M953 Homework 3 (Click here for solutions)

Due Friday, February 15, 2002

Macaulay2 is a symbolic algebra program to do commutative algebra. To use Macaulay2, logon to a Linux machine. Using a terminal window, type in M2. At the beginning of each line you input the program puts i5: (or i#:, # being the number of the current input). Each output line starts with o# . Anything from "--" to the end of a line is a comment and will be ignored by Macaulay2. Documentation is on-line at http://www.math.uiuc.edu/Macaulay2/Manual.

To get started, you must define the polynomial ring you will be using. Here are some examples:
i1 : R = ZZ/101[x,y,z] -- polynomials in x, y, and z over the integers mod the prime 101
i1 : R = QQ[x,y] -- polynomials in x and y over the rationals

Assuming you've already defined your ring with variables x and y, here's how to define an ideal named I:
i2 : I = ideal(x,y,x*y-x^2, y^2-x^3)
Here's how to find the square of I:
i3 : I^2 -- or use (ideal(x,y,x*y-x^2, y^2-x^3))^2
Note that the output for I^2 is not a minimal set of generators. Here's how to get a minimal set:
i4 : mingens(I^2) -- the output is an array, with each generator an entry in the array
Given an ideal, here's how to find the ideals of the irreducible components of V(I), with output given as a list of ideals:
i5 : decompose(ideal(x*y-x^2, y^2-x^3)) -- or decompose(I) or decompose I if I is the ideal
Here's how to find the radical of I:
i6 : radical I -- or mingens radical I if you want simpler output
Here's a useful operation: given ideals I and J, I:J is the ideal of all f in your ring R such that fg is in I for all g in J. So, to find the least power of x-y that is in ideal(x*y-x^2, y^2-x^3), keep trying ideal(x*y-x^2, y^2-x^3) : ideal((x-y)^n) until you find an n such that the output is the ideal (1) (you may want to use mingens to simplify the output):
i7 : mingens (ideal(y-x^2, y^2-x^3) : ideal((x-y)^1))
Here's how to intersect a list of ideals:
i8 : intersect(I,J)
When you're done, just quit:
i7 : quit

So here's the homework:

[1] In Problem [1](b)(i) from Homework 2, use Macaulay2 to find the minimum n and m that work. (Pick a polynomial ring over a field of your choice.)

[2] Given an algebraic set X, a theorem in commutative algebra says that Radical(I(X)) is the intersection of the ideals of the irreducible components of X. Demonstrate this in Macaulay2 with X = V(x*y-x^2, y^2-x^3) in A2 by comparing Radical(I(X)) with the intersection of the ideals of the irreducible components of X.

[3] Do problems #25(b) on p. 17 and #31 on p. 20. Use Macaulay2 (over a field of your choice) and compare its output with your answer.