1.1 A brief introduction to computational algebraic geometry  (Page 13/20)

In our above example, it turns out though that we would have been better off first trying to cancel the leading terms of $f_2$ and $f_3$ : the polynomial

has a leading term which properly divides that of $f_4$ , and in fact we see that once we have $f_5\in I$ , the polynomial $f_4$ is redundant since $f_4=x{f}_5$ . We must then check the $S$ -polynomials of $f_5$ with $f_1$ , $f_2$ , and $f_3$ (this is enough since we've already looked at every pair from $f_1,f_2,f_3$ ):

and we see that the remainders are all zero, so $\{f_1,f_2,f_3,f_5\}$ is a Gröbner basis for $I$ . The set $\{f_1,f_2,f_3,f_4,f_5\}$ is also a Gröbner basis for this ideal, but $f_4$ is redundant, so we may as well leave it out.

Elimination of variables

We may want to try to find elements of an ideal $I\subseteq \mathbf{R}[x_1,...,x_n]$ which only involve some of the variables $x_i,...,x_n$ . For example, to find an implicit equation for the curve given parametrically by $\left(\frac{a\left(t\right)}{p\left(t\right)},,,\frac{b\left(t\right)}{q\left(t\right)}\right)$ , we would like to find an element of the ideal $⟨p\left(t\right)x-a\left(t\right),q\left(t\right)y-b\left(t\right)⟩$ that only involves $x$ and $y$ and not $t$ .

It turns out that to do this, all we need to do is find a Gröbner basis for the ideal in Lex order. This is because in Lex order, if the leading term of a polynomial only involves the variables $x_i,...,x_n$ , then in fact all of its terms involve only $x_i,...,x_n$ . Thus we have

and if $G$ is a Gröbner basis for $I$ in Lex order, then $G\cap \mathbf{R}[x_i,...,x_n]$ is a Gröbner basis (and hence a generating set!) for $I\cap \mathbf{R}[x_i,...,x_n]$ . See 3.1 in Cox, Little, O'Shea for more details.

Thus, to “eliminate” variables from our ideal (i.e. find the polynomials in the ideal which only involve the other variables), we just need to put the variables to be eliminated first in Lex order and find a Gröbner basis for the ideal. This is something very special about Lex order: none of the other orders we've looked at are “elimination orders” in this sense.

Sample Code

Here is a sample session in Macaulay 2 computing a Gröbner basis for $I=⟨x^{2}y+y^{2}z,x{y}^2+z^2⟩\subseteq \mathbf{R}[x,y,z]$ in graded lex order:

Here is some code computing the same Gröbner basis in Singular: