One part was overlooked that a+b - ab = 0 or (a-1)(b-1) =1 this is pssible for positive a, b ( for example a = 1.5, b= 3) but not for given condition...
The coefficients of $f$ are elements of $F(S)$. Thus each coefficient of $f$ looks like a ratio of two polynomials over $F$ evaluated on some members...
I have have shown that $k$ is a common multiple of the $\exp(H_i)$. Since $m$ is the least common multiple of the $\exp(H_i)$, then $k$ is a multiple...