Coverage for local/lib/python2.7/site-packages/sage/rings/padics/morphism.pyx : 59%

""" 

Frobenius endomorphisms on p-adic fields 

""" 

#***************************************************************************** 

# Copyright (C) 2013 Xavier Caruso <xavier.caruso@normalesup.org> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License 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 sage.rings.integer cimport Integer 

from sage.rings.infinity import Infinity 

from sage.rings.ring import CommutativeRing 

from sage.categories.homset import Hom 

from sage.structure.element cimport Element 

from sage.structure.richcmp cimport (richcmp, rich_to_bool, 

richcmp_not_equal) 

from sage.rings.morphism cimport RingHomomorphism 

from .padic_generic import pAdicGeneric 

from sage.categories.morphism cimport Morphism 

cdef class FrobeniusEndomorphism_padics(RingHomomorphism): 

""" 

A class implementing Frobenius endomorphisms on padic fields. 

""" 

def __init__ (self,domain,n=1): 

""" 

INPUT: 

- ``domain`` -- an unramified padic field 

- ``n`` -- an integer (default: 1) 

.. NOTE:: 

`n` may be negative. 

OUTPUT: 

The `n`-th power of the absolute (arithmetic) Frobenius 

endomorphism on ``domain`` 

TESTS:: 

sage: from sage.rings.padics.morphism import FrobeniusEndomorphism_padics 

sage: K.<a> = Qq(5^3) 

sage: FrobeniusEndomorphism_padics(K) 

Frobenius endomorphism on Unramified Extension ... lifting a |--> a^5 on the residue field 

sage: FrobeniusEndomorphism_padics(K,2) 

Frobenius endomorphism on Unramified Extension ... lifting a |--> a^(5^2) on the residue field 

sage: FrobeniusEndomorphism_padics(K,a) 

Traceback (most recent call last): 

... 

TypeError: n (=a + O(5^20)) is not an integer 

sage: K = Qp(5) 

sage: L.<pi> = K.extension(x^2 - 5) 

sage: FrobeniusEndomorphism_padics(L) 

Traceback (most recent call last): 

... 

TypeError: The domain must be unramified 

sage: FrobeniusEndomorphism_padics(QQ) 

Traceback (most recent call last): 

... 

TypeError: The domain must be an instance of pAdicGeneric 

""" 

if not isinstance(domain, pAdicGeneric): 

raise TypeError("The domain must be an instance of pAdicGeneric") 

if domain.e() != 1: 

raise TypeError("The domain must be unramified") 

try: 

n = Integer(n) 

except (ValueError, TypeError): 

raise TypeError("n (=%s) is not an integer" % n) 

self._degree = domain.f() 

self._power = n % self._degree 

self._order = self._degree / domain.degree().gcd(self._power) 

RingHomomorphism.__init__(self, Hom(domain, domain)) 

def _repr_(self): 

""" 

Return a string representation of this endomorphism. 

EXAMPLES:: 

sage: K.<a> = Qq(5^3) 

sage: Frob = K.frobenius_endomorphism(); Frob 

Frobenius endomorphism on Unramified Extension ... lifting a |--> a^5 on the residue field 

sage: Frob._repr_() 

'Frobenius endomorphism on Unramified Extension ... lifting a |--> a^5 on the residue field' 

""" 

name = self.domain().variable_name() 

if self._power == 0: 

s = "Identity endomorphism of %s" % self.domain() 

elif self._power == 1: 

s = "Frobenius endomorphism on %s lifting %s |--> %s^%s on the residue field" % (self.domain(), name, name, self.domain().residue_characteristic()) 

else: 

s = "Frobenius endomorphism on %s lifting %s |--> %s^(%s^%s) on the residue field" % (self.domain(), name, name, self.domain().residue_characteristic(), self._power) 

return s 

def _repr_short(self): 

""" 

Return a short string representation of this endomorphism. 

EXAMPLES:: 

sage: K.<a> = Qq(5^3) 

sage: Frob = K.frobenius_endomorphism(); Frob 

Frobenius endomorphism on Unramified Extension ... lifting a |--> a^5 on the residue field 

sage: Frob._repr_short() 

'Frob' 

""" 

name = self.domain().variable_name() 

if self._power == 0: 

s = "Identity" 

elif self._power == 1: 

s = "Frob" 

else: 

s = "Frob^%s" % self._power 

return s 

cpdef Element _call_ (self, x): 

""" 

TESTS:: 

sage: K.<a> = Qq(5^3) 

sage: Frob = K.frobenius_endomorphism(); 

sage: Frob(a) == a.frobenius() 

True 

""" 

res = x 

for i in range(self._power): 

res = res.frobenius() 

return res 

def order(self): 

""" 

Return the order of this endomorphism. 

EXAMPLES:: 

sage: K.<a> = Qq(5^12) 

sage: Frob = K.frobenius_endomorphism() 

sage: Frob.order() 

12 

sage: (Frob^2).order() 

6 

sage: (Frob^9).order() 

4 

""" 

if self._order == 0: 

return Infinity 

else: 

return Integer(self._order) 

def power(self): 

""" 

Return the smallest integer `n` such that this endomorphism 

is the `n`-th power of the absolute (arithmetic) Frobenius. 

EXAMPLES:: 

sage: K.<a> = Qq(5^12) 

sage: Frob = K.frobenius_endomorphism() 

sage: Frob.power() 

1 

sage: (Frob^9).power() 

9 

sage: (Frob^13).power() 

1 

""" 

return self._power 

def __pow__(self,n,modulus): 

""" 

Return the `n`-th iterate of this endomorphism. 

EXAMPLES:: 

sage: K.<a> = Qq(5^12) 

sage: Frob = K.frobenius_endomorphism() 

sage: Frob^2 

Frobenius endomorphism on Unramified Extension ... lifting a |--> a^(5^2) on the residue field 

The result is simplified if possible:: 

sage: Frob^15 

Frobenius endomorphism on Unramified Extension ... lifting a |--> a^(5^3) on the residue field 

sage: Frob^36 

Identity endomorphism of Unramified Extension ... 

""" 

return self.__class__(self.domain(), self.power()*n) 

def _composition(self,right): 

""" 

Return self o right. 

EXAMPLES:: 

sage: K.<a> = Qq(5^12) 

sage: f = K.frobenius_endomorphism(); f 

Frobenius endomorphism on Unramified Extension ... lifting a |--> a^5 on the residue field 

sage: g = K.frobenius_endomorphism(2); g 

Frobenius endomorphism on Unramified Extension ... lifting a |--> a^(5^2) on the residue field 

sage: f * g 

Frobenius endomorphism on Unramified Extension ... lifting a |--> a^(5^3) on the residue field 

The result is simplified if possible:: 

sage: f = K.frobenius_endomorphism(9) 

sage: g = K.frobenius_endomorphism(10) 

sage: f * g 

Frobenius endomorphism on Unramified Extension ... lifting a |--> a^(5^7) on the residue field 

""" 

if isinstance(right,FrobeniusEndomorphism_padics): 

return self.__class__(self.domain(), self._power+right.power()) 

else: 

return RingHomomorphism._composition(self,right) 

def is_injective(self): 

""" 

Return true since any power of the Frobenius endomorphism 

over an unramified padic field is always injective. 

EXAMPLES:: 

sage: K.<a> = Qq(5^3) 

sage: Frob = K.frobenius_endomorphism() 

sage: Frob.is_injective() 

True 

""" 

return True 

def is_surjective(self): 

""" 

Return true since any power of the Frobenius endomorphism 

over an unramified padic field is always surjective. 

EXAMPLES:: 

sage: K.<a> = Qq(5^3) 

sage: Frob = K.frobenius_endomorphism() 

sage: Frob.is_surjective() 

True 

""" 

return True 

def is_identity(self): 

""" 

Return true if this morphism is the identity morphism. 

EXAMPLES:: 

sage: K.<a> = Qq(5^3) 

sage: Frob = K.frobenius_endomorphism() 

sage: Frob.is_identity() 

False 

sage: (Frob^3).is_identity() 

True 

""" 

return self.power() == 0 

def __hash__(self): 

""" 

Return a hash of this morphism. 

It is the hash of ``(domain, codomain, ('Frob', power)`` 

where ``power`` is the smalles integer `n` such that 

this morphism acts by `x \mapsto x^(p^n)` on the 

residue field 

""" 

domain = self.domain() 

codomain = self.codomain() 

return hash((domain,codomain,('Frob',self._power))) 

cpdef _richcmp_(left, right, int op): 

""" 

Compare left and right 

""" 

if left is right: 

return rich_to_bool(op, 0) 

l_domain = left.domain() 

r_domain = right.domain() 

if l_domain != r_domain: 

return richcmp_not_equal(l_domain, r_domain, op) 

l_codomain = left.codomain() 

r_codomain = right.codomain() 

if l_codomain != r_codomain: 

return richcmp_not_equal(l_codomain, r_codomain, op) 

if isinstance(right, FrobeniusEndomorphism_padics): 

return richcmp(left._power, (<FrobeniusEndomorphism_padics>right)._power, op) 

try: 

for x in l_domain.gens(): 

lx = left(x) 

rx = right(x) 

if lx != rx: 

return richcmp_not_equal(lx, rx, op) 

return rich_to_bool(op, 0) 

except (AttributeError, NotImplementedError): 

raise NotImplementedError