13 14 23 1. (10 points) Let S = k[11, 12, 13, 14) and define the monomial ideal I = (x112.13.12.1314,1114). Find the Hilbert function and Hilbert polynomial for S/I. 2. (15 points) Find the syzygies of each of the following monomial ideals. (a) I1 = (x34x3, 21:1) in S = k[71, 12] (b) 12 = (21, 262, 21x2, 2133, 14) in S = k[7], [2] (c) 13 = (2182, 1113, 12:13) in S = k[r1, 12, 13] Hint: Use the Lemma proved in class. 3. (10 points) Let S = k[31, 12, 13, 14) and let 1. J CS be the ideals generated by the 2 x 2 minors of the matrices 11 12 13 and 12 21 22 23 24 21 respectively. Find Grbner bases for I and I with respect to reverse lexicographic order on monomials. 4. (15 points) Let F be a free module (of finite rank) over S = k[21...., 2] with monomial order >. Let MCF be a submodule and let B = {91, ..., ge} be a Grbner basis for M. We make the following definitions. 1. B is a minimal Grbner basis if in(91),..., in(9t) is a minimal set of generators for in(M). 2. B is a reduced Grbner basis if, for each i, 1. Let MCF be a submodule and let B = {91, ..., ge} be a Grbner basis for M. We make the following definitions. 1. B is a minimal Grbner basis if in(91),..., in(9t) is a minimal set of generators for in(M). 2. B is a reduced Grbner basis if, for each i, 1