Let's see how Worksheet 1 could have been solved with Macaulay 2.

Exercise 1 
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Exercise 10

Proposition 3.4.5 [elimination ideal]

Let \( \pi:\mathbb{K}^{r_1+r_2}\rightarrow \mathbb{K}^{r_1} \) be the coordinate projection

\((a_1,\dots,a_{r_1},b_1,\dots,b_{r_2})\mapsto (a_1,\dots,a_{r_1})\).

Let  \( V \subseteq \mathbb{K}^{r_1+r_2}\) be a variety and

\( I:=I(V)\subseteq \mathbb{K}[p_1,\dots,p_{r_1},q_{1},\dots,q_{r_2}] \) be its vanishing ideal.

Then

\( I(\pi(V))=I\cap \mathbb{K}[p] \) is its elimination ideal.

   (Behind the scenes elimination is done using Gröbner bases, see Theorem 3.4.6).

   R=QQ[p,q]
   I = ideal (p-q,p^2-q)
   J = eliminate(I,q)

  Exercise 4 

   restart
   R=QQ[p1,p2,p3,t]
   I = ideal (p1-t,p2-t^2,p3-t^3)
   J = eliminate(I,t)

(this follows from Proposition 3.4.7 in Sullivant. Alternatively, check Theorem 1 in Chapter 3 of Cox, Little, O'Shea)

 Exercise 6 & 7

   restart

   R=QQ[p1,p2]

   I = ideal(p1^2,p2)

   J = radical I

   I==J

Exercise 9
   R=QQ[p]
   I = ideal (p^6-p, p^4-p)
   f = p^3-p
  
   To check if f is in the ideal
   f%I 

In general, Groebner basis depends on a term order, so cannot check easily computationally if it is a Groebner basis. However, for a fixed term order, there is Buchberger's algorithm (not in the scope of this course, but covered in Computational Algebraic Geometry).

Last modified: Friday, 11 September 2020, 1:38 PM