math.NT — Number Theory

The Mesirov-Sweet property over F_2 up to degree 33, with two quotients of degree two up to degree 28

Contributed by David Niedbala Giraudin

For a monic polynomial g in F_2[T] of degree d, the conjecture of Mesirov and Sweet asserts that there is f in F_2[T], coprime to g with deg f < d, such that every partial quotient of the continued fraction of f/g has degree at most 2. We check by exhaustive computation that this holds for every g of degree at most 33. We also check that for every g of degree at most 28 one can choose f so that all partial quotients are linear except at most two, which have degree 2. The computations combine exhaustive enumeration of fixed degree patterns, explicit certificates for the polynomials not reached by the patterns, and two separately written enumeration programs whose outputs coincide bit for bit. All programs are provided.

Author comments: 3 pages, 2 tables. Ancillary files: C sources of the enumeration programs, of the certificate search and of the checker (stored as text in c_sources.py), cell lists, run logs, a numpy verifier and samples of 10^5 certificates in degrees 28 and 33.