Let $f(x\_1,\dots,x\_n)$ be squarefree polynomial with integer
coefficients. Assume $f$ at integers is not always divisible
by a fixed square $m^2 > 1$.


Is it possible $f$ to never be squarefree at integers?


I suspect this is impossible.



Multivariate polynomial is never squarefree at integers
This is a supplement to Ilya Bogdanov's answer.


For univariate polynomials Granville (Int. Math. Res. Not. 1998, 991-1009) deduced from the $abc$-conjecture that there are infinitely many natural numbers $x$ such that $f(x)$ is square-free; in fact these numbers have positive density. The same is known unconditionally when $\deg f\leq 3$, by the work of Hooley (Mathematika 14 (1967), 21-26). In addition, Granville's conditional result was extended to multivariate polynomials by Poonen (Duke Math. J. 118 (2003), 353-373).


