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)[source]

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)                                       # needs sage.libs.pari
66944986927

>>> from sage.all import *
>>> from sage.rings.polynomial.cyclotomic import bateman_bound
>>> bateman_bound(Integer(2)**Integer(8) * Integer(1234567893377))                                       # needs sage.libs.pari
66944986927
sage.rings.polynomial.cyclotomic.cyclotomic_coeffs(nn, sparse=None)[source]

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 nonzero 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

>>> from sage.all import *
>>> from sage.rings.polynomial.cyclotomic import cyclotomic_coeffs
>>> cyclotomic_coeffs(Integer(30))
[1, 1, 0, -1, -1, -1, 0, 1, 1]
>>> cyclotomic_coeffs(Integer(10)**Integer(5))
{0: 1, 10000: -1, 20000: 1, 30000: -1, 40000: 1}
>>> R = QQ['x']
>>> R(cyclotomic_coeffs(Integer(30)))
x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

Check that it has the right degree:

sage: euler_phi(30)                                                             # needs sage.libs.pari
8
sage: R(cyclotomic_coeffs(14)).factor()                                         # needs sage.libs.pari
x^6 - x^5 + x^4 - x^3 + x^2 - x + 1

>>> from sage.all import *
>>> euler_phi(Integer(30))                                                             # needs sage.libs.pari
8
>>> R(cyclotomic_coeffs(Integer(14))).factor()                                         # needs sage.libs.pari
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]

>>> from sage.all import *
>>> cyclotomic_coeffs(Integer(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

>>> from sage.all import *
>>> v = cyclotomic_coeffs(Integer(1181895))
>>> max(v)
14102773

The polynomial is a palindrome for any n:

sage: n = ZZ.random_element(50000)
sage: v = cyclotomic_coeffs(n, sparse=False)
sage: v == list(reversed(v))
True

>>> from sage.all import *
>>> n = ZZ.random_element(Integer(50000))
>>> v = cyclotomic_coeffs(n, sparse=False)
>>> 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)[source]

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

>>> from sage.all import *
>>> cyclotomic_value(Integer(51), Integer(3))
1282860140677441
>>> cyclotomic_polynomial(Integer(51))(Integer(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

>>> from sage.all import *
>>> cyclotomic_value(Integer(144), Integer(4)/Integer(3))
79148745433504023621920372161/79766443076872509863361
>>> cyclotomic_polynomial(Integer(144))(Integer(4)/Integer(3))
79148745433504023621920372161/79766443076872509863361