I have a polynomial fraction which results in a polynomial $$\frac{f(x)}{g(x)}=q(x)$$ with $f$ $g$ and $q$ being polynomials. I have formulas for the coefficients of $f(x)$ and $g(x)$ dependent on the degree of $f$ and of $g$.

Now I searched for a way to express the coefficients of $q(x)$ by algebraic expressions of the coefficients of $f$ and $g$.

One way I think I found until now is the "subresultant PRS algorithm" which allows to calculate the coefficients of $q(x)$ by appropriate determinants of matrices with coefficients of $f$ and $g$.

But these determinants seem not to be calculable in a non-computeralgebra situation.

Are there other methods ( e.g. algebraic calculus complex analysis ) how to tackle such a general problem ?

  • $\begingroup$ This is probably very simplistic but how about the polynomial version of Euclidean division? $\endgroup$ – GeorgSaliba Aug 5 '15 at 18:59

This may not be what you want, but ...

You have $f(x) =g(x) q(x) $ where $f(x) = \sum_{i=0}^n A_ix^i$, $g(x) = \sum_{j=0}^m B_jx^j$, $q(x) = \sum_{k=0}^{n-m} C_kx^k$.

Then, by the standard polynomial multiplication, $A_i =\sum_{k=0}^i B_{i-k} C_k $, so $C_i =\frac1{B_0}\left(A_i - \sum_{k=0}^{i-1} B_{i-k} C_k \right) $ with $C_0 =\frac{A_0}{B_0} $.

This is just standard division that gives an iterative method of getting the $C_i$.

  • $\begingroup$ I think I have then to "solve" a system of linear equations and then I would have to use determinants ( which I hardly cant write down ) or perhaps use the Gauss elimination in such a general situation which might be viable if general formulas were available. Otherwise : "but...". $\endgroup$ – Wolfgang Tintemann Aug 5 '15 at 18:19
  • $\begingroup$ This is a method of computing the coefficients, not a method of getting a formula for the coefficients. That is why I said that this may not be what you want. However, it is a reasonably fast method for computing them. $\endgroup$ – marty cohen Aug 5 '15 at 20:51

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