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# -*- coding: utf-8 -*- r""" Miscellaneous Functions
This file contains several miscellaneous functions used by `p`-adics.
- ``gauss_sum`` -- compute Gauss sums using the Gross-Koblitz formula. - ``min`` -- a version of ``min`` that returns `\infty` on empty input. - ``max`` -- a version of ``max`` that returns `-\infty` on empty input.
AUTHORS:
- David Roe - Adriana Salerno - Ander Steele - Kiran Kedlaya (modified gauss_sum 2017/09) """ #***************************************************************************** # Copyright (C) 2007-2013 David Roe <roed.math@gmail.com> # William Stein <wstein@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # # http://www.gnu.org/licenses/ #***************************************************************************** from __future__ import absolute_import
from six.moves.builtins import min as python_min from six.moves.builtins import max as python_max from six.moves.builtins import range, zip from sage.rings.infinity import infinity
def gauss_sum(a, p, f, prec=20, factored=False): r""" Return the Gauss sum `g_q(a)` as a `p`-adic number.
The Gauss sum `g_q(a)` is defined by
.. MATH::
g_q(a)= \sum_{u\in F_q^*} \omega(u)^{-a} \zeta_q^u,
where `q = p^f`, `\omega` is the Teichmüller character and `\zeta_q` is some arbitrary choice of primitive `q`-th root of unity. The computation is adapted from the main theorem in Alain Robert's paper *The Gross-Koblitz formula revisited*, Rend. Sem. Mat. Univ. Padova 105 (2001), 157--170.
Let `p` be a prime, `f` a positive integer, `q=p^f`, and `\pi` be the unique root of `f(x) = x^{p-1}+p` congruent to `\zeta_p - 1` modulo `(\zeta_p - 1)^2`. Let `0\leq a < q-1`. Then the Gross-Koblitz formula gives us the value of the Gauss sum `g_q(a)` as a product of `p`-adic Gamma functions as follows:
.. MATH::
g_q(a) = -\pi^s \prod_{0\leq i < f} \Gamma_p(a^{(i)}/(q-1)),
where `s` is the sum of the digits of `a` in base `p` and the `a^{(i)}` have `p`-adic expansions obtained from cyclic permutations of that of `a`.
INPUT:
- ``a`` -- integer
- ``p`` -- prime
- ``f`` -- positive integer
- ``prec`` -- positive integer (optional, 20 by default)
- ``factored`` - boolean (optional, False by default)
OUTPUT:
If ``factored`` is ``False``, returns a `p`-adic number in an Eisenstein extension of `\QQ_p`. This number has the form `pi^e * z` where `pi` is as above, `e` is some nonnegative integer, and `z` is an element of `\ZZ_p`; if ``factored`` is ``True``, the pair `(e,z)` is returned instead, and the Eisenstein extension is not formed.
.. NOTE::
This is based on GP code written by Adriana Salerno.
EXAMPLES:
In this example, we verify that `g_3(0) = -1`::
sage: from sage.rings.padics.misc import gauss_sum sage: -gauss_sum(0,3,1) 1 + O(pi^40)
Next, we verify that `g_5(a) g_5(-a) = 5 (-1)^a`::
sage: from sage.rings.padics.misc import gauss_sum sage: gauss_sum(2,5,1)^2-5 O(pi^84) sage: gauss_sum(1,5,1)*gauss_sum(3,5,1)+5 O(pi^84)
Finally, we compute a non-trivial value::
sage: from sage.rings.padics.misc import gauss_sum sage: gauss_sum(2,13,2) 6*pi^2 + 7*pi^14 + 11*pi^26 + 3*pi^62 + 6*pi^74 + 3*pi^86 + 5*pi^98 + pi^110 + 7*pi^134 + 9*pi^146 + 4*pi^158 + 6*pi^170 + 4*pi^194 + pi^206 + 6*pi^218 + 9*pi^230 + O(pi^242) sage: gauss_sum(2,13,2,prec=5,factored=True) (2, 6 + 6*13 + 10*13^2 + O(13^5))
.. SEEALSO::
- :func:`sage.arith.misc.gauss_sum` for general finite fields - :meth:`sage.modular.dirichlet.DirichletCharacter.gauss_sum` for prime finite fields - :meth:`sage.modular.dirichlet.DirichletCharacter.gauss_sum_numerical` for prime finite fields """
def min(*L): r""" Return the minimum of the inputs, where the minimum of the empty list is `\infty`.
EXAMPLES::
sage: from sage.rings.padics.misc import min sage: min() +Infinity sage: min(2,3) 2 """
def max(*L): r""" Return the maximum of the inputs, where the maximum of the empty list is `-\infty`.
EXAMPLES::
sage: from sage.rings.padics.misc import max sage: max() -Infinity sage: max(2,3) 3 """ L = L[0]
def precprint(prec_type, prec_cap, p): """ String describing the precision mode on a p-adic ring or field.
EXAMPLES::
sage: from sage.rings.padics.misc import precprint sage: precprint('capped-rel', 12, 2) 'with capped relative precision 12' sage: precprint('capped-abs', 11, 3) 'with capped absolute precision 11' sage: precprint('floating-point', 1234, 5) 'with floating precision 1234' sage: precprint('fixed-mod', 1, 17) 'of fixed modulus 17^1' """ 'capped-abs':'with capped absolute precision %s'%prec_cap, 'floating-point':'with floating precision %s'%prec_cap, 'fixed-mod':'of fixed modulus %s^%s'%(p, prec_cap), 'lattice-cap':'with lattice-cap precision', 'lattice-float':'with lattice-float precision'}
def trim_zeros(L): r""" Strips trailing zeros/empty lists from a list.
EXAMPLES::
sage: from sage.rings.padics.misc import trim_zeros sage: trim_zeros([1,0,1,0]) [1, 0, 1] sage: trim_zeros([[1],[],[2],[],[]]) [[1], [], [2]] sage: trim_zeros([[],[]]) [] sage: trim_zeros([]) []
Zeros are also trimmed from nested lists (one deep):
sage: trim_zeros([[1,0]]) [[1]] sage: trim_zeros([[0],[1]]) [[], [1]] """ # strip trailing zeros from the sublists else: else: |