# abc conjecture and surjective polynomials

Let $$a,b$$ be coprime multivariate polynomials with integer coefficients and $$\deg(a) > \deg(\rm{rad}(a b))$$.

Let $$c=a+b$$ and assume $$c$$ is either surjective or $$c$$ represents infinitely many integers with small radicals at integer values.

Let $$c(P_i)=A_i$$ be the integral points with $$A_i$$ with small radical.

We have the abc triples $$a(P_i) , b(P_i), c(P_i) = a(P_i)+b(P_i)$$. The radical at integers is $$\rm{rad}(A_i a(P_i) b(P_i))$$.

abc implies either $$\gcd(a(P_i),b(P_i)$$ is rather large or roughly speaking the degree argument $$\deg(a) > \deg(\rm{rad}(a b))$$ fails at integers.

Are these implications of abc known to hold?

Example: For $$D > 3$$ define $$a=x^Dy$$ and $$b=z^Dt$$. Then $$c=x^D y + z^D t$$ is surjective since it is linear in $$y,t$$. For fixed coprime $$x,z$$ abc implies lack of small solutions to linear diophantine equation.