Need These Questions Urgent Thanks in Advance

Hint

Just need answer of Q#5

7 Problem 5.a) Suppose your friend ran the Division Algorithm in two variables to divide f = xy + ry2 + y2 by J1 = xy-1 and 12 = y2 - 1, using the lexicographic order. Your friend reports that the result is the following: 1 = (x+y) f1 +0/2 + (x + y2 + y), or, in other words, ry + xy + y)= (x + y)(xy - 1) +0(72 - 1) + (x + y2 + y). Assuming that your friend's calculations are correct thus far, is she done with division? If yes, what is the remainder? If not, why not? Finish the run of the division algorithm for her. 8 Problem 5.b) Let f = yx2 + 4y2x + yaz, /2 = 3yx3+yr?, 13 = 4y2x + yr? CR[x, y, z). I am proposing an ordering on monomials that chooses LT() = 4yr and LT (12) = 3yx. Does there exist such a monomial oder? If yes, state its name; if not, state why. (Hint below.) 9 Extra credit problem. Consider the ideal 1 = (x4y + 132%y, 29 y2 y, y) C R(x, y). (a) Is I a monomial ideal? (b) In the (n, m)-plane, plot the set of exponent vectors (nm) of monomials "y" appearing in elements of I (c) Suppose you apply the Division Algorithm to some / R[x,y) using {r^y, x*y, y} as divisors. What terms can appear in the remainder? 2 LT (f2) = y2x2 7 Problem 5.a) Suppose your friend ran the Division Algorithm in two variables to divide f = xy + ry2 + y2 by J1 = xy-1 and 12 = y2 - 1, using the lexicographic order. Your friend reports that the result is the following: 1 = (x+y) f1 +0/2 + (x + y2 + y), or, in other words, ry + xy + y)= (x + y)(xy - 1) +0(72 - 1) + (x + y2 + y). Assuming that your friend's calculations are correct thus far, is she done with division? If yes, what is the remainder? If not, why not? Finish the run of the division algorithm for her. 8 Problem 5.b) Let f = yx2 + 4y2x + yaz, /2 = 3yx3+yr?, 13 = 4y2x + yr? CR[x, y, z). I am proposing an ordering on monomials that chooses LT() = 4yr and LT (12) = 3yx. Does there exist such a monomial oder? If yes, state its name; if not, state why. (Hint below.) 9 Extra credit problem. Consider the ideal 1 = (x4y + 132%y, 29 y2 y, y) C R(x, y). (a) Is I a monomial ideal? (b) In the (n, m)-plane, plot the set of exponent vectors (nm) of monomials "y" appearing in elements of I (c) Suppose you apply the Division Algorithm to some / R[x,y) using {r^y, x*y, y} as divisors. What terms can appear in the remainder? 2 LT (f2) = y2x2