1.1 A brief introduction to computational algebraic geometry  (Page 14/20)

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

Exercises

You may wish to use a computer to do most of the work in the following calculations. One advantage of Macaulay 2 is the getChangeMatrix command to express the generators of the Gröbner basis as linear combinations of the original generators, something I'm not sure how to do in the other programs.

1. 1. Determine whether of not $f=x{y}^3-z^2+y^5-z^3$ is in the ideal
      $$I=〈-,x^3,+,y,,,x^2,y,-,z〉.$$
   2. Determine whether or not $f=x^{3}z-2y^2$ is in the ideal
      $$I=〈x,z,-,y,,,x,y,+,2,z^2,,,y,-,z〉.$$
2. 1. Find the points on the variety
      $$V\left(x^2,+,y^2,+,z^2,-,1,,,x^2,+,y^2,+,z^2,-,2,x,,,2,x,-,3,y,-,z\right).$$
   2. Find the points on the variety
      $$V\left(x^2,y,-,z^3,,,2,x,y,-,4,z,-,1,,,z,-,y^2,,,x^3,-,4,z,y\right).$$
3. 1. Find an implicit equation for the surface parametrized by:
      $$\begin{array}{cc}\hfill x& =ut\hfill \\ \hfill y& =1-u\hfill \\ \hfill z& =u+t-ut\hfill \end{array}$$
   2. Find an implicit equation for the surface parametrized by:
      $$\begin{array}{cc}\hfill x& =t+u\hfill \\ \hfill y& =t^2+2tu\hfill \\ \hfill z& =t^3+3t^{2}u\hfill \end{array}$$

Reduced gröbner bases

Last time we found that

was a Gröbner basis for $I=⟨x^{2}y+y^{2}z,x{y}^2+z^2⟩\subset \mathbf{R}[x,y,z]$ in grlex order with $x>y>z$ .

Of course, we'd like to be able to say our Gröbner bases are unique. As a first step, we noticed last time that one element of the Gröbner basis was redundant: $x^3{z}^2+xy{z}^3=x(x^2{z}^2+y{z}^{)}$ , so we could remove it and still have a Gröbner basis

More generally, if $G$ is a Gröbner basis with $f,g\in G$ and $LT\left(f\right)$ is a multiple of $LT\left(g\right)$ , then $f$ will be redundant and can be removed, i.e. the set $G-\left\{f\right\}$ is still a Gröbner basis for the same ideal. (Why?)

Of course, multiplying any element of a Gröbner basis by a scalar will give a different Gröbner basis, so if we want uniqueness, we should require that each leading coefficient be 1. Since a monomial ideal certainly has a unique minimal monomial generating set, we might hope that forcing constant leading coefficients and removing redundant elements would be enough to get a Gröbner basis which is unique, but that is not quite the case. The problem is that we could still add a multiple of one generator to another. For example, for any $a,b\in \mathbf{R}$

is another Gröbner basis for the same ideal as above. To avoid non-uniqueness arising in this way, we say that a reduced Gröbner basis $G$ is a Gröbner basis where the leading coefficient of every $f\in G$ is 1 and no term of any $f\in G$ is divisible by the leading term of any $g\in G$ with $g\ne f$ .