User peter scholl - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:00:22Z http://mathoverflow.net/feeds/user/13965 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59859/computing-a-polynomial-product-over-roots-of-unity Computing a polynomial product over roots of unity Peter Scholl 2011-03-28T16:30:13Z 2011-03-29T23:08:45Z <p>I'm trying to compute the coefficients of the following polynomial, where $\omega$ is a primitive $p$-th root of unity, for $p$ prime:</p> <p>$$a(x) = \prod_{i=0}^{p-1} f(\omega^ix).$$</p> <p>It turns out that the $i$-th coefficient is always an integer, and non-zero only when $i$ is a multiple of $p$. So it seems to me like there should be an elementary expression for $a$.</p> <p>So far I've got this expression for the $i$-th coefficient: $$a_i = x^i\sum_{k_0 + \ldots + k_{p-1} = i}f_{k_0} \cdots f_{k_{p-1}} \omega^{k_1 + 2k_2 + \ldots + (p-1)k_{p-1}}$$</p> <p>where each $k_i$ is non-negative and bounded by the degree of $f$.</p> <p>Clearly the roots of unity all cancel out somehow, but I can't figure out how to get a 'nice' expression out. Any suggestions?</p> http://mathoverflow.net/questions/59859/computing-a-polynomial-product-over-roots-of-unity/59890#59890 Comment by Peter Scholl Peter Scholl 2011-03-29T12:36:29Z 2011-03-29T12:36:29Z That's a nice result, but I think computing the resultant here is quite slow when $f$ is large, which I need it to be. What I'm really looking for (perhaps I should have made it more clear) is an efficient way to recover $a$, rather than just an expression. http://mathoverflow.net/questions/59859/computing-a-polynomial-product-over-roots-of-unity Comment by Peter Scholl Peter Scholl 2011-03-28T17:18:19Z 2011-03-28T17:18:19Z No, unfortunately I don't know the roots of $f$. Thanks anyway though, that's interesting to know.