Abstract:

The existence is shown of infinitely many nonsplitting perfect polynomials over GF(2d), GF(3d), GF(5d) for each odd integer d > 1, and over GF(2d)
for each (even) integer d 1 0 (mod 3). Stronger results show that each unitary perfect polynomial over GF(q) determines an infinite equivalence class of unitary perfect polynomials over GF(q). The number SUP(q) of distinct equivalence classes of splitting unitary perfect polynomials over GF(q) is calculated for q = p and shown to be infinite for q # p. The number
NSUP(q) of distinct equivalence classes of nonsplitting unitary perfect polynomials over GF(q) remains undetermined, but is shown to be infinite whenever there are two relatively prime unitary perfect polynomials over GF(q) and one of them does not split. In particular NSUP(2d), NSUP(3d), and NSUP(5d) are infinite for each odd integer d > 1, and NSUP(2d) is
infinite for each (even) integer d 1 0 (mod 3). Examples are given to establish NSUP(2) 33, NSUP(3) 16, and NSUP(5) 6. It is conjectured that for all
primes p and odd integers d 1, NSUP(pd) is infinite. 