Daniel Berend

Let N* (m) be the minimal length of a polynomial with ±1 coefficients divisible by (x - 1)^m. Byrnes noted that N* (m) ≤ 2^m for each m, and asked whether in fact N*(m) = 2 ^m. Boyd showed that N*(m) = 2^m for all m ≤ 5, but N* (6) = 48. He further showed that N* (7) = 96, and that N* (8) is one of the 5 numbers 96, 144, 160, 176, or 192. Here we prove that N* (8) = 144. Similarly, let m* (N) be the maximal power of (x - 1) dividing some polynomial of degree N - 1 with ±1 coefficients. Boyd was able to find m* (N) for W < 88. In this paper we determine m* (N) for N < 168.