MATH 441/751/819 Assignment 3. Spring 2014. Due Tuesday February 25th at 2:30pm. Late policy: -20% for up to 48 hours late. Zero for more than 48 hours late. Problems marked * are for MATH 819 students only. Enjoy, Michael Monagan Problems from the book. 2.7 #1,2,3,7 2.8 #1,5a,8,11 2.9 #13a-b 3.1 #1,2,4,5*,6b*,7* 3.3 #8 Use Maple for 2.7 #1. For exercises #2 and #3 in 2.7, assume x>y>z. Do parts (a) and (b) by hand and check your results using Maple. For part (c) use Maple as appropriate. To reduce the number of steps, you may, if you wish, apply the "very useful lemma" and proposition 4 of 2.9. Show your working. For the exercises in 2.8, 2.9, 3.1, and 3.3 use Maple to compute Groebner bases. 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. For exercise 8 of 2.8 and 8 of 3.3, plot the surfaces in Maple. Plot the surfaces using BOTH the parametric representation using plot3d AND and the implicit representation using plots[implicitplot3d] to check that it really is the same surface. Additional exercises. 1: 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 > 2: By hand, compute a REDUCED Groebner basis for the following linear system using lexicographical 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? 3: PROGRAM THE DIVISION ALGORITHM in Maple and test your program on the same examples in exercises #1, 2 from section 2.3. Program the division algorithm as follows DIVALG := proc(f::polynom,B::list,LT::procedure) ... end; where f (the dividend) is a polynomial in Q[x1,x2,...,xn] and B (divisors) is a Maple list of polynomials in Q[x1,x2,...,xn] and LT is a Maple procedure which computes the leading term of a polynomial in Q[x1,x2,...,xn]. See my programming notes for how to program the LT command for the lex and grlex orderings. Your algorithm should return the remainder r and an array q of quotients as shown below. DIVALG := proc(f::polynom,B::list,LT::procedure) local n,a,r, ... ; n := numelems(B); q := Array(1..n); # array of quotients r := 0; # the remainder ... rest of the code ... return(r,q); end: