# 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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### Is a monomial basis a Groebner basis?

Given a monomial ideal $$I = \langle x^{\alpha} \mid \alpha \in A \subset \Bbb Z^n_{\geq 0}\rangle,$$ I want to know if the rest of the polynomial division of $f \in \mathbb{K}[x_1,...,x_n]$ by the ...
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### Gröbner Basis consisting of 3 polynomials with exactly 13 common solutions

I'm studying Gröbner bases and am using the following definitions and results: Let $I$ be an ideal in $R = K[x_1,...,x_n]$ for some field $K$ and some $n>0$ and fix a monomial ordering on $R$. A ...
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### Groebner basis for system of integral quadratic forms

I have a system of quadratic forms (homogeneous polynomials of degree $2$) with integer coefficients. Each quadratic form is the trace of a product of matrices. I'm solving for the matrix entries. ...
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### Are zero dimensional Ideals radical?

I have a question about Theorem 3.7.25. of Computational commutative algebra I by M. Kreuzer and L. Robbiano. Let $K$ be a perfect field, $I \subseteq K[x_1, \ldots, x_n]$, be a zero dimensional ...
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### Finding redundant equations in a underdetermined system of multivariate polynomial equations over $\Bbb R$

Starting from a geometric problem, I came up with a system of multivariate (many lines and points) polynomial equations where some equations are redundant (because they correspond to redundant ...
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### Non uniqueness of solution to system of polynomial equations using Gröbner bases

Consider the system of polynomial equations \begin{align} x + y + z &= a,\\ xy &= b,\\ z&=c, \end{align} where $x, y, z$ are the unknowns, and $a,b,c$ are known real numbers. It is clear ...
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### Property of remainders of Grobner basis

Let $G$ be a Gröbner basis for an ideal $I \subseteq \mathbb{F}[x_{1}, \ldots x_{n}]$. Let $f,g$ be in $\mathbb{F}[x_{1}, \ldots x_{n}]$ and recall that $\overline{f}^{G}$ is the remainder of $f$ ...
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### Write an ideal as an intersection of maximal ideals.

I'm trying to solve the following problem: Write $$I = (x^2+2y^2-2,x^2+xy+y^2-2) \subset \mathbb{R}[x,y]$$ as an intersection of maximal ideals. I have used the Gröbner basis and I found that the ...
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### Multiplicative basis gets inherited for an algebra modulo a compatible ideal

Let $A$ be an associative $k$-algebra with unit. Definition. A set $B \subseteq A$ is called a multiplicative basis of $A$ if $B$ is a $k$-basis of $A$ and if $B \cup \{0\}$ is closed under ...
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### Are minimal Groebner bases minimized bases?

With "minimal Groebner basis" I mean, fixed an ordering, a Groebner basis $G$ such that any proper subset of $G$ is no more a Groebner basis for the ideal $I(G)$ generated by $G$. With "...
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### Doubt in computation of groebner basis and proving a set is a groebner basis.

I have a set $G$ , and I am trying to prove that the set is a groebner basis . From chapter $2$ of Ideals,Varieties and Algorithm I have the following theorem, A basis $G = \{g_1,\cdots, g_n \}$ for ...
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### Buchberger algorithm with three variables

Consider $>_{degrevlex}$ with $x>y>z$ in $\mathbb{C}[x,y,z]$. Let $f_{1} = y^2 - xz$, $f_{2} = x^2y - z^2$, $f_{3} = x^3 - yz$. Show (manually) that $\lbrace f_{1}, f_{2}, f_{3}\rbrace$ is a ...
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### Insights into the property of a "generic" matrix.

This is in fact related to the previous question that I have asked but I decided that it would be better if I post this as a new question 1)What is a generic matrix(the definitions in net dont really ...
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### Confusion in grobner basis and ideals generated by matrices.

https://drive.google.com/file/d/1MzElOy1tgJGzZkCpzY8d_msKHuV-03kx/view?usp=sharing Here is a link to Strumfels 1990 paper(Grobner Basis and stanley decomposition of determinantal rings (pdf pg-137)). ...
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### How is the replacement by the remainder gives a reduced Grobner basis?

Here is the proof I am stuck in understanding: But how is the replacement by the remainder gives a reduced Grobner basis? 