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- R.<x, y, a, b, K> = QQ[]
- a
- S1 = R.quotient(x - a)
- f = (x - a) + b^2
- assert f(x=a) == b^2
- W1 = b
- # This expression is 0 at x = 0
- # => (x - a)^1 is a factor
- assert (W1^2 - f)(x=a) == 0
- # Therefore 0 in the quotient ring
- assert S1(W1^2) == S1(f)
- W_prev = W1
- n = 1
- S2 = R.quotient((x - a)^(n + 1))
- W_next = W_prev + K*(x - a)^n
- # The term k^2*(x - a)^(2*n) disappears in the quotient ring
- assert S2(K^2*(x - a)^(2*n)) == 0
- assert S2(W_next^2 - f) == S2(W_prev^2 - f + 2*K*(x - a)^n*W_prev)
- # Remember from the last step that (x - a)^n is a factor of
- # W_prev^2 - f
- P = (W_prev^2 - f) / (x - a)^n
- k = -P/2
- W_next = W_prev + k*(x - a)^n
- assert S2(W_next^2) == S2(W_prev^2 - W_prev^3 + f*W_prev)
- #assert S(W_next^2 - f) == 0
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