I suppose a conjecture implies this so there might be an unconditional proof.

Let $F(x,y)=0$ be a curve with infinitely many integral points $(u^m,v^n)$ where $\gcd(u,v)=1$ infinitely often and $m \ge 3,n \ge 2$. Such curves are easy to construct by starting with a parametrization for example.

For a bivariate polynomial $F$ define $\operatorname{High}(F)$ to be the sum of the highest degree monomials (i.e., $\operatorname{High}(F) = F$ iff $F$ is homogeneous and $\operatorname{High}(2x^3+3y^3+x+y)=2x^3+3y^3$). Let $\gcd(\operatorname{High}(F),xy)=1$.

Under these conditions is $\operatorname{High}(F)$ not square-free?

 

Counterexamples?

This can't be relaxed to $m \ge 2 $

I suppose a conjecture implies this so there might be an unconditional proof.

Let $F(x,y)=0$ be a curve with infinitely many integral points $(u^m,v^n)$ where $\gcd(u,v)=1$ infinitely often and $m \ge 3,n \ge 2$. Such curves are easy to construct by starting with a parametrization for example.

For a bivariate polynomial $F$ define $\operatorname{High}(F)$ to be the sum of the highest degree monomials (i.e., $\operatorname{High}(F) = F$ iff $F$ is homogeneous and $\operatorname{High}(2x^3+3y^3+x+y)=2x^3+3y^3$). Let $\gcd(\operatorname{High}(F),xy)=1$.

Under these conditions is $\operatorname{High}(F)$ not square-free?

Counterexamples?

This can't be relaxed to $m \ge 2 $

I suppose a conjecture implies this so there might be an unconditional proof.

Let $F(x,y)=0$ be a curve with infinitely many integral points $(u^m,v^n)$ where $\gcd(u,v)=1,m \ge 3,n \ge 2$. Such curves are easy to construct by starting with a parametrization for example.

For a bivariate polynomial $F$ define $\operatorname{High}(F)$ to be the sum of the highest degree monomials (i.e., $\operatorname{High}(F) = F$ iff $F$ is homogeneous). Let $\gcd(\operatorname{High}(F),xy)=1$.

Under these conditions is $\operatorname{High}(F)$ not square-free?

I suppose a conjecture implies this so there might be an unconditional proof.

Let $F(x,y)=0$ be a curve with infinitely many integral points $(u^m,v^n)$ where $\gcd(u,v)=1$ infinitely often and $m \ge 3,n \ge 2$. Such curves are easy to construct by starting with a parametrization for example.

For a bivariate polynomial $F$ define $\operatorname{High}(F)$ to be the sum of the highest degree monomials (i.e., $\operatorname{High}(F) = F$ iff $F$ is homogeneous and $\operatorname{High}(2x^3+3y^3+x+y)=2x^3+3y^3$). Let $\gcd(\operatorname{High}(F),xy)=1$.

Under these conditions is $\operatorname{High}(F)$ not square-free?

Counterexamples?

This can't be relaxed to $m \ge 2 $