Fast calculation of cyclotomic polynomials
This module provides a function cyclotomic_coeffs(), which calculates the coefficients of cyclotomic polynomials. This is not intended to be invoked directly by the user, but it is called by the method cyclotomic_polynomial() method of univariate polynomial ring objects and the top-level cyclotomic_polynomial() function.
This calculates the coefficients of the n-th cyclotomic polynomial by using the formula
where \(\mu(d)\) is the Moebius function that is 1 if d has an even number of distinct prime divisors, -1 if it has an odd number of distinct prime divisors, and 0 if d is not squarefree.
Multiplications and divisions by polynomials of the form \(1-x^n\) can be done very quickly in a single pass.
If sparse is True, the result is returned as a dictionary of the non-zero entries, otherwise the result is returned as a list of python ints.
EXAMPLES:
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}
sage: R = QQ['x']
sage: R(cyclotomic_coeffs(30))
x^8 + x^7 - x^5 - x^4 - x^3 + x + 1
Check that it has the right degree:
sage: euler_phi(30)
8
sage: R(cyclotomic_coeffs(14)).factor()
x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
The coefficients are not always +/-1:
sage: cyclotomic_coeffs(105)
[1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1]
In fact the height is not bounded by any polynomial in n (Erdos), although takes a while just to exceed linear:
sage: v = cyclotomic_coeffs(1181895)
sage: max(v)
14102773
The polynomial is a palindrome for any n:
sage: n = ZZ.random_element(50000)
sage: factor(n)
3 * 10009
sage: v = cyclotomic_coeffs(n, sparse=False)
sage: v == list(reversed(v))
True
AUTHORS:
Returns the value of the \(n\)-th cyclotomic polynomial evaulated at \(x\).
INPUT:
OUTPUT:
ALGORITHM:
where \(q\) is the radical of \(n\).
where \(\mu\) is the Moebius function.
EXAMPLES:
sage: cyclotomic_value(51, 3)
1282860140677441
sage: cyclotomic_polynomial(51)(3)
1282860140677441
It works for non-integral values as well:
sage: cyclotomic_value(144, 4/3)
79148745433504023621920372161/79766443076872509863361
sage: cyclotomic_polynomial(144)(4/3)
79148745433504023621920372161/79766443076872509863361
TESTS:
sage: K.<i> = NumberField(polygen(QQ)^2 + 1)
sage: R.<x> = QQ[]
sage: for y in [-1, 0, 1, 2, 1/2, mod(3, 8), GF(9,'a').gen(), Zp(3)(54), i, x^2+2]:
... for n in range(1, 61):
... val1 = cyclotomic_value(n, y)
... val2 = cyclotomic_polynomial(n)(y)
... assert val1 == val2 and val1.parent() is val2.parent()
sage: cyclotomic_value(20, I)
5
sage: a = cyclotomic_value(10, mod(3, 11)); a
6
sage: a.parent()
Ring of integers modulo 11
sage: cyclotomic_value(30, -1.0)
1.00000000000000
sage: S.<t> = R.quotient(R.cyclotomic_polynomial(15))
sage: cyclotomic_value(15, t)
0
sage: cyclotomic_value(30, t)
2*t^7 - 2*t^5 - 2*t^3 + 2*t
sage: S.<t> = R.quotient(x^10)
sage: cyclotomic_value(2^128-1, t)
-t^7 - t^6 - t^5 + t^2 + t + 1
sage: cyclotomic_value(10,mod(3,4))
1