# Tagged Questions

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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### 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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### 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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### 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 ...