Articles of groebner basis

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 the generators form a Gröbner basis for $I$ w.r.t. some monomial order. So I guess one should try the standard […]

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 $f_i\in I?$ The answer is Yes in some cases. The question can be checked with Macaulay 2: when the remainder is zero with respect to the Gröbner basis like (R=QQ[x,y,z]; fi=…; I=ideal(x*z-y^2,x^3-z^2); G=gb(I);f%G returning zero, $f_i\not\in I$. Division with respect to the elements in ideal […]

Prove that determinant of a matrix (with polynomial entries) is non-zero

For $\mathbf x\in\mathbb (0,1)^n$ with $n>2$ and a positive integer $1\le k<n$, define the mapping $f_{i,j}\colon (0,1)^n\rightarrow (0,1)$ by, $$f_{i,j}(\mathbf x)= \sum\limits_{\substack{\mathcal A \subset \{ 1,\ldots,n\}\backslash \{i,j\}\\ |\mathcal A|=k-1}}\quad\left( \prod\limits_{l\not\in \mathcal A\cup \{i,j\} }x_l\,\,\cdot\!\!\!\! \prod\limits_{ l\in \mathcal A}(1-x_l)\right).$$ Convention: if the set of indexes is empty, the product is one. I’m trying to prove that […]

Basis of matrices with a variable

So I have these bunch of matrices I want to find the value of a to find the basis $$ \begin{pmatrix} 2 & 2 \\ 1 & -2 \\ \end{pmatrix} $$ $$ \begin{pmatrix} 0 & 0 \\ 1 & 1 \\ \end{pmatrix} $$ $$ \begin{pmatrix} 1 & a \\ 2 & -2 \\ \end{pmatrix} $$ […]

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 corresponding edge-adjacency matrices can be constructed, as shown here (in a crude way, I admit…). Planarity assures the existence […]

Applications of Gröbner bases

I would like to present an application of Gröbner bases. The audience is a class of first year graduate students who are taking first year algebra. Does anyone have suggestions on a specific application that the audience would appreciate?