ASKSAGE: Sage Q&A Forum - Latest question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 23 Mar 2019 06:26:52 -0500Evaluating polynomials in Z[x] at roots of unity.http://ask.sagemath.org/question/45880/evaluating-polynomials-in-zx-at-roots-of-unity/Suppose that I have a polynomial $p(x)$ with integer coeffcients, then I can easily evaluate this at some integer to get an exact answer. That is, I can write something like
sage: p=1+x+2*x^2+x^3+x^4
sage: p(2)
35
But If I instead want to evaluate this polynomial at a root of unity (I take $w$ to be a primitive $5$:th root of unity in the example below but any root of unity would do), then I may write say
sage: w = e^(2*pi*i/5)
sage: p(w)
1/256*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^4 + 1/64*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^3 + 1/8*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1)^2 + 1/4*sqrt(5) + 1/4*I*sqrt(2*sqrt(5) + 10) + 3/4
Which is a correct answer but written in a very complicated form. Alternatively, I can write
sage: K.<z> = CyclotomicField(5)
sage: w=CC(z)
sage: p(w)
-0.809016994374947 + 0.587785252292473*I
which is an approximation of the correct answers. In the above case, we actually have $p(w)=w^2$ and I am wondering if there is an easy way to always obtain such an answer. For a general $p(x)$ with integer coefficients and $w$ that is a primitive $n$:th root of unity, we can write $p(w)$ on the form
$$a_0+a_1 w + a_2 w^2 + \dots +a_{n-1} w^{n-1}$$
where the $a_i$:s are integers. Is there a good way to get the $p(w)$ on this form in Sage?joakim_uhlinSat, 23 Mar 2019 06:26:52 -0500http://ask.sagemath.org/question/45880/doubt about cyclotomic polynomialshttp://ask.sagemath.org/question/34467/doubt-about-cyclotomic-polynomials/ sage: from sage.rings.polynomial.cyclotomic import cyclotomic_coeffs
sage: cyclotomic_coeffs(30)
[1, 1, 0, -1, -1, -1, 0, 1, 1]
sage: cyclotomic_coeffs(10^5)
{0: 1, 10000: -1, 40000: 1, 30000: -1, 20000: 1}
I don't understand the notation. What does the last line mean ?nebuckandazzerSun, 14 Aug 2016 03:45:13 -0500http://ask.sagemath.org/question/34467/