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.
- sage.rings.polynomial.cyclotomic.bateman_bound(nn)¶
Reference:
Bateman, P. T.; Pomerance, C.; Vaughan, R. C. On the size of the coefficients of the cyclotomic polynomial.
EXAMPLES:
sage: from sage.rings.polynomial.cyclotomic import bateman_bound sage: bateman_bound(2**8*1234567893377) 66944986927
- sage.rings.polynomial.cyclotomic.cyclotomic_coeffs(nn, sparse=None)¶
Return the coefficients of the n-th cyclotomic polynomial by using the formula
\[\Phi_n(x) = \prod_{d|n} (1-x^{n/d})^{\mu(d)}\]where \(\mu(d)\) is the Möbius 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, 20000: 1, 30000: -1, 40000: 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:
Robert Bradshaw (2007-10-27): initial version (inspired by work of Andrew Arnold and Michael Monagan)
REFERENCE:
- sage.rings.polynomial.cyclotomic.cyclotomic_value(n, x)¶
Return the value of the \(n\)-th cyclotomic polynomial evaluated at \(x\).
INPUT:
\(n\) – an Integer, specifying which cyclotomic polynomial is to be evaluated
\(x\) – an element of a ring
OUTPUT:
the value of the cyclotomic polynomial \(\Phi_n\) at \(x\)
ALGORITHM:
Reduce to the case that \(n\) is squarefree: use the identity
\[\Phi_n(x) = \Phi_q(x^{n/q})\]where \(q\) is the radical of \(n\).
Use the identity
\[\Phi_n(x) = \prod_{d | n} (x^d - 1)^{\mu(n / d)},\]where \(\mu\) is the Möbius function.
Handles the case that \(x^d = 1\) for some \(d\), but not the case that \(x^d - 1\) is non-invertible: in this case polynomial evaluation is used instead.
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