MATH 439/739/819 Assignment 3. Spring 2007. Due February 19th at 9:30am. 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. 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. Which monomial ordering would be a good choice? 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 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 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 and G = {x-y^2,x*y-x,y^3-y^2,x^2-x} for . So by adding enough "redundant" polynomials in I to a given Groebner basis we can make it a Groebner basis for all monomial orderings.