MATH 495/800 Summer 2006.
Assignment 6.
Problems marked * are for MATH 800 students only.
You must use Maple when appropriate.
Due Wednesday August 9th at 10am in my office.
Late policy: -10% for each day late.
Michael Monagan

Exercises from the text

5.2 # 3, 6, 14*
5.3 # 1, 5, 8, 10(c)
6.4 # 6, 8, 10, 11

Notes on exercises

For problem #5 of section 5.3 use Maple to construct the multiplication table.
Compute also the inverse of 1+y in R[x,y]/I.
Is the quotient ring R[x,y]/I a field or not?  Justify your answer.

Additional exercises

1: Let I be an ideal in k[x1,...,xn] and R = k[x1,...,xn]/I.
   Prove that R is an integral domain if and only if I is prime (over k).
   Prove that R is a field if and only if I is maximal (over k).

2: Is Q[x,y]/<x^2-y^2+1> an integral domain?  Justify your answer.
   Is Q[x,y]/<x^2+1,y^2+1> an integral domain?  Justify your answer.

3: Consider the ideal I = <x^2+y+z-1, x+y^2+z-1, x+y+z^2-1> in C[x,y,z]
   There are five points in the variety V(I) -- see section 3.1.
   Verify that parts (i) and (ii) of Proposition 8 of section 5.3 hold for
   this example as follows.
   Let J be the radical of I.  Compute a basis for J -- see handout.
   Determine the monomial basis for the vector spaces C[x,y,z]/I and C[x,y,z]/J
   using both lex and grevlex (tdeg) orderings -- see Example 2 in section 5.3.
   Hence determine the dimension of C[x,y,z]/I and of C[x,y,z]/J.

4: The 10 Circles Problem.
   Find the value of m numerically for the distance between the n=10 points
   in the unit square in the four arrangements shown at the top of page 2
   in the article "The History of Packing Circles into Squares".
   Use Groebner bases and factorization to find the minimal polynomial f(m)
   for m then solve f(m)=0 numerically for m using the Maple command fsolve.
   The root we want will be the smallest positive real root.
   Using the relation r = m/(2*m+2), check if your result for m agrees with
   those given in the literature for r, namely,
   10(a) r=0.14777, 10(b) r=0.14792, 10(c) r=0.14818, and 10(d) r=0.148204 .

   Make use of symmetry in the four arrangements to specify additional
   (possibly redundant) equations to simplify the system to be solved.
   When you set up the systems of equations, if there are linear equations
   e.g.  x[1] = 1, y[1] = x[3], then eliminate the linear variables from the
   system first by making the assignments  x[1] := 1; y[1] := x[3];

   Note: you do not need the whole Grobner basis, only f(m), hence, an
   elimination ordering will easier to compute than a lex Groebner basis.

   Note: for one of the cases you may need to state explicitly that m<>0
   as otherwise the variety of the Groebner basis will contain an infinite
   component (corresponding to m=0) which makes working with the Groebner
   basis much more difficult.  Do you remember how to impose m<>0 ?