Questions

$(1)$

Show that Equation (2) has a solution if and only if $g$ divides $c$ .

$(2)$

Assume that $g$ divides $c$ . Show that Equation (2) may admit several solutions $(u,v)$ .

$(3)$

Assume that $g$ divides $a$ and that ${\deg}(a) + {\deg}(b) - {\deg}(g) > {\deg}(c)$ . Show that Equation (2) admits a unique solution $(u,v)$ such that ${\deg}(u) < {\deg}(a) - {\deg}(g)$ and ${\deg}(v) < {\deg}(b) - {\deg}(g)$ both hold.