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Abstract

Let $f\in\mathbb Q[X]$ and $\mathop{\rm deg}f\leq 3$. We prove that if $\mathop{\rm deg}f=2$, then the diophantine equation $f(x)f(y)=f(z)^2$ has infinitely many nontrivial solutions in $\mathbb Q(t)$. In the case when $\mathop{\rm deg}f=3$ and $f(X)=X(X^2+aX+b)$ we show that for all but finitely many $a,b\in\mathbb Z$ satisfying $ab\neq 0$ and additionally, if $p\mid a$, then $p^2\nmid b$, the equation $f(x)f(y)=f(z)^2$ has infinitely many nontrivial solutions in rationals.