-Adic Polynomials and Partial Fraction Decomposition of Proper Rational Functions over or

Algorithm 1

Main algorithm.

  Output: for and for .

for to do

  Let and find using the Taylor shift (Proposition 9). Compute :

    using a binary splitting scheme (Lemma  4.1 on [2]), Proposition 6 and the -adic Cauchy product formula (Proposition 12).

end

for to do

  Let and . Using the Taylor shift (Proposition 9), find . Using the -adic shift

    (Proposition 11(a)), find

    where and in Proposition 11(b). Compute :

    using a binary splitting scheme (Lemma  4.1 on [2]), Proposition 6 and the -adic Cauchy product formula (Proposition 12).

(6) end