Let $K$ be a field of characteristic zero. Let $f,g,h \in K[t]$ be three separable polynomials, with no common zeros. Denote $\deg(f)=a,\deg(g)=b,\deg(h)=c$, $a \geq 2, b \geq 1, c \geq 1$, and denote the roots of $f,g,h$ ,respectively, by $a_i,b_j,c_k$, $1 \leq i \leq a$, $1 \leq j \leq b$, $1 \leq k \leq c$.
Let $u,v,w \in K[t]$, with $\deg(u) < a, \deg(v) < a$.
If $ug+vh=wf$, then is there something interesting to say about $f,g,h,u,v,w$ and their degrees?
The only thing I have managed to obtain thus far is that, in the special case where $w=0$, we have: $ug+vh=0$, so $ug=-vh$, and since $g$ and $h$ are relatively prime, it follows that $g$ divides $v$ and $h$ divides $u$. Hence, $v=g\hat{v}$ and $u=h\hat{u}$, and then $h\hat{u}g+g\hat{v}h=0$, so $\hat{u}=-\hat{v}$, so $v=g\hat{v}$ and $u=-h\hat{v}$, which shows that $\gcd(u,v)=\hat{v}$.
I do not know what is the answer I am looking for, but I hope that it would be 'nice'.
I prefer not to assume that $w=0$.
Any hints and comments are welcome!