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I just read this excellent Mathematics Stack Exchange article: [Integrating Reciprocals of Polynomials

However, my goal in using is to implement a parallizable polynomial division algorithm. So far, I have read that polynomial division can be done in nearly linear time using Hensel lifting to compute the inverse of the denominator polynomial and multiplying it by the numerator polynomial function.

I remember that inverse of a polynomial function if it exists is quite different from the reciprocal of a polynomial function.

How do I find the reciprocal of any polynomial without use Heavyside partial fraction decompostion or Egyptian fractions? If so, what is the nearly linear running time recurrence relation for this step?

One way to this by completing the square is shown here: How do I integrate (1/polynomial) without using partial fractions?.

But it seems to be a roundabout way to divide polynomials by completing the square of the divisor polynomial. Is there a more computationally efficient way to do this?

   Thank you.
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Frank
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    I still can't parse how in the world would the Heaviside function and/or egyptian fractions help to get "the reciprocal of a polynomial" (??) In fact, what does " finding the reciprocal of a polynomial" mean? If the intention is to ask how to factor it in order to have a chan ce to integrate then that's way another thing, imo. – DonAntonio Feb 18 '17 at 09:52
  • @DonAntonio, You are completely correct.. The answer to my question is found in this uRL, http://mathoverflow.net/questions/53384/power-series-of-the-reciprocal-does-a-recursive-formula-exist-for-the-coeffic#. Please let me about any comments or questions regarding this. Thank you. – Frank Feb 18 '17 at 11:27
  • @DonAntonio, How do I parallelize your answer using C/C++ programming? Thank you. – Frank Feb 18 '17 at 17:40
  • I've no idea whatsoever what "to parallelize an answer" could possibly be, either using C, CC+ or a shovel. Honest. – DonAntonio Feb 18 '17 at 18:05

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