I have the radical Ideal of $\mathbb{C}$ [x,y,z] generated by $\{x - 3 y - z + 9, z^2 -3z + 2, yz -2y - 3z + 6, y^2 - 5y + 6\}$ which form a Gröbner basis with TO deglex (x>y>z) and I want to compute the points in the corresponding variety but am unsure how to do so.
I started by computing the monomial Basis of $\mathbb{C} [x,y,z] / I$ which is $B = ${1,y,z} if I'm not mistaken. Then I considered the polynomial x which by the excercise has different values for all points in the Variety and wanted to compute the matrix $m_x$ relative to B. This is where I got lost because I don't understand how to compute $m_x$ if x is not an element of B. For example $m_y$ would be
\begin{pmatrix}0&-6&-6\\1&5&2\\0&0&3\end{pmatrix}
For context: I tried to understand this topic with "D. Cox, J. Little, D. O'shea, Using Algebraic Geometry", especially Proposition 4.7
- Thanks in advance
The first generator of the ideal gives a linear relation between $ x, y, z, 1$ in the quotient, so $x$ can be expressed in the basis $B$ (it is a basis, after all), and using this expression for $x$ you can express multiplication-by-$x$ as a matrix.
In general, to express multiplication-by-$f$ as a matrix, one first computes the reduction of $f$ with respect to the given Groebner basis $G$ (i.e. the remainder after division by that set of polynomials), which will be exactly a linear combination of the monomials in the normal basis $\operatorname{NB}(I) = \mathbb{M} \setminus \operatorname{LM}(G)$, and by construction $f$ is equal to this reduction in the quotient $\mathbb{C}[x,y,z]/I$.