I've a question on 2 problems in this book:
2.4. Let $S = K[x_1, . . . , x_6]$. Let $f = x_1x_5 − x_2x_4$, $g = x_1x_6 − x_3x_4$ and $h = x_2x_6 − x_3x_5$.
(a) Find a monomial order $<$ on $S$ such that {$f, g, h$} is a Grobner basis of $I = (f, g, h)$ with respect to $<$.2.8. Let $S = K[x_1, x_2, . . . , x_8]$ and $I$ the ideal of $S$ generated by $f_1 = x_2x_8 − x_4x_7, f_2 = x_1x_6 − x_3x_5, f_3 = x_1x_3 − x_2x_4$.
(a) Show that there exists no monomial order $<$ on $S$ such that ${f_1, f_2, f_3}$ is a Grobner basis of $I$ with respect to $<$.
I don't want a solution but I wish to know how these examples made. How can I reach an answer? Example 2.1.10 has a solution to a similar problem, which I can understand, but I don't understand how the author reaches the answer.
As others have said, the title should be modified to refer to the content.
In both questions, a monomial order will pick a monomial out of your polynomials. A priori there are $2^3$ possibilities, since you have three binomials.
The theorem to know is that when $I$ is fixed, we can restrict from arbitrary term orders to ones based on a weighting $\vec w$, where ${\bf x}^{\vec e} < {\bf x}^{\vec f} \iff \vec e \cdot \vec w < \vec f \cdot \vec w$. Now each statement like "$x_1x_5 < x_2 x_4$" becomes an inequality on $\vec w$. Finding a $\vec w$ satisfying a bunch of such conditions is now a linear programming problem.