On a conjecture of Beard, O'Connell and West concerning perfect polynomials

Abstract

Following Beard, O'Connell and West [J.T.B. Beard Jr., J.R. O'Connell 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 F q perfect if it coincides with the sum of its monic divisors. The study of perfect polynomials was initiated in 1941 by Carlitz's doctoral student Canaday in the case q = 2 , who proposed the still unresolved conjecture that every perfect polynomial over F 2 has a root in F 2 . 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–O'Connell–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 Schinzel's Hypothesis H, we argue that if there is a single perfect polynomial over the finite field F q with no linear factor, then there are infinitely many. Lastly, we prove without any hypothesis that there are infinitely many perfect polynomials over F 11 with no linear factor.