MS-E1621 - Algebraic Statistics D, 09.09.2020-04.12.2020
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Worksheet 1 in Macaulay 2
Let's see how Worksheet 1 could have been solved with Macaulay 2.
Exercise 1- Solve in Mathematica https://reference.wolfram.com/language/ref/Solve.html
- In simple cases, solve in WolframAlpha https://www.wolframalpha.com/examples/mathematics/algebra/equation-solving/
- Similar functions in Maple, etc
- In Macaulay2 you have the package NumericalAlgebraicGeometry https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.12/share/doc/Macaulay2/NumericalAlgebraicGeometry/html/index.html
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
restartR=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 9I = 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).