On a problem of Byrnes concerning polynomials with restricted coefficients

As in the earlier paper with this title, we consider a question of Byrnes concerning the minimal length N*(m) of a polynomial with all coefficients in {-1, 1} which has a zero of a given order m at x = 1. In that paper we showed that N*(m) = 2 m for all m < 5 and showed that the extrernal polynomials for were those conjectured by Byrnes, but for m = 6 that N*(6) = 48 rather than 64. A polynomial with N = 48 was exhibited for m = 6, but it was not shown there that this extremal was unique. Here we show that the extremal is unique. In the previous paper, we showed that N*(7) is one of the 7 values 48, 56, 64, 72, 80, 88 or 96. Here we prove that N* (7) = 96 without determining all extremal polynomials. We also make some progress toward determining N*(8). As in the previous paper, we use a combination of number theoretic ideas and combinatorial computation. The main point is that if ζ p is a primitive pth root of unity where p < m + 1 is a prime, then the condition that all coefficients of P be in {-1, 1}, together with the requirement that P(x) be divisible by (x - 1) m puts severe restrictions on the possible values for the cyclotomic integer P(ζ p ).