On a conjecture of Beard, OʹConnell and West concerning perfect polynomials

Following Beard, OʹConnell and West [J.T.B. Beard Jr., J.R. OʹConnell Jr., K.I. West, Perfect polynomials over GF(q), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 62 (1977) 283–291] we call a polynomial over a finite field Fq perfect if it coincides with the sum of its monic divisors. The study of perfect polynomials was initiated in 1941 by Carlitzʹs doctoral student Canaday in the case q=2, who proposed the still unresolved conjecture that every perfect polynomial over F2 has a root in F2. Beard et al. later proposed the analogous hypothesis for all finite fields. Counterexamples to this general conjecture were found by Link (in the cases q=11,17) and Gallardo & Rahavandrainy (in the case q=4). Here we show that the Beard–OʹConnell–West conjecture fails in all cases except possibly when q is prime. When q=p is prime, utilizing a construction of Link we exhibit a counterexample whenever p≡11 or 17 (mod 24). On the basis of a polynomial analog of Schinzelʹs Hypothesis H, we argue that if there is a single perfect polynomial over the finite field Fq with no linear factor, then there are infinitely many. Lastly, we prove without any hypothesis that there are infinitely many perfect polynomials over F11 with no linear factor.