# # For the following question in the textbook: # # -------------------- # Chapter 6, Exercise 4, page 274, GCL # -------------------- # # we need a generalization of a p-adic expansion to an I-adic expansion, # for an ideal such as I = . # Recall that the concept of a p-adic expansion of a polynomial u(x) in Z[x] # is that for a prime integer p, we represent u(x) in the form # # # u(x) = u0(x) + u1(x)*p + u2(x)*p^2 + ... + un(x)*p^n # # where uk(x) lies in Zp[x] . # # As a consequence, we have the following facts: # # u(x) mod p = u0(x) # # u(x) mod p^2 = u0(x) + u1(x)*p # # u(x) mod p^3 = u0(x) + u1(x)*p + u2(x)*p^2 # . . . # . . . # etc. # # # # I-adic expansion # ================ # (See section 6.2 of the textbook for this information.) # # Consider a polynomial u(x,y) in Z[x,y] and an ideal I = for some # integer a in Z. # For an ideal such as this with just one variable, the concept of the I-adic # expansion of u(x,y) is simply the Taylor series expansion of u(x,y) about # the point y=a. # For example, if I = then the I-adic expansion of the following # polynomial u is the Taylor series seen below: # ----------------------------------- -------------------------------------------------------------------------------- -------------------------------------------------------------------------------- # > u := 25*x^2*y^3 - 13*x^2*y^2 + x^2*y - 751*x*y + 31*x - 11*y + 77; 2 3 2 2 2 u := 25 x y - 13 x y + x y - 751 x y + 31 x - 11 y + 77 > taylor(u, y=2, 4); 2 2 2 2 (150 x - 1471 x + 55) + (- 751 x + 249 x - 11) (y - 2) + 137 x (y - 2) 2 3 + 25 x (y - 2) # ----------------------------------- # Then it follows that # u(x,y) mod I = the term of degree 0 in (y-2) # u(x,y) mod I^2 = terms through degree 1 in (y-2) # u(x,y) mod I^3 = terms through degree 2 in (y-2) # etc. # # # Similarly, for a polynomial a(x,y,z) as in Exercise 4, and for an # ideal such as I = which contains two variables, we need to use # the concept of a *multivariate* Taylor series about the point y=1,z=0. # Maple's "mtaylor" command is very useful for this purpose. # For example, # ----------------------------------- -------------------------------------------------------------------------------- -------------------------------------------------------------------------------- > v := expand(u*(z+3)^3); 2 2 3 2 3 2 v := - 351 x y - 20277 x y + 31 x z + 279 x z + 837 x z - 11 y z - 99 y z 2 2 2 3 - 297 y z + 2079 z + 693 z + 27 x y + 837 x - 297 y + 675 x y + 2079 2 3 3 2 3 2 2 2 3 3 2 + 25 x y z + 225 x y z - 13 x y z - 751 x y z - 6759 x y z 2 3 3 2 2 2 2 3 2 2 2 2 + 675 x y z + 77 z - 117 x y z + x y z + 9 x y z - 351 x y z 2 + 27 x y z - 20277 x y z > mtaylor(v, [y=1,z=0], 4); 2 2 1782 + 351 x - 19440 x + (1782 + 351 x - 19440 x) z 2 + (- 20277 x - 297 + 1350 x ) (y - 1) 2 2 2 + (- 20277 x - 297 + 1350 x ) (y - 1) z + 1674 x (y - 1) 2 2 2 2 + (117 x + 594 - 6480 x) z + (450 x - 99 - 6759 x) (y - 1) z 2 3 2 3 2 2 + (- 720 x + 13 x + 66) z + 675 x (y - 1) + 1674 x (y - 1) z -------------------------------------------------------------------------------- > -------------------------------------------------------------------------------- -------------------------------------------------------------------------------- # ----------------------------------- # This is the I-adic expansion with respect to the ideal I = . # It takes the general form: # v(x,y,z) = v00(x) # + v10(x)*(y-1) + v01(x)*z # + v20(x)*(y-1)^2 + v11(x)*(y-1)*z + v02(x)*z^2 # + v30(x)*(y-1)^3 + v21(x)*(y-1)^2*z + v12(x)*(y-1)*z^2 + v03(x)*z^3 # + . . . # Here, on the first line is the term of total degree 0 -- namely, it is of # degree (0,0) with repect to (y-1) and z; # on the second line are all terms of total degree 1; # on the third line are all terms of total degree 2; # etc. # As before, the reason why we will want this "I-adic expansion" of a # polynomial is because it follows that # v(x,y,z) mod I = the term of total degree 0 --> first line # v(x,y,z) mod I^2 = terms through total degree 1 --> first two lines # v(x,y,z) mod I^3 = terms through total degree 2 --> first three lines # etc. # These operations "mod I^k" work this way because of the definition of # taking the k-th power of an ideal: # I = < y-1, z > # I^2 = < (y-1)^2, (y-1)*z, z^2 > # I^3 = < (y-1)^3, (y-1)^2*z, (y-1)*z^2, (y-1)^3 > # etc. #