# I would like to know if there is a closed form expression from taking the reciprocal of a polynomial

I would like to know if there is a closed form expression from taking the reciprocal of a polynomial so that I can apply polynomial division to deconvolution using parallel fork-join multithreading.

My idea is to multiply the numerator polynomial by the reciprocal of the denominator polynomial using Furer's algorithm to multiply dense polyomials with integer coefficients in a parallel fashion targeting multi-core processor architectures.

The URL , https://math.stackexchange.com/questions/469415/general-formula-for-polynomial-division , states that the algebra for deriving a general Formula for Polynomial Division is too tedious and complicated when the author substitutes 1/x for x in the numerator and denominator polymonials and then multiply the numerator and denominator by 1/x^(m+v) and factor the denominator by x^v.

• I don't understand your question, or how it relates to computer science. How is, e.g., $1/(x^2+1)$ not already a closed form for the reciprocal of $x^2+1$? – David Richerby Mar 4 '17 at 12:15
• What do you mean by "closed form" in this case? If you want to be sure that the inverse of every polynomial is still whatever you mean by "closed", you have to take something with group structure, e.g. the Laurent series. – quicksort Mar 4 '17 at 13:00
• Thank you for answering your own question! However, we want the question to be clear when viewed on its own, even for self-answered questions. For instance, the question should specify what you mean by a "closed form" expression, and what criteria should be used to evaluate candidate answers. – D.W. Mar 4 '17 at 17:40
• D.W., Thank you for your comment. I thought a "closed form " expression included the real coefficients and integer non-negative exponents such as in the Taylor series expansion of exp(x) where exp is the exponential function. – Frank Mar 4 '17 at 21:13
• Is there a relationship between this question and arxiv.org/abs/1612.05778? If so, what is that relationship? – D.W. Mar 29 '17 at 6:42