MATH 441/819 Assignment 6. Spring 2014. Due Thursday April 10th by 2:30pm Note there are no classes on April 10th. Late policy: -20% for up to 24 hours late. Zero for more than 24 hours late. On this assignment, use Maple for computations where appropriate. Michael Monagan Exercises from the text 5.2 # 6, 14 5.3 # 1, 5, 8, 10(b+c) 6.4 # 6, 8 Notes on exercises For problem #5 of 5.3 use Maple to construct the multiplication table. Is the quotient ring R[x,y]/I a field or not? Justify your answer. Compute also the inverse of 1+y in R[x,y]/I by solving a linear system. For problems #6 and #8 of 6.4, also compute the appropriate Groebner basis in Maple to complete the proof of the theorems. Additional exercises Note, Q = field of rational numbers, R = real numbers, C = complex numbers. 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]/ an integral domain? A field? Is Q[x,y]/ an integral domain? A field? Is Q[x,y]/ an integral domain? A field? Justify your answers briefly. 3: Consider the ideal I = 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 using Maple. Determine the monomial basis for the vector spaces C[x,y,z]/I and C[x,y,z]/J using both plex and tdeg orderings -- see Example 2 in section 5.3. Hence determine the dimension of C[x,y,z]/I and C[x,y,z]/J. 4: Using Groebner bases, determine the minimal polynomial m(x) in Q[x] for (i) alpha = sqrt(2)+sqrt(3)+sqrt(5) and (ii) beta = 1+2^(1/2)+2^(1/4). 5: The 10 Circles Problem (worth 30% of mark) 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 . For the arrangement 10(d), the circle closest to the origin has x coordinate 0 and y coordinate > 0. Now solve for the y coordinate to verify that there really is a "gap" there. For each problem, please identify and show the minimal polynomial for m. 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]; In particular, for 10(d), I can determine FOUR equations by symmetry. 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 ?