Bounded base-p digit complexity for polynomials with rational roots

Determine whether, for every prime p, there exists a constant c_p such that every monic polynomial f(x) in Z[x] with only rational roots and no coefficients divisible by p has at most c_p nonzero digits in the base-p expansion of deg(f)+1; in particular, determine whether one may take c_p=p-1.

Background

Fine’s theorem establishes the corresponding statement for p=2, where deg(f)+1 has at most one nonzero binary digit, and the result of Hajdu, Tijdeman, and Varga establishes a related bound for p=3. The paper uses Walnut to verify additional cases for p=5 and for polynomials over the field with four elements, but the general assertion for every prime remains unresolved.

The problem asks whether the absence of coefficients divisible by a prime p forces a uniformly bounded number of nonzero base-p digits in the degree-plus-one parameter. The sharper proposed bound c_p=p-1 would extend the observed p=2 and p=3 behavior.

References

Hajdu et al ask if a far reaching generalization of these results holds: is it true that for every prime $p$ there exists a constant $c_p$ such that any monic polynomial $f(x)\in\mathbb Z[x]$ with only rational roots and no coefficients divisible by $p$ has at most $c_p$ nonzero digits in the base $p$ expansion of $\mathrm{deg}(f)+1$? In particular, is this true for $c_p=p-1$?

— Using Walnut to solve problems from the OEIS  (2503.04122 - Bosma et al., 6 Mar 2025) in Section 2, immediately following Theorem 2.1 (Hajdu et al.)