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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relationship between independence of multivariate polynomials, generating sets of polynomial ideals

I am studying something that touches on Groebner algorithms at the moment and It seems like i am missing something obvious about the relationship between three definitions that feel like they should ...
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Is reverse lexicographic order the same as graded reverse lexicographic order?

I want to make sure whether the two monomial orderings are actually the same thing. I am confused because the Cox book on Ideals, Varieties and Algorithms mentions only the graded reverse ...
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Is calculating the order ideal easier than calculating a Groebner basis?

Given an ideal $I = \langle f_1,\cdots,f_k \rangle \subset K[x_1,\cdots,x_n]$ and a monomial order I am interested in calculating the order ideal of $I$ with respect to that monomial order. This is ...
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Separate translation from plane rotation

Given $$\cos(\omega)a_x-\sin(\omega)a_y+b_x=c_x$$ and $$\sin(\omega)a_x+\cos(\omega)a_y+b_y=c_y$$ we have that $$a_x^2 + a_y^2 - b_x^2 + 2b_xc_x - b_y^2 + 2b_yc_y - c_x^2 - c_y^2=0$$ Notice that $\...
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Deriving Formulae for the roots of the quartic and cubic polynomials

I have seen derivations of the general solution for the roots of fourth and third degree polynomials of 1 variable in Dummit & Foote's Abstract Algebra; however, it was by no means simple to me. I ...
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Why is Grobner basis useful? [duplicate]

I don't get the motivation for calculating Grobner bases. What's good by computing a Grobner basis for an ideal of $k[X_1,...,X_n]$? Moreover, is there any theorem whose proof relies on the use of ...
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Gröbner basis is not a vector basis?

We use the same notation for Gröbner basis and vector basis. I recall that $\langle 1\rangle_{GR}$ is the largest Gröbner basis while $\langle 1\rangle_{vector}$ is the smallest vector basis. So for ...
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Compute a Gröbner basis for $I=\langle f_1,f_2,f_3\rangle$.

Using lexicographic order compute a Gröbner basis for $$I=\langle f_1=xy^2-xy+y,f_2=xy-z^2,f_3=x-yz^4\rangle\subset \Bbb R[x,y,z]$$ I was strictly using these notes to compute a Gröbner basis. ...
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Does $c(f) = \gcd(\{ f(n) | n \in \mathbb{Z} \})$?

Consider $\sum_{i = 0}^n a_i x^i \in \mathbb{Z}[x]$. Recall that the content of a polynomial is the gcd of its coefficients. I'm wondering whether the content is equal to $\gcd ( \{ \sum_{i = 0}^n a_i ...
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Buchberger algorithm and ideals

I'm working on Groebner bases using the book Ideals, Varieties and Algorithms. I'm interested in this problem : Let $\mathbb{Q}[x,y,z]$ with the graded lexicographic order with $x>y>z$. For ...
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Find $g\in I$ such that $LT(g)\notin \langle LT(g_1),LT(g_2),LT(g_3)\rangle$.

Let $I=\langle g_1,g_2,g_3\rangle\subset \Bbb R[x,y,z]$ where $$g_1=xy^2-xy+y,\qquad g_2=xy-z^2, \text{ and } g_3=x-yz^4$$ Using lexicographic order find $g\in I$ such that $LT(g)\notin \langle LT(...
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Characterization of Groebner Bases in terms of uniqueness of remainders

Let $I$ be an ideal of a polynomial ring $R=k[x_1,\ldots,x_n]$ 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 ...
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Testing if a submodule is free

This is hopefully a very simple question. In Gröbner 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 $K$, $S=K[x_1\ldots ...
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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 $1 \leq i \leq m$. Then ...
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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 k[x_{l+1}...
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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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128 views

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): $0=2abc+a^2+2ad+b+3d,...
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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 finite-...
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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 $\{f_1, \ldots, f_s\}$ form a ...
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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 (r+...