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 ?
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 $C_i$.