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