Coverage for local/lib/python2.7/site-packages/sage/rings/padics/CA_template.pxi : 78%
Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
|
""" Capped absolute template for complete discrete valuation rings
In order to use this template you need to write a linkage file and gluing file. For an example see mpz_linkage.pxi (linkage file) and padic_capped_absolute_element.pyx (gluing file).
The linkage file implements a common API that is then used in the class CAElement defined here. See the documentation of mpz_linkage.pxi for the functions needed.
The gluing file does the following:
- ctypedef's celement to be the appropriate type (e.g. mpz_t) - includes the linkage file - includes this template - defines a concrete class inheriting from CAElement, and implements any desired extra methods
AUTHORS:
- David Roe (2012-3-1) -- initial version """
#***************************************************************************** # Copyright (C) 2012 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/ #*****************************************************************************
# This file implements common functionality among template elements include "padic_template_element.pxi"
from sage.structure.element cimport Element from sage.rings.padics.common_conversion cimport comb_prec, _process_args_and_kwds from sage.rings.integer_ring import ZZ from sage.rings.rational_field import QQ from sage.categories.sets_cat import Sets from sage.categories.sets_with_partial_maps import SetsWithPartialMaps from sage.categories.homset import Hom
cdef class CAElement(pAdicTemplateElement): cdef int _set(self, x, long val, long xprec, absprec, relprec) except -1: """ Sets the value of this element from given defining data.
This function is intended for use in conversion, and should not be called on an element created with :meth:`_new_c`.
INPUT:
- ``x`` -- data defining a `p`-adic element: int, long, Integer, Rational, other `p`-adic element...
- ``val`` -- the valuation of the resulting element
- ``xprec -- an inherent precision of ``x``
- ``absprec`` -- an absolute precision cap for this element
- ``relprec`` -- a relative precision cap for this element
TESTS::
sage: R = ZpCA(5) sage: a = R(17,5); a #indirect doctest 2 + 3*5 + O(5^5) sage: a = R(75, absprec = 5, relprec = 4); a #indirect doctest 3*5^2 + O(5^5) sage: a = R(25/9, absprec = 5); a #indirect doctest 4*5^2 + 2*5^3 + O(5^5) sage: a = R(25/9, absprec = 5, relprec = 4); a #indirect doctest 4*5^2 + 2*5^3 + O(5^5) """ else: cshift(self.value, (<CAElement>x).value, 0, self.absprec, self.prime_pow, True) else:
cdef CAElement _new_c(self): """ Creates a new element with the same basic info.
TESTS::
sage: R = ZpCA(5); R(6,5) * R(7,8) #indirect doctest 2 + 3*5 + 5^2 + O(5^5) """
cdef pAdicTemplateElement _new_with_value(self, celement value, long absprec): """ Creates a new element with a given value and absolute precision.
Used by code that doesn't know the precision type. """
cdef int _get_unit(self, celement value) except -1: """ Sets ``value`` to the unit of this p-adic element. """
cdef int check_preccap(self) except -1: """ Checks that this element doesn't have precision higher than allowed by the precision cap.
TESTS::
sage: ZpCA(5)(1).lift_to_precision(30) # indirect doctest Traceback (most recent call last): ... PrecisionError: Precision higher than allowed by the precision cap. """
def __copy__(self): """ Return a copy of this element.
EXAMPLES::
sage: a = ZpCA(5,6)(17); b = copy(a) sage: a == b True sage: a is b False """
def __dealloc__(self): """ Deallocate the underlying data structure.
TESTS::
sage: R = ZpCA(5) sage: a = R(17) sage: del(a) """
def __reduce__(self): """ Return a tuple of a function and data that can be used to unpickle this element.
TESTS::
sage: a = ZpCA(5)(-3) sage: type(a) <type 'sage.rings.padics.padic_capped_absolute_element.pAdicCappedAbsoluteElement'> sage: loads(dumps(a)) == a True """
cpdef _neg_(self): """ Return the additive inverse of this element.
EXAMPLES::
sage: R = Zp(5, prec=10, type='capped-abs') sage: a = R(1) sage: -a #indirect doctest 4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10) """
cpdef _add_(self, _right): """ Return the sum of this element and ``_right``.
EXAMPLES::
sage: R = ZpCA(13, 4) sage: R(2) + R(3) #indirect doctest 5 + O(13^4) sage: R(12) + R(1) 13 + O(13^4)
Check that :trac:`20245` is resolved::
sage: R(1,1) + R(169,3) 1 + O(13) """
cpdef _sub_(self, _right): """ Return the difference of this element and ``_right``.
EXAMPLES::
sage: R = ZpCA(13, 4) sage: R(10) - R(10) #indirect doctest O(13^4) sage: R(10) - R(11) 12 + 12*13 + 12*13^2 + 12*13^3 + O(13^4) """
def __invert__(self): """ Return the multiplicative inverse of this element.
.. NOTE::
The result always lives in the fraction field, even if this element is a unit.
EXAMPLES::
sage: R = ZpCA(17) sage: ~R(-1) == R(-1) True sage: ~R(5) * 5 1 + O(17^20) sage: ~R(5) 7 + 3*17 + 10*17^2 + 13*17^3 + 6*17^4 + 3*17^5 + 10*17^6 + 13*17^7 + 6*17^8 + 3*17^9 + 10*17^10 + 13*17^11 + 6*17^12 + 3*17^13 + 10*17^14 + 13*17^15 + 6*17^16 + 3*17^17 + 10*17^18 + 13*17^19 + O(17^20) sage: ~R(-1) == R(-1) #indirect doctest True """
cpdef _mul_(self, _right): """ Return the product of this element and ``_right``.
EXAMPLES::
sage: R = ZpCA(5) sage: a = R(20,5); b = R(75, 4); a * b #indirect doctest 2*5^3 + 2*5^4 + O(5^5) """ cdef long vals, valr else:
cpdef _div_(self, right): """ Return the quotient of this element and ``right``.
.. NOTE::
The result always lives in the fraction field, even if ``right`` is a unit.
EXAMPLES::
sage: R = ZpCA(13, 4) sage: R(2) / R(3) # indirect doctest 5 + 4*13 + 4*13^2 + 4*13^3 + O(13^4) sage: a = R(169 * 2) / R(13); a 2*13 + O(13^3) sage: R(13) / R(169 * 2) 7*13^-1 + 6 + O(13) sage: ~a 7*13^-1 + 6 + O(13) sage: 1 / a 7*13^-1 + 6 + O(13) """
def __pow__(CAElement self, _right, dummy): """ Exponentiation.
When ``right`` is divisible by `p` then one can get more precision than expected. See the documentation in :mod:`sage.rings.padics.CR_template.pxi` for more details.
For `p`-adic exponents, `a^b` is defined as `\exp(b \log(a))`. Since the `p`-adic logarithm is defined for `a` a unit, the same is true of exponentiation.
INPUT:
- ``_right`` -- currently integers and `p`-adic exponents are supported.
- ``dummy`` -- not used (Python's ``__pow__`` signature includes it)
EXAMPLES::
sage: R = ZpCA(11, 5) sage: R(1/2)^5 10 + 7*11 + 11^2 + 5*11^3 + 4*11^4 + O(11^5) sage: R(1/32) 10 + 7*11 + 11^2 + 5*11^3 + 4*11^4 + O(11^5) sage: R(1/2)^5 == R(1/32) True sage: R(3)^1000 1 + 4*11^2 + 3*11^3 + 7*11^4 + O(11^5)
`p`-adic exponents are supported::
sage: R = ZpCA(11, 5, print_mode='terse') sage: a = R(3/14, 3); b = R(8/9); c = R(11,2) sage: a 1046 + O(11^3) sage: b 35790 + O(11^5) sage: a^b 177 + O(11^3) sage: a^35790 177 + O(11^3) sage: a^c 848 + O(11^3) sage: (a.log()*c).exp() 848 + O(11^3)
sage: R = ZpCA(19, 5, print_mode='series') sage: a = R(8/5,4); a 13 + 7*19 + 11*19^2 + 7*19^3 + O(19^4) sage: a^(R(19/7)) 1 + 14*19^2 + 11*19^3 + 13*19^4 + O(19^5) sage: (a // R.teichmuller(13))^(R(19/7)) 1 + 14*19^2 + 11*19^3 + 13*19^4 + O(19^5) sage: (a.log() * 19/7).exp() 1 + 14*19^2 + 11*19^3 + 13*19^4 + O(19^5) """ cdef long relprec, val, rval cdef mpz_t tmp cdef Integer right cdef CAElement pright, ans cdef bint exact_exp base = ~self return base.__pow__(-_right, dummy) ## For extension elements, we need to switch to the ## fraction field sometimes in highly ramified extensions. else: self, _right = canonical_coercion(self, _right) return self.__pow__(_right, dummy) # return 1 to maximum precision # We may assume from above that right > 0 if exact. # So we return a zero of precision right * self.ordp. ans.absprec = 0 else: else: ans.absprec = mpz_get_si(tmp) else: if not exact_exp and self.absprec > 0: raise ValueError("in order to raise to a p-adic exponent, base must be a unit") raise PrecisionError("Need more precision") else: # exact_pow_helper is defined in padic_template_element.pxi ans.absprec = self.prime_pow.ram_prec_cap csetzero(ans.value, ans.prime_pow) else: else: # We may assume that val = 0 since the following will quickly raise an error otherwise. # padic_pow_helper is defined in padic_template_element.pxi
cdef pAdicTemplateElement _lshift_c(self, long shift): """ Multiplies by `\pi^{\mbox{shift}}`.
Negative shifts may truncate the result.
TESTS::
sage: R = ZpCA(5); a = R(17); a << 2 2*5^2 + 3*5^3 + O(5^20) sage: a << -1 3 + O(5^19) sage: a << 0 == a True sage: a << 400 O(5^20) sage: a << -400 O(5^0) """ else:
cdef pAdicTemplateElement _rshift_c(self, long shift): """ Divides by ``\pi^{\mbox{shift}}``.
Positive shifts may truncate the result.
TESTS::
sage: R = ZpCA(5); a = R(77); a >> 1 3*5 + O(5^19) sage: a >> -1 2*5 + 3*5^3 + O(5^20) sage: a >> 0 == a True sage: a >> 400 O(5^0) sage: a >> -400 O(5^20) """ else:
def add_bigoh(self, absprec): """ Returns a new element with absolute precision decreased to ``absprec``. The precision never increases.
INPUT:
- ``absprec`` -- an integer or infinity
OUTPUT:
``self`` with precision set to the minimum of ``self's`` precision and ``prec``
EXAMPLES::
sage: R = Zp(7,4,'capped-abs','series'); a = R(8); a.add_bigoh(1) 1 + O(7)
sage: k = ZpCA(3,5) sage: a = k(41); a 2 + 3 + 3^2 + 3^3 + O(3^5) sage: a.add_bigoh(7) 2 + 3 + 3^2 + 3^3 + O(3^5) sage: a.add_bigoh(3) 2 + 3 + 3^2 + O(3^3)
TESTS:
Verify that :trac:`13591` has been resolved::
sage: k(3).add_bigoh(-1) O(3^-1)
""" cdef long aprec, newprec else: absprec = Integer(absprec) raise ValueError("absprec must fit into a signed long") else: else:
cpdef bint _is_exact_zero(self) except -1: """ Tests whether this element is an exact zero, which is always False for capped absolute elements.
This function exists for compatibility with capped relative elements.
EXAMPLES::
sage: ZpCA(5)(0)._is_exact_zero() False """
cpdef bint _is_inexact_zero(self) except -1: """ Determines whether this element is indistinguishable from zero.
EXAMPLES::
sage: R = ZpCA(7, 5) sage: R(7^5)._is_inexact_zero() True sage: R(0,4)._is_inexact_zero() True sage: R(0)._is_inexact_zero() True """
def is_zero(self, absprec = None): r""" Determines whether this element is zero modulo `\pi^{\mbox{absprec}}`.
If ``absprec is None``, returns ``True`` if this element is indistinguishable from zero.
INPUT:
- ``absprec`` -- an integer, infinity, or ``None``
EXAMPLES::
sage: R = ZpCA(17, 6) sage: R(0).is_zero() True sage: R(17^6).is_zero() True sage: R(17^2).is_zero(absprec=2) True sage: R(17^6).is_zero(absprec=10) Traceback (most recent call last): ... PrecisionError: Not enough precision to determine if element is zero """ raise PrecisionError("Not enough precision to determine if element is zero") if iszero and absprec > self.absprec: raise PrecisionError("Not enough precision to determine if element is zero") return val >= absprec absprec = Integer(absprec) else:
def __nonzero__(self): """ Whether this element should be considered true in a boolean context.
For most applications, explicitly specifying the power of p modulo which the element is supposed to be nonzero is preferable.
EXAMPLES::
sage: R = ZpCA(5); a = R(0); b = R(0,5); c = R(75) sage: bool(a), bool(b), bool(c) (False, False, True) """
def is_equal_to(self, _right, absprec=None): r""" Determines whether the inputs are equal modulo `\pi^{\mbox{absprec}}`.
INPUT:
- ``right`` -- a `p`-adic element with the same parent
- ``absprec`` -- an integer, infinity, or ``None``
EXAMPLES::
sage: R = ZpCA(2, 6) sage: R(13).is_equal_to(R(13)) True sage: R(13).is_equal_to(R(13+2^10)) True sage: R(13).is_equal_to(R(17), 2) True sage: R(13).is_equal_to(R(17), 5) False sage: R(13).is_equal_to(R(13+2^10),absprec=10) Traceback (most recent call last): ... PrecisionError: Elements not known to enough precision """ raise PrecisionError("Elements not known to enough precision") cdef CAElement right cdef long aprec, rprec, sval, rval else: right = self.parent()(_right) else: absprec = Integer(absprec) if mpz_sgn((<Integer>absprec).value) < 0: return True else: raise PrecisionError("Elements not known to enough precision")
cdef int _cmp_units(self, pAdicGenericElement _right) except -2: """ This function is used in comparing `p`-adic elements.
EXAMPLES::
sage: R = ZpCA(37) sage: R(17) == R(17+37^6) # indirect doctest False """ return 0
cdef pAdicTemplateElement lift_to_precision_c(self, long absprec): """ Returns an arbitrary lift of this element to higher precision.
If ``absprec`` is less than the absolute precision of this element this function will return the input element.
INPUT:
- ``absprec`` -- an integer, at most the precision cap of the parent.
EXAMPLES::
sage: R = ZpCA(19) sage: a = R(19, 7); a 19 + O(19^7) sage: a.lift_to_precision(12) # indirect doctest 19 + O(19^12) sage: a.lift_to_precision(4) is a True """ cdef CAElement ans
def _cache_key(self): r""" Return a hashable key which identifies this element for caching.
TESTS::
sage: R.<a> = ZqCA(9) sage: (9*a)._cache_key() (..., ((), (), (0, 1)), 20)
.. SEEALSO::
:meth:`sage.misc.cachefunc._cache_key` """
def _teichmuller_set_unsafe(self): """ Sets this element to the Teichmuller representative with the same residue.
.. WARNING::
This function modifies the element, which is not safe. Elements are supposed to be immutable.
EXAMPLES::
sage: R = ZpCA(17,5); a = R(11) sage: a 11 + O(17^5) sage: a._teichmuller_set_unsafe(); a 11 + 14*17 + 2*17^2 + 12*17^3 + 15*17^4 + O(17^5) sage: E = a.expansion(lift_mode='teichmuller'); E 17-adic expansion of 11 + 14*17 + 2*17^2 + 12*17^3 + 15*17^4 + O(17^5) (teichmuller) sage: list(E) [11 + 14*17 + 2*17^2 + 12*17^3 + 15*17^4 + O(17^5), O(17^5), O(17^5), O(17^5), O(17^5)]
Note that if you set an element which is congruent to 0 you get 0 to maximum precision::
sage: b = R(17*5); b 5*17 + O(17^5) sage: b._teichmuller_set_unsafe(); b O(17^5) """ raise ValueError("not enough precision") else:
def polynomial(self, var='x'): """ Returns a polynomial over the base ring that yields this element when evaluated at the generator of the parent.
INPUT:
- ``var`` -- string, the variable name for the polynomial
EXAMPLES::
sage: R.<a> = ZqCA(5^3) sage: a.polynomial() (1 + O(5^20))*x + (O(5^20)) sage: a.polynomial(var='y') (1 + O(5^20))*y + (O(5^20)) sage: (5*a^2 + R(25, 4)).polynomial() (5 + O(5^4))*x^2 + (O(5^4))*x + (5^2 + O(5^4)) """ else: L = [R(c, (prec - i - 1) // e + 1) for i, c in enumerate(L)]
def precision_absolute(self): """ The absolute precision of this element.
This is the power of the maximal ideal modulo which this element is defined.
EXAMPLES::
sage: R = Zp(7,4,'capped-abs'); a = R(7); a.precision_absolute() 4 """
def precision_relative(self): """ The relative precision of this element.
This is the power of the maximal ideal modulo which the unit part of this element is defined.
EXAMPLES::
sage: R = Zp(7,4,'capped-abs'); a = R(7); a.precision_relative() 3 """
cpdef pAdicTemplateElement unit_part(CAElement self): r""" Returns the unit part of this element.
EXAMPLES::
sage: R = Zp(17,4,'capped-abs', 'val-unit') sage: a = R(18*17) sage: a.unit_part() 18 + O(17^3) sage: type(a) <type 'sage.rings.padics.padic_capped_absolute_element.pAdicCappedAbsoluteElement'> sage: R(0).unit_part() O(17^0) """
cdef long valuation_c(self): """ Returns the valuation of this element.
TESTS::
sage: R = ZpCA(5) sage: R(5^5*1827).valuation() 5 sage: R(1).valuation() 0 sage: R(2).valuation() 0 sage: R(5).valuation() 1 sage: R(10).valuation() 1 sage: R(25).valuation() 2 sage: R(50).valuation() 2 sage: R(0).valuation() 20 sage: R(0,6).valuation() 6 """
cpdef val_unit(self): """ Returns a 2-tuple, the first element set to the valuation of this element, and the second to the unit part of this element.
For a zero element, the unit part is ``O(p^0)``.
EXAMPLES::
sage: R = ZpCA(5) sage: a = R(75, 6); b = a - a sage: a.val_unit() (2, 3 + O(5^4)) sage: b.val_unit() (6, O(5^0)) """
def __hash__(self): """ Hashing.
.. WARNING::
Hashing of `p`-adic elements will likely be deprecated soon. See :trac:`11895`.
EXAMPLES::
sage: R = ZpCA(11, 5) sage: hash(R(3)) == hash(3) True """
cdef class pAdicCoercion_ZZ_CA(RingHomomorphism): """ The canonical inclusion from the ring of integers to a capped absolute ring.
EXAMPLES::
sage: f = ZpCA(5).coerce_map_from(ZZ); f Ring morphism: From: Integer Ring To: 5-adic Ring with capped absolute precision 20
TESTS::
sage: TestSuite(f).run()
""" def __init__(self, R): """ Initialization.
EXAMPLES::
sage: f = ZpCA(5).coerce_map_from(ZZ); type(f) <type 'sage.rings.padics.padic_capped_absolute_element.pAdicCoercion_ZZ_CA'> """
cdef dict _extra_slots(self): """ Helper for copying and pickling.
EXAMPLES::
sage: f = ZpCA(5).coerce_map_from(ZZ) sage: g = copy(f) # indirect doctest sage: g == f True sage: g(6) 1 + 5 + O(5^20) sage: f(6) == g(6) True """
cdef _update_slots(self, dict _slots): """ Helper for copying and pickling.
EXAMPLES::
sage: f = ZpCA(5).coerce_map_from(ZZ) sage: g = copy(f) # indirect doctest sage: g == f True sage: g(6) 1 + 5 + O(5^20) sage: f(6) == g(6) True """
cpdef Element _call_(self, x): """ Evaluation.
EXAMPLES::
sage: f = ZpCA(5).coerce_map_from(ZZ) sage: f(0).parent() 5-adic Ring with capped absolute precision 20 sage: f(5) 5 + O(5^20) """
cpdef Element _call_with_args(self, x, args=(), kwds={}): """ This function is used when some precision cap is passed in (relative or absolute or both).
See the documentation for :meth:`pAdicCappedAbsoluteElement.__init__` for more details.
EXAMPLES::
sage: R = ZpCA(5,4) sage: type(R(10,2)) <type 'sage.rings.padics.padic_capped_absolute_element.pAdicCappedAbsoluteElement'> sage: R(10,2) 2*5 + O(5^2) sage: R(10,3,1) 2*5 + O(5^2) sage: R(10,absprec=2) 2*5 + O(5^2) sage: R(10,relprec=2) 2*5 + O(5^3) sage: R(10,absprec=1) O(5) sage: R(10,empty=True) O(5^0) """ cdef long val, aprec, rprec cdef CAElement ans else: else:
def section(self): """ Returns a map back to the ring of integers that approximates an element by an integer.
EXAMPLES::
sage: f = ZpCA(5).coerce_map_from(ZZ).section() sage: f(ZpCA(5)(-1)) - 5^20 -1 """ import copy self._section = copy.copy(self._section)
cdef class pAdicConvert_CA_ZZ(RingMap): """ The map from a capped absolute ring back to the ring of integers that returns the smallest non-negative integer approximation to its input which is accurate up to the precision.
Raises a ``ValueError`` if the input is not in the closure of the image of the ring of integers.
EXAMPLES::
sage: f = ZpCA(5).coerce_map_from(ZZ).section(); f Set-theoretic ring morphism: From: 5-adic Ring with capped absolute precision 20 To: Integer Ring """ def __init__(self, R): """ Initialization.
EXAMPLES::
sage: f = ZpCA(5).coerce_map_from(ZZ).section(); type(f) <type 'sage.rings.padics.padic_capped_absolute_element.pAdicConvert_CA_ZZ'> sage: f.category() Category of homsets of sets """ else:
cpdef Element _call_(self, _x): """ Evaluation.
EXAMPLES::
sage: f = ZpCA(5).coerce_map_from(ZZ).section() sage: f(ZpCA(5)(-1)) - 5^20 -1 sage: f(ZpCA(5)(0)) 0 """
cdef class pAdicConvert_QQ_CA(Morphism): """ The inclusion map from the rationals to a capped absolute ring that is defined on all elements with non-negative `p`-adic valuation.
EXAMPLES::
sage: f = ZpCA(5).convert_map_from(QQ); f Generic morphism: From: Rational Field To: 5-adic Ring with capped absolute precision 20 """ def __init__(self, R): """ Initialization.
EXAMPLES::
sage: f = ZpCA(5).convert_map_from(QQ); type(f) <type 'sage.rings.padics.padic_capped_absolute_element.pAdicConvert_QQ_CA'> """
cdef dict _extra_slots(self): """ Helper for copying and pickling.
EXAMPLES::
sage: f = ZpCA(5).convert_map_from(QQ) sage: g = copy(f) # indirect doctest sage: g == f # todo: comparison not implemented True sage: g(1/6) 1 + 4*5 + 4*5^3 + 4*5^5 + 4*5^7 + 4*5^9 + 4*5^11 + 4*5^13 + 4*5^15 + 4*5^17 + 4*5^19 + O(5^20) sage: g(1/6) == f(1/6) True """
cdef _update_slots(self, dict _slots): """ Helper for copying and pickling.
EXAMPLES::
sage: f = ZpCA(5).convert_map_from(QQ) sage: g = copy(f) # indirect doctest sage: g == f # todo: comparison not implemented True sage: g(1/6) 1 + 4*5 + 4*5^3 + 4*5^5 + 4*5^7 + 4*5^9 + 4*5^11 + 4*5^13 + 4*5^15 + 4*5^17 + 4*5^19 + O(5^20) sage: g(1/6) == f(1/6) True """
cpdef Element _call_(self, x): """ Evaluation.
EXAMPLES::
sage: f = ZpCA(5,4).convert_map_from(QQ) sage: f(1/7) 3 + 3*5 + 2*5^3 + O(5^4) sage: f(0) O(5^4) """
cpdef Element _call_with_args(self, x, args=(), kwds={}): """ This function is used when some precision cap is passed in (relative or absolute or both).
See the documentation for :meth:`pAdicCappedAbsoluteElement.__init__` for more details.
EXAMPLES::
sage: R = ZpCA(5,4) sage: type(R(10/3,2)) <type 'sage.rings.padics.padic_capped_absolute_element.pAdicCappedAbsoluteElement'> sage: R(10/3,2) 4*5 + O(5^2) sage: R(10/3,3,1) 4*5 + O(5^2) sage: R(10/3,absprec=2) 4*5 + O(5^2) sage: R(10/3,relprec=2) 4*5 + 5^2 + O(5^3) sage: R(10/3,absprec=1) O(5) sage: R(10/3,empty=True) O(5^0) sage: R(3/100,relprec=3) Traceback (most recent call last): ... ValueError: p divides denominator """ cdef long val, aprec, rprec cdef CAElement ans ans = self._zero._new_c() csetzero(ans.value, ans.prime_pow) ans.absprec = aprec else: else:
cdef class pAdicCoercion_CA_frac_field(RingHomomorphism): """ The canonical inclusion of Zq into its fraction field.
EXAMPLES::
sage: R.<a> = ZqCA(27, implementation='FLINT') sage: K = R.fraction_field() sage: f = K.coerce_map_from(R); f Ring morphism: From: Unramified Extension in a defined by x^3 + 2*x + 1 with capped absolute precision 20 over 3-adic Ring To: Unramified Extension in a defined by x^3 + 2*x + 1 with capped relative precision 20 over 3-adic Field
TESTS::
sage: TestSuite(f).run()
""" def __init__(self, R, K): """ Initialization.
EXAMPLES::
sage: R.<a> = ZqCA(27, implementation='FLINT') sage: K = R.fraction_field() sage: f = K.coerce_map_from(R); type(f) <type 'sage.rings.padics.qadic_flint_CA.pAdicCoercion_CA_frac_field'> """
cpdef Element _call_(self, _x): """ Evaluation.
EXAMPLES::
sage: R.<a> = ZqCA(27, implementation='FLINT') sage: K = R.fraction_field() sage: f = K.coerce_map_from(R) sage: f(a) a + O(3^20) """
cpdef Element _call_with_args(self, _x, args=(), kwds={}): """ This function is used when some precision cap is passed in (relative or absolute or both).
See the documentation for :meth:`pAdicCappedAbsoluteElement.__init__` for more details.
EXAMPLES::
sage: R.<a> = ZqCA(27, implementation='FLINT') sage: K = R.fraction_field() sage: f = K.coerce_map_from(R) sage: f(a, 3) a + O(3^3) sage: b = 9*a sage: f(b, 3) a*3^2 + O(3^3) sage: f(b, 4, 1) a*3^2 + O(3^3) sage: f(b, 4, 3) a*3^2 + O(3^4) sage: f(b, absprec=4) a*3^2 + O(3^4) sage: f(b, relprec=3) a*3^2 + O(3^5) sage: f(b, absprec=1) O(3) sage: f(R(0)) O(3^20) """ cdef long aprec, rprec
def section(self): """ Returns a map back to the ring that converts elements of non-negative valuation.
EXAMPLES::
sage: R.<a> = ZqCA(27, implementation='FLINT') sage: K = R.fraction_field() sage: f = K.coerce_map_from(R) sage: f(K.gen()) a + O(3^20) """
cdef dict _extra_slots(self): """ Helper for copying and pickling.
TESTS::
sage: R.<a> = ZqCA(27, implementation='FLINT') sage: K = R.fraction_field() sage: f = K.coerce_map_from(R) sage: g = copy(f) # indirect doctest sage: g Ring morphism: From: Unramified Extension in a defined by x^3 + 2*x + 1 with capped absolute precision 20 over 3-adic Ring To: Unramified Extension in a defined by x^3 + 2*x + 1 with capped relative precision 20 over 3-adic Field sage: g == f True sage: g is f False sage: g(a) a + O(3^20) sage: g(a) == f(a) True
"""
cdef _update_slots(self, dict _slots): """ Helper for copying and pickling.
TESTS::
sage: R.<a> = ZqCA(9, implementation='FLINT') sage: K = R.fraction_field() sage: f = K.coerce_map_from(R) sage: g = copy(f) # indirect doctest sage: g Ring morphism: From: Unramified Extension in a defined by x^2 + 2*x + 2 with capped absolute precision 20 over 3-adic Ring To: Unramified Extension in a defined by x^2 + 2*x + 2 with capped relative precision 20 over 3-adic Field sage: g == f True sage: g is f False sage: g(a) a + O(3^20) sage: g(a) == f(a) True
"""
def is_injective(self): r""" Return whether this map is injective.
EXAMPLES::
sage: R.<a> = ZqCA(9, implementation='FLINT') sage: K = R.fraction_field() sage: f = K.coerce_map_from(R) sage: f.is_injective() True
"""
def is_surjective(self): r""" Return whether this map is surjective.
EXAMPLES::
sage: R.<a> = ZqCA(9, implementation='FLINT') sage: K = R.fraction_field() sage: f = K.coerce_map_from(R) sage: f.is_surjective() False
"""
cdef class pAdicConvert_CA_frac_field(Morphism): """ The section of the inclusion from `\ZZ_q`` to its fraction field.
EXAMPLES::
sage: R.<a> = ZqCA(27, implementation='FLINT') sage: K = R.fraction_field() sage: f = R.convert_map_from(K); f Generic morphism: From: Unramified Extension in a defined by x^3 + 2*x + 1 with capped relative precision 20 over 3-adic Field To: Unramified Extension in a defined by x^3 + 2*x + 1 with capped absolute precision 20 over 3-adic Ring """ def __init__(self, K, R): """ Initialization.
EXAMPLES::
sage: R.<a> = ZqCA(27, implementation='FLINT') sage: K = R.fraction_field() sage: f = R.convert_map_from(K); type(f) <type 'sage.rings.padics.qadic_flint_CA.pAdicConvert_CA_frac_field'> """
cpdef Element _call_(self, _x): """ Evaluation.
EXAMPLES::
sage: R.<a> = ZqCA(27, implementation='FLINT') sage: K = R.fraction_field() sage: f = R.convert_map_from(K) sage: f(K.gen()) a + O(3^20) """ else:
cpdef Element _call_with_args(self, _x, args=(), kwds={}): """ This function is used when some precision cap is passed in (relative or absolute or both).
See the documentation for :meth:`pAdicCappedAbsoluteElement.__init__` for more details.
EXAMPLES::
sage: R.<a> = ZqCA(27, implementation='FLINT') sage: K = R.fraction_field() sage: f = R.convert_map_from(K); a = K(a) sage: f(a, 3) a + O(3^3) sage: b = 9*a sage: f(b, 3) a*3^2 + O(3^3) sage: f(b, 4, 1) a*3^2 + O(3^3) sage: f(b, 4, 3) a*3^2 + O(3^4) sage: f(b, absprec=4) a*3^2 + O(3^4) sage: f(b, relprec=3) a*3^2 + O(3^5) sage: f(b, absprec=1) O(3) sage: f(K(0)) O(3^20) """ cdef long aprec, rprec else:
cdef dict _extra_slots(self): """ Helper for copying and pickling.
TESTS::
sage: R.<a> = ZqCA(27, implementation='FLINT') sage: K = R.fraction_field() sage: f = R.convert_map_from(K) sage: a = K(a) sage: g = copy(f) # indirect doctest sage: g Generic morphism: From: Unramified Extension in a defined by x^3 + 2*x + 1 with capped relative precision 20 over 3-adic Field To: Unramified Extension in a defined by x^3 + 2*x + 1 with capped absolute precision 20 over 3-adic Ring sage: g == f True sage: g is f False sage: g(a) a + O(3^20) sage: g(a) == f(a) True """
cdef _update_slots(self, dict _slots): """ Helper for copying and pickling.
TESTS::
sage: R.<a> = ZqCA(9, implementation='FLINT') sage: K = R.fraction_field() sage: f = R.convert_map_from(K) sage: a = f(a) sage: g = copy(f) # indirect doctest sage: g Generic morphism: From: Unramified Extension in a defined by x^2 + 2*x + 2 with capped relative precision 20 over 3-adic Field To: Unramified Extension in a defined by x^2 + 2*x + 2 with capped absolute precision 20 over 3-adic Ring sage: g == f True sage: g is f False sage: g(a) a + O(3^20) sage: g(a) == f(a) True
"""
def unpickle_cae_v2(cls, parent, value, absprec): """ Unpickle capped absolute elements.
INPUT:
- ``cls`` -- the class of the capped absolute element.
- ``parent`` -- the parent, a `p`-adic ring
- ``value`` -- a Python object wrapping a celement, of the kind accepted by the cunpickle function.
- ``absprec`` -- a Python int or Sage integer.
EXAMPLES::
sage: from sage.rings.padics.padic_capped_absolute_element import unpickle_cae_v2, pAdicCappedAbsoluteElement sage: R = ZpCA(5,8) sage: a = unpickle_cae_v2(pAdicCappedAbsoluteElement, R, 42, int(6)); a 2 + 3*5 + 5^2 + O(5^6) sage: a.parent() is R True """ |