Counting perfect polynomials

Abstract Let A ∈ F 2 [ T ] . We say A is perfect if A coincides with the sum of all of its divisors in F 2 [ T ] . We prove that the number of perfect polynomials A with | A | ≤ x is O ϵ ( x 1 / 12 + ϵ ) for all ϵ > 0 , where | A | = 2 deg ⁡ A . We also prove that every perfect polynomial A with 1 | A | ≤ 1.6 × 10 60 is divisible by T or T + 1 ; that is, there are no small “odd” perfect polynomials.