## 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
A^{2} 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.