# 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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### Method for solving polynomial system without multilinear form?

I am an engineer who is currently working with some network optimization problem during my post graduate study. During my study time, I see that sometimes I need to look for solution of polynomial ...
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### Proving that an ideal is primary

Problem 1: Show that the ideal $$I:=(af+d^2,bf+de,cf+e^2)+(a,b,c)^2+(a,b,c)(d,e)$$ is primary in the polynomial ring $k[a,b,c,d,e,f]$ where $k$ is a field. Macaulay2 confirmed the answer. But I ...
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1 vote
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### $n$ quadratic forms in $n+1$ variables

This answer seems to imply that there is something special about a system of $n$ quadratic forms $q_1, \dots, q_n$ in $n+1$ unknowns $x_1, \dots, x_{n+1}$. I want to understand better why this ...
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### If $G$ is a Gröbner basis of $I$ with respect to $<_w$, then $\{in_w(g)\ |\ g\in G\}$ is a Gröbner basis for $in_w(I)$ with respect to $<$

Let $I \subset R = K[x_1,...,x_n]$ be an ideal and $<$ a monomial term ordering on $R$. Let $w\in \mathbb{R}_{\geq 0}^n$ be a weight vector. Then for each $f = \sum c_\alpha x^\alpha \in I$ we can ...
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### Declaring constants in Sagemath

I want to find the Groebner base of a ideal,the ideal is generated by some polynomials with constant coefficients, but they do not have numerical values. ...
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### Strange Sage behavior grobner bases

I have the following code in Sage ...
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### Applications of Gröbner bases for beginners

What are some applications of Gröbner bases that could be interesting to a group of students that more or less only studied Chapter 1 and Chapter 2 of Ideals, Varieties, and Algorithms by David A. Cox ...
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### How to compute the intersection of ideals $(w,y,xz+x+z), (w,x,yz+y+z), (z,y,xw+x+w), (z,x,yw+y+w), (x+y,zw+z+w,y^2)$ over $\mathbb{F}_p$?

Let $p$ be any prime number in $\mathbb{Z}$, and consider the ideals $(w,y,xz+x+z)$, $(w,x,yz+y+z)$, $(z,y,xw+x+w)$, $(z,x,yw+y+w)$, $(x+y,zw+z+w,y^2)$ in $\mathbb{F}_p[x,y,z,w]$. My question is, how ...
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### Algorithm for writing an image of a polynomial in a quotient ring in terms of a given basis of the quotient ring.

I need to calculate the equivariant Chern classes of certain vector bundles on the classifying spaces of complex algebraic groups. In order to do this I am looking for a way to do the following ...
1 vote
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### Solve system of polynomial equations based on distance?

I am trying to find the solution for this polynomial system. For some context it represents a game where every player has to choose a location in the x-line, given that it is best to be closer to 0, ...
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### Example of initial ideal

I have a problem with understand some example, which I present below. Let ideal $I = (x_1^2 + 3x_1x_2, 2x_1^2 + x_2^2)$. The initial monomial of both generators is $x_1^2$. However, twice the first ...
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### Find (a generating set for) $\mathbb{Q}[x]\cap I$ where $I=\langle x^2-y,y^2-x,x^5-x^2\rangle$ (generate gröbner basis).

Consider the polynomial ring $\mathbb{Q}[x,y]$ and the ideal $I=\langle x^2-y,y^2-x,x^5-x^2\rangle$ in $\mathbb{Q}[x,y]$. $G=(x^2-y,y^2-x)$ is a (reduced) gröbner basis for $I$ wrt. graded ...
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### When does the system $x^n-1=y^n-1=x(y-1)^2+y(x-1)^2=0$ have only the trivial solution $(1,1)$ over the prime field of order $p$?
Let $F_p$ be the prime field of order $p$, $\overline{F_p}$ be its algebraic closure, and $n$ be an integer such that $\gcd(n,p)=1$. Consider the following three polynomials in $F_p[x,y]$:  p_1(x,y)=...