactly $6$ ...
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Krull dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$

What is the Krull dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$ where $A$ is a Noetherian, commutative ring and $\mathfrak{a} = \langle f_1, \ldots, f_s \rangle$, where $\{f_1, \ldots, f_s\}$ ...
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Translate a geometric theorem into polynomial equations — Theorem of the orthocenter of a triangle

This is Exercise 13 of Chapter 6 of Ideals, Varieties, and Algorithms by Cox et al. The problem asks to translate the following geometric theorem into polynomials and using Groebner basis to test ...
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What is a good lex order to compute the Groebner basis of this ideal?

This comes from chapter 6 of Ideals, Varieties and Algorithms by Cox et al. The equations are from a planar robot with three joints and one prismatic joint. See the following picture: Given a ...
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Grobner bases of a determinantal ideal

I've been studying algebraic geometry recently and there is a problem I'm struggling with: Suppose $A$ is a $m\times n$ complex matrix of rank $\leq r$, this is equivalent to all its $(r+1)\times ...
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What is the “projective limit” of a polynomial?

Bayer and Mumford, What can be computed in algebraic geometry, reads (in part): Let $S = k[x_0, \ldots, x_n]$ be the homogeneous coordinate ring of $\mathbb{P}^n$. [. . .] Choose a ...
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Show that the number of points of $V(I)$ is at most $m_1m_2…m_n$ if $x_i^{m_i}\in \left\langle \text{LT}(I) \right\rangle$.

This is an exercise from Ideals, Varieties and Algorithms by Cox et al. Let $I\subset \mathbb{C}[x_1,...,x_n]$ be an ideal such that for each $i$, some power $x_i^{m_i}\in \left\langle ...
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How to convert the parametric equation into implicit form?

This problem is generated from another Green's theorem related question of mine. The original equation of the plane curve is not in rational parametric form. In order to calculate the symbolic ...
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How can one show that an ideal with some property is zero-dimensional?

Let $\mathfrak{a}$ be an ideal in $\mathbb{k}[x_1, \ldots, x_n]$ and a Gröbner basis of the ideal be $\{g_1, \ldots, g_t\}$. For each $i = 1, \ldots,n$, there exists $j \in \{1, \ldots, t\}$ such that ...
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Help with computation and Gröbner basis

Hi guys I am learning a new software and a new topic (Gröbner basis) I have this problem $$ \begin{cases} 6-21(x_1x_2+x_1x_3+x_1x_4)=0 \\ 10-21(x_2x_1+x_2x_3+x_2x_4)=0 \\ ...
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Show that $G$ is a Groebner bases of $I$ if division of $f$ on $G$ is zero for all $f\in I$.

Let $I=\langle g_1,\dots, g_t\rangle$ be an ideal in $k[x_1,\dots,x_n]$ with $k$ a field. Let $G=\{g_1,\dots,g_t\}$. Show that if the remainder of $f$ on division by $G$ is $0$ for all $f\in I$, ...
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Prove Theorem with Groebner Basis

I'm trying to prove some theorems using Groebner Basis (as described in Cox, Little and O'Shea Link ) The mentioned book gives as an excercise to prove Pappus theorem using the given methodology, ...
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Help me understand Gröbner basis result please

I'm practicing a bit with Gröbner bases but I'm not understanding the following result I obtain from Mathematica: ...
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Gröbner Basis and Division Algorithm

I recently read a lemma on a course in Commutative Algebra that states, If $G$ is a Gröbner Basis for an Ideal $I$ in $k[x_{1},...,x_{n}]$, then a polynomial $f$ belongs to $I$ if and only if $f$ on ...
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How to check if a polynomial is inside an ideal using a Groebner basis

I'm given that an ideal $I=\langle F_1, F_2, F_3, F_4, F_5, F_6, F_7\rangle$ $F_1=a+b+c-d-e-f$ $F_2=a+b+c-g-h-i$ $F_3=a+b+c-g-e-c$ $F_4=a+b+c-a-e-i$ $F_5=a+d+g-a-e-i$ $F_6=a+d+g-c-f-i$ ...
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Is the mentioned basis a Gröbner basis?

It's mentioned into my notes that if the ideal given as $I=\langle x+y+z, 3x-2y\rangle$, then $\{x+y+z, 5y+3z\}$ is a Gröbner basis for the ideal. I can see how $I=\langle x+y+z, 3x-2y\rangle=\langle ...