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.


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