What is the elementary proof of Weil's polynomial theorem of decomposition?

Question

André Weil sometimes glosses his Theorem of Decomposition in a simplified polynomial form:

If $P(x,y)$ and $Q(x,y)$ are homogeneous polynomials algebraically prime to each other, with integer coefficients, and $x,y$ are integers prime to each other, then $P(x,y)$ and $Q(x,y)$ are ``almost" prime to each other, that is to say, their GCD admits a finite number of possible values.  (The Apprenticeship of a Mathematician (1992), p. 46)

I think this must have an elementary proof which i am just not able to see.  Can someone help me?

Answer 1

If $P(x,y),Q(x,y)$ are relatively prime, then so are the one-variable polynomials $p(x)=P(x,1),q(x)=Q(x,1)$ (since we can homogenize any common factor of $p,q$ to a common factor of $P,Q$). It follows that in $\mathbb Q[x]$ there are two polynomials $a(x),b(x)$ such that $a(x)p(x)+b(x)q(x)=1$. Dehomogenizing and multiplying by a common denominator $M$ we get $$A(x,y)P(x,y)+B(x,y)Q(x,y)=My^k.$$ Hence every common factor of $P(x,y),Q(x,y)$ divides $My^k$. Similarly for some integer $N$ it must divide $Nx^l$. If $x,y$ are relatively prime, we get that $\gcd(P(x,y),Q(x,y))$ must divide $MN$ (by looking at exponent of each prime power separately).