About some time I am struggling with the following interesting problem:
There is a well-known theorem of Mignotte which says that for a polynomial $f\in\mathbb{Z}[x]$ of degree $n$ and height (coefficient size) $2^\tau$, the height of its divisors is bounded by:
$2^n\mathcal{M}(f)=\mathcal{O}(2^{n+\tau})$,
see, e.g., http://arxiv.org/abs/0904.3057
In other words, the height of polynomial's divisors can be larger than the height of a polynomial itself. Example could be:
$x^5+3x^4+2x^3-2x^2-3x-1=(x^4+4x^3+6x^2+4x+1)(x-1)$.
My conjecture is that this cannot happen for polynomial's square-free part. To be precise, for a polynomial $f\in\mathbb{Z}[x]$ of height $2^\tau$, its square-free part $f^*=f / \gcd(f,f')$ cannot have height larger than $2^\tau$.
Simple example: $f=(x-1)^3=x^3-3x^2+3x-1$ and $f^*=x-1$.
I appreciate if someone has ideas how to prove that or find a counterexample.
Why it's an interesting problem is because theoretical complexity of many algorithms from algebraic geometry use such bounds. On a similar note, modular approaches use these bounds to estimate the number of primes needed for computation.