Coupled systems of equations with entire and polynomial functions

We consider the coupled system F(x,y)=G(x,y)=۰, whereF(x, y)=\sum_{k=۰}^{m_۱}   A_k(y)x^{m_۱-k} \quad \text{ and }\quad G(x, y)=\sum_{k=۰}^{m_۲}  B_k(y)x^{m_۲-k}with entire functions A_k(y), B_k(y). We derive a priory estimate for the sums of the roots of the considered system and for the counting function of roots. 

Keywords:

coupled systems , entire and polynomial functions , a priory estimates , resultant