This looks like an interesting puzzle. His sum of k = z^-3 + ... + z^3 can also be read as the sum of a few cosines, or as the sum of a geometric series. It should be easy to gather more data. Rich ----- Forwarded message from boston@MATH.WISC.EDU ----- Date: Wed, 29 Apr 2009 13:32:25 -0400 From: Nigel Boston <boston@MATH.WISC.EDU> Reply-To: Nigel Boston <boston@MATH.WISC.EDU> Subject: almost minimal polynomial To: NMBRTHRY@LISTSERV.NODAK.EDU When computing the Koetter-Vontobel lower bound for minimum pseudoweight of the binary Golay code, a certain term came out as 15.9996. My engineering collaborator rounded this to 16 but the fact that it's so close to an integer but not actually an integer is in itself interesting. The term is actually 4+2k, where k = z^-3 + z^-2 + z^-1 + 1 + z + z^2 + z^3 and z is the 23rd root of unity exp(2\pi i/23). k is 5.99977967. Is there some underlying reason why k is so close to 6 ? Alternatively, why is x^6 + x^5 + x^4 - 5x^3 + x^2 + x + 1 "almost" a minimal polynomial for z over Q ? I searched for similar examples but there are no better until you get up to 77th and 79th roots of unity (z = exp(2\pi i 12/77), sum from z^-20 to z^20 is 1.999833364548; z = exp(2\pi i 3/79), sum from z^-62 to z^62 is 5.9999365499). ----- End forwarded message -----