Let $$ f(x,y)=p_1(x,y)-cq_1(x,y),\\ g(x,y)=p_2(x,y)-dq_2(x,y), $$ be polynomials, where $c,d\in\left(0,1\right)$. Coefficient $x,y$ of polynomials represent probabilities, so $x,y\in\left<0,1\right>$. Polynomials $p_1(x,y)$ and $q_1(x,y)$ are coprime. Also polynomials $p_2(x,y)$ and $q_2(x,y)$ are coprime. How to prove that for fixed $c\in\left(0,1\right)$ there are finite number of $d\in\left(0,1\right)$ such that polynomials $f(x,y)$ and $g(x,y)$ are not coprime?
Any help will be appreciated. Thank you very much.