MATH 439/739/819 Assignment 3.  Spring 2010.
Due Monday March 1st at 10:30am.
Late policy: -20% for up to 24 hours late.  Zero for more than 24 hours late.
Problems marked * are for MATH 819 students only.
Enjoy, Michael Monagan

Problems from the book.
2.7 #2,3,7
2.8 #1,5a,11 
2.9 #13a-b
3.1 #1,2,4,5*,6b*,7*
3.3 #6,8,14a

For exercises #2 and #3 in 2.7, assume x>y>z.
Do parts (a) and (b) by hand.
To reduce the number of steps, apply the "very useful lemma" and
proposition 4 of 2.9 where appropriate.
For part (c) use Maple as appropriate.

For the exercises in 2.8, 2.9, 3.1, and 3.3 use Maple to compute Groebner bases.
For exercise 2.8 #1, you are free to use any monomial ordering.  Pick a good one.
For exercise 13 of 2.9, don't hand in your answer.
For 3.1 #7c, check your answer using the lexdeg([t],[x,y,z]) ordering in Maple

Additional exercises.

1: Section 2.4 exercise 5 concerned the ideal <f1,f2> where f1=x^2*y-z, 
   and f2=x*y-1.  In parts b) and d) you found two polynomials r and g
   in I for which division by {f1,f2} resulted in a non-zero remainder.
   Apply Buchberger's algorithm to compute a reduced Grobner basis G
   for <f1,f2> by hand using lex order with x>y>z.
   To reduce the number of steps, apply the "very useful lemma."
   Now verify that r divided by G and g divided G have remainder 0.

2: Using Maple, determine which of the following ideals are the same.
   a) < y^3-z^2, x*z-y^2, x*y-z, x^2-y >
   b) < x*y-z^2, x*z-y^2, x*y-z, x^2-y >
   c) < x*z-y^2, x+y^2-z-1, x*y*z-1 >
   d) < y^2-x^2*y, z-x*y, y-x^2 >

3: By hand, compute a REDUCED Grobner basis for the following
   linear system using lexicograhical ordering with x > y > z.
      S = { x + y + z = 1, x - 2*y - z = 2, y + 2*z = 5 }
   Use proposition 4 from section 2.9 to skip S-polynomial calculations.
   What would happen if we use the grlex ordering with x > y > z?

Something for interest only.

   Let I be an ideal in k[x1,...,xn] which is non-trivial.
   Let T = {all possible monomial orderings on k[x1,...,xn]}.
   I gave in class an example of an infinite class of monomial orderings for
   k[x,y], the prime power orderings, so T is infinite when n > 1.
   Since we know that a Groebner basis for I depends on the monomial ordering
   this suggests that there might be an infinite number of different
   reduced Groebner bases for a given ideal.  Surprise.  This is not the case!
   Let R = {all reduced Groebner bases for I for each monomial ordering in T}
   Theorem: R is finite.
   It follows from the above that a finite set F in k[x1,...,xn] exists which
   is a Groebner basis for I for EVERY monomial ordering -- just take F to be
   the union of R.  Such a Groebner basis is called a universal Groebner basis.
   Some examples of universal Groebner bases.
   F = {x,y} for <x,y> and G = {x-y^2,x*y-x,y^3-y^2,x^2-x} for <x-y^2,x*y-x>.
   So by adding enough "redundant" polynomials in I to a given Groebner basis
   we can make it a Groebner basis for all monomial orderings.