Fast calculation of cyclotomic polynomials

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)
sage.rings.polynomial.cyclotomic.cyclotomic_coeffs(nn, sparse=None)

This calculates 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 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, 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)
sage.rings.polynomial.cyclotomic.cyclotomic_value(n, x)

Returns the value of the \(n\)-th cyclotomic polynomial evaulated 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 Moebius 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

TESTS:

sage: R.<x> = QQ[]
sage: K.<i> = NumberField(x^2 + 1)
sage: for y in [-1, 0, 1, 2, 1/2, Mod(3, 8), Mod(3,11), GF(9,'a').gen(), Zp(3)(54), i, x^2+2]:
....:     for n in [1..60]:
....:         val1 = cyclotomic_value(n, y)
....:         val2 = cyclotomic_polynomial(n)(y)
....:         if val1 != val2:
....:             print "Wrong value for cyclotomic_value(%s, %s) in %s"%(n,y,parent(y))
....:         if val1.parent() is not val2.parent():
....:             print "Wrong parent for cyclotomic_value(%s, %s) in %s"%(n,y,parent(y))

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
sage.rings.polynomial.cyclotomic.my_cmp(a, b)

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