A question about co-prime polynomials in $\Bbb{C}[x,y]$

Question

Say $f$ and $g$ are two co-prime polynomials in $\Bbb{C}[x,y]$. Can the following expression always be written $$af+bg=1$$ where $a,b,f,g\in\Bbb{C}[x,y]$? I realise that the Euclidean algorithm is not valid here. However, I thought of an algorithm to make this possible (although I'm not sure whether it is indeed correct).

Answer 1

So this isn't a constructive argument but if $f$ and $g$ define curves that have no common solutions then Hilbert's Nullstellensatz tells us that $(f,g) = 1$ and also that this is the only time that this is the case. So to answer your question you can write $af + bg = 1$ only when $f$ and $g$ share no common solutions.