$p$ -Adic `ZZ_pX` CR Element.¶

This file implements elements of Eisenstein and unramified extensions of $\mathbb{Z}_p$ and $\mathbb{Q}_p$ with capped relative precision.

For the parent class see padic_extension_leaves.pyx.

The underlying implementation is through NTL’s ZZ_pX class. Each element contains the following data:

ordp (long) – A power of the uniformizer to scale the unit by. For unramified extensions this uniformizer is $p$ , for Eisenstein extensions it is not. A value equal to the maximum value of a long indicates that the element is an exact zero.
relprec (long) – A signed integer giving the precision to which this element is defined. For nonzero relprec, the absolute value gives the power of the uniformizer modulo which the unit is defined. A positive value indicates that the element is normalized (ie unit is actually a unit: in the case of Eisenstein extensions the constant term is not divisible by $p$ , in the case of unramified extensions that there is at least one coefficient that is not divisible by $p$ ). A negative value indicates that the element may or may not be normalized. A zero value indicates that the element is zero to some precision. If so, ordp gives the absolute precision of the element. If ordp is greater than maxordp, then the element is an exact zero.
unit (ZZ_pX_c) – An ntl ZZ_pX storing the unit part. The variable $x$ is the uniformizer in the case of Eisenstein extensions. If the element is not normalized, the unit may or may not actually be a unit. This ZZ_pX is created with global ntl modulus determined by the absolute value of relprec. If relprec is 0, unit is not initialized, or destructed if normalized and found to be zero. Otherwise, let $r$ be relprec and $e$ be the ramification index over $\mathbb{Q}_p$ or $\mathbb{Z}_p$ . Then the modulus of unit is given by $p^{ceil(r/e)}$ . Note that all kinds of problems arise if you try to mix moduli. ZZ_pX_conv_modulus gives a semi-safe way to convert between different moduli without having to pass through ZZX (see sage/libs/ntl/decl.pxi and c_lib/src/ntl_wrap.cpp)
prime_pow (some subclass of PowComputer_ZZ_pX) – a class, identical among all elements with the same parent, holding common data.
- prime_pow.deg – The degree of the extension
- prime_pow.e – The ramification index
- prime_pow.f – The inertia degree
- prime_pow.prec_cap – the unramified precision cap. For Eisenstein extensions this is the smallest power of p that is zero.
- prime_pow.ram_prec_cap – the ramified precision cap. For Eisenstein extensions this will be the smallest power of $x$ that is indistinguishable from zero.
- prime_pow.pow_ZZ_tmp, prime_pow.pow_mpz_t_tmp``, prime_pow.pow_Integer – functions for accessing powers of $p$ . The first two return pointers. See sage/rings/padics/pow_computer_ext for examples and important warnings.
- prime_pow.get_context, prime_pow.get_context_capdiv, prime_pow.get_top_context – obtain an ntl_ZZ_pContext_class corresponding to $p^n$ . The capdiv version divides by prime_pow.e as appropriate. top_context corresponds to $p^{prec_cap}$ .
- prime_pow.restore_context, prime_pow.restore_context_capdiv, prime_pow.restore_top_context – restores the given context.
- prime_pow.get_modulus, get_modulus_capdiv, get_top_modulus – Returns a ZZ_pX_Modulus_c* pointing to a polynomial modulus defined modulo $p^n$ (appropriately divided by prime_pow.e in the capdiv case).

EXAMPLES:

An Eisenstein extension:

sage: R = Zp(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f); W
Eisenstein Extension of 5-adic Ring with capped relative precision 5 in w defined by (1 + O(5^5))*x^5 + (O(5^6))*x^4 + (3*5^2 + O(5^6))*x^3 + (2*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6))*x^2 + (5^3 + O(5^6))*x + (4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6))
sage: z = (1+w)^5; z
1 + w^5 + w^6 + 2*w^7 + 4*w^8 + 3*w^10 + w^12 + 4*w^13 + 4*w^14 + 4*w^15 + 4*w^16 + 4*w^17 + 4*w^20 + w^21 + 4*w^24 + O(w^25)
sage: y = z >> 1; y
w^4 + w^5 + 2*w^6 + 4*w^7 + 3*w^9 + w^11 + 4*w^12 + 4*w^13 + 4*w^14 + 4*w^15 + 4*w^16 + 4*w^19 + w^20 + 4*w^23 + O(w^24)
sage: y.valuation()
4
sage: y.precision_relative()
20
sage: y.precision_absolute()
24
sage: z - (y << 1)
1 + O(w^25)
sage: (1/w)^12+w
w^-12 + w + O(w^13)
sage: (1/w).parent()
Eisenstein Extension of 5-adic Field with capped relative precision 5 in w defined by (1 + O(5^5))*x^5 + (O(5^6))*x^4 + (3*5^2 + O(5^6))*x^3 + (2*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6))*x^2 + (5^3 + O(5^6))*x + (4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + O(5^6))

An unramified extension:

sage: g = x^3 + 3*x + 3
sage: A.<a> = R.ext(g)
sage: z = (1+a)^5; z
(2*a^2 + 4*a) + (3*a^2 + 3*a + 1)*5 + (4*a^2 + 3*a + 4)*5^2 + (4*a^2 + 4*a + 4)*5^3 + (4*a^2 + 4*a + 4)*5^4 + O(5^5)
sage: z - 1 - 5*a - 10*a^2 - 10*a^3 - 5*a^4 - a^5
O(5^5)
sage: y = z >> 1; y
(3*a^2 + 3*a + 1) + (4*a^2 + 3*a + 4)*5 + (4*a^2 + 4*a + 4)*5^2 + (4*a^2 + 4*a + 4)*5^3 + O(5^4)
sage: 1/a
(3*a^2 + 4) + (a^2 + 4)*5 + (3*a^2 + 4)*5^2 + (a^2 + 4)*5^3 + (3*a^2 + 4)*5^4 + O(5^5)

Different printing modes:

sage: R = Zp(5, print_mode='digits'); S.<x> = R[]; f = x^5 + 75*x^3 - 15*x^2 + 125*x -5; W.<w> = R.ext(f)
sage: z = (1+w)^5; repr(z)
'...4110403113210310442221311242000111011201102002023303214332011214403232013144001400444441030421100001'
sage: R = Zp(5, print_mode='bars'); S.<x> = R[]; g = x^3 + 3*x + 3; A.<a> = R.ext(g)
sage: z = (1+a)^5; repr(z)
'...[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 4, 4]|[4, 3, 4]|[1, 3, 3]|[0, 4, 2]'
sage: R = Zp(5, print_mode='terse'); S.<x> = R[]; f = x^5 + 75*x^3 - 15*x^2 + 125*x -5; W.<w> = R.ext(f)
sage: z = (1+w)^5; z
6 + 95367431640505*w + 25*w^2 + 95367431640560*w^3 + 5*w^4 + O(w^100)
sage: R = Zp(5, print_mode='val-unit'); S.<x> = R[]; f = x^5 + 75*x^3 - 15*x^2 + 125*x -5; W.<w> = R.ext(f)
sage: y = (1+w)^5 - 1; y
w^5 * (2090041 + 19073486126901*w + 1258902*w^2 + 674*w^3 + 16785*w^4) + O(w^100)

You can get at the underlying ntl unit:

sage: z._ntl_rep()
[6 95367431640505 25 95367431640560 5]
sage: y._ntl_rep()
[2090041 19073486126901 1258902 674 16785]
sage: y._ntl_rep_abs()
([5 95367431640505 25 95367431640560 5], 0)

NOTES:

If you get an error ``internal error: can't grow this
_ntl_gbigint,`` it indicates that moduli are being mixed
inappropriately somewhere.  For example, when calling a function
with a ``ZZ_pX_c`` as an argument, it copies.  If the modulus is not
set to the modulus of the ``ZZ_pX_c``, you can get errors.

AUTHORS:

David Roe (2008-01-01) initial version

sage.rings.padics.padic_ZZ_pX_CR_element.make_ZZpXCRElement(parent, unit, ordp, relprec, version)¶

Unpickling.

EXAMPLES:

sage: R = Zp(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: y = W(775, 19); y
w^10 + 4*w^12 + 2*w^14 + w^15 + 2*w^16 + 4*w^17 + w^18 + O(w^19)
sage: loads(dumps(y)) #indirect doctest
w^10 + 4*w^12 + 2*w^14 + w^15 + 2*w^16 + 4*w^17 + w^18 + O(w^19)

class sage.rings.padics.padic_ZZ_pX_CR_element.pAdicZZpXCRElement¶

Bases: sage.rings.padics.padic_ZZ_pX_element.pAdicZZpXElement

is_equal_to(right, absprec=None)¶

Returns whether self is equal to right modulo self.uniformizer()^absprec.

If absprec is None, returns if self is equal to right modulo the lower of their two precisions.

EXAMPLES:

sage: R = Zp(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: a = W(47); b = W(47 + 25)
sage: a.is_equal_to(b)
False
sage: a.is_equal_to(b, 7)
True

is_zero(absprec=None)¶

Returns whether the valuation of self is at least absprec. If absprec is None, returns if self is indistinguishable from zero.

If self is an inexact zero of valuation less than absprec, raises a PrecisionError.

EXAMPLES:

sage: R = Zp(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: O(w^189).is_zero()
True
sage: W(0).is_zero()
True
sage: a = W(675)
sage: a.is_zero()
False
sage: a.is_zero(7)
True
sage: a.is_zero(21)
False

lift_to_precision(absprec)¶

Returns a pAdicZZpXCRElement congruent to self but with absolute precision at least absprec. If setting absprec that high would violate the precision cap, raises a precision error. If self is an inexact zero and absprec is greater than the maximum allowed valuation, raises an error.

Note that the new digits will not necessarily be zero.

EXAMPLES:

sage: R = Zp(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: a = W(345, 17); a
4*w^5 + 3*w^7 + w^9 + 3*w^10 + 2*w^11 + 4*w^12 + w^13 + 2*w^14 + 2*w^15 + O(w^17)
sage: b = a.lift_to_precision(19); b
4*w^5 + 3*w^7 + w^9 + 3*w^10 + 2*w^11 + 4*w^12 + w^13 + 2*w^14 + 2*w^15 + w^17 + 2*w^18 + O(w^19)
sage: c = a.lift_to_precision(24); c
4*w^5 + 3*w^7 + w^9 + 3*w^10 + 2*w^11 + 4*w^12 + w^13 + 2*w^14 + 2*w^15 + w^17 + 2*w^18 + 4*w^19 + 4*w^20 + 2*w^21 + 4*w^23 + O(w^24)
sage: a._ntl_rep()
[19 35 118 60 121]
sage: b._ntl_rep()
[19 35 118 60 121]
sage: c._ntl_rep()
[19 35 118 60 121]

list(lift_mode='simple')¶

Returns a list giving a series representation of self.

If lift_mode == 'simple' or 'smallest', the returned list will consist of integers (in the Eisenstein case) or a list of lists of integers (in the unramified case). self can be reconstructed as a sum of elements of the list times powers of the uniformiser (in the Eisenstein case), or as a sum of powers of the times polynomials in the generator (in the unramified case).
- If lift_mode == 'simple', all integers will be in the interval $[0,p-1]$ .
- If lift_mode == 'smallest' they will be in the interval $[(1-p)/2, p/2]$ .
If lift_mode == 'teichmuller', returns a list of pAdicZZpXCRElements, all of which are Teichmuller representatives and such that self is the sum of that list times powers of the uniformizer.

Note that zeros are truncated from the returned list if self.parent() is a field, so you must use the valuation function to fully reconstruct self.

EXAMPLES:

sage: R = Zp(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: y = W(775, 19); y
w^10 + 4*w^12 + 2*w^14 + w^15 + 2*w^16 + 4*w^17 + w^18 + O(w^19)
sage: (y>>9).list()
[0, 1, 0, 4, 0, 2, 1, 2, 4, 1]
sage: (y>>9).list('smallest')
[0, 1, 0, -1, 0, 2, 1, 2, 0, 1]
sage: w^10 - w^12 + 2*w^14 + w^15 + 2*w^16 + w^18 + O(w^19)
w^10 + 4*w^12 + 2*w^14 + w^15 + 2*w^16 + 4*w^17 + w^18 + O(w^19)
sage: g = x^3 + 3*x + 3
sage: A.<a> = R.ext(g)
sage: y = 75 + 45*a + 1200*a^2; y
4*a*5 + (3*a^2 + a + 3)*5^2 + 4*a^2*5^3 + a^2*5^4 + O(5^6)
sage: y.list()
[[], [0, 4], [3, 1, 3], [0, 0, 4], [0, 0, 1]]
sage: y.list('smallest')
[[], [0, -1], [-2, 2, -2], [1], [0, 0, 2]]
sage: 5*((-2*5 + 25) + (-1 + 2*5)*a + (-2*5 + 2*125)*a^2)
4*a*5 + (3*a^2 + a + 3)*5^2 + 4*a^2*5^3 + a^2*5^4 + O(5^6)
sage: W(0).list()
[]
sage: W(0,4).list()
[0]
sage: A(0,4).list()
[[]]

log(branch=None, same_ring=False)¶

Compute the $p$ -adic logarithm of any unit. (See below for normalization.)

INPUT:

branch – pAdicZZpXFMElement (default None). The log of the uniformizer. If None, then an error is raised if self is not a unit.
same_ring – bool or pAdicGeneric (default True). When $e > p$ it is possible (even common) that the image of the log map is not contained in the ring of integers. If same_ring is True, then this function will return a value in self.parent() or raise an error if the answer would have negative valuation. If same_ring is False, then this function will raise an error (this behavior is for consistency with other $p$ -adic types). If same_ring is a $p$ -adic field into which this fixed mod ring can be successfully cast, then self is cast into that field and the log is taken there. Note that this casting will assume that self has the full precision possible.

OUTPUT:

The $p$ -adic log of self.

Let $K$ be the parent of self, $\pi$ be a uniformizer of $K$ and $w$ be a generator for the group of roots of unity in $K$ . The usual power series for log with values in the additive group of $K$ only converges for 1-units (units congruent to 1 modulo $\pi$ ). However, there is a unique extension of log to a homomorphism defined on all the units. If $u = a v$ is a unit with $v \equiv 1 \pmod{p}$ , then we define $log(u) = log(v)$ . This is the correct extension because the units $U$ of $K$ split as a product $U = V \times <w>$ , where $V$ is the subgroup of 1-units. The $<w>$ factor is torsion, so must go to 0 under any homomorphism to the torsion free group $(K, +)$ .

NOTES:

What some other systems do with regard to non-1-units:

PARI: Seems to define log the same way as we do.
MAGMA: Gives an error when unit is not a 1-unit.

In addition, if branch is specified, then the log map will work on non-units.

..math

log(pi^k \cdot u) = k \cdot branch + log(u)

ALGORITHM:

Input: Some unit u.

Check that the input is really a unit (i.e., valuation 0), or that branch is specified.
Let $1-x = u^{q-1}$ , which is a 1-unit, where $q$ is the order of the residue field of $K$ .
Use the series expansion

..math

\log(1-x) = F(x) = -x - 1/2*x^2 - 1/3*x^3 - 1/4*x^4 - 1/5*x^5 - \cdots

to compute the logarithm $log(u^{q-1})$ .

Add on terms until $x^k$ is zero modulo the precision cap, and then determine if there are further terms that contribute to the sum (those where $k$ is slightly above the precision cap but divisible by $p$ ).

Then

..math

\log(u) = log(u^{q-1})/(q-1) = F(1-u^{q-1})/(q-1).``

EXAMPLES:

First, the Eisenstein case.:

sage: R = ZpCR(5,5)
sage: S.<x> = R[]
sage: f = x^4 + 15*x^2 + 625*x - 5
sage: W.<w> = R.ext(f)
sage: z = 1 + w^2 + 4*w^7; z
1 + w^2 + 4*w^7 + O(w^20)

Log is actually not implemented completely yet in this case!:

sage: z.log()   # not tested -- what we wish would happen
4*w^2 + 3*w^4 + w^6 + w^7 + w^8 + 4*w^9 + 3*w^10 + w^12 + w^13 + 3*w^14 + w^15 + 4*w^16 + 4*w^17 + 3*w^18 + 3*w^19 + O(w^20)
sage: z.log()   # what does happen
...
NotImplementedError: log is not quite working yet

Check that log is multiplicative:

sage: y = 1 + 3*w^4 + w^5
sage: y.log() + z.log() - (y*z).log()  # not tested -- what we wish would happen
O(w^20)
sage: y.log() + z.log() - (y*z).log()  # what does happen
...
NotImplementedError: log is not quite working yet

Now an unramified example.:

sage: g = x^3 + 3*x + 3
sage: A.<a> = R.ext(g)
sage: b = 1 + 5*(1 + a^2) + 5^3*(3 + 2*a)
sage: b.log()          # not tested -- what should happen
(4*a^2 + 4)*5 + (a^2 + a + 2)*5^2 + (a^2 + 2*a + 4)*5^3 + (a^2 + 2*a + 2)*5^4 + O(5^5)
sage: b.log()          # what does happen
...
NotImplementedError: log is not quite working yet

Check that log is multiplicative:

sage: c = 3 + 5^2*(2 + 4*a)
sage: b.log() + c.log() - (b*c).log()   # not tested -- what should happen
O(5^5)
sage: b.log() + c.log() - (b*c).log()   # what *actually* does happen
...
NotImplementedError: log is not quite working yet

AUTHORS:

David Roe: initial version

TODO:

Currently implemented as $O(N^2)$ . This can be improved to soft- $O(N)$ using algorithm described by Dan Bernstein: http://cr.yp.to/lineartime/multapps-20041007.pdf

matrix_mod_pn()¶

Returns the matrix of right multiplication by the element on the power basis $1, x, x^2, \ldots, x^{d-1}$ for this extension field. Thus the rows of this matrix give the images of each of the $x^i$ . The entries of the matrices are IntegerMod elements, defined modulo p^(self.absprec() / e).

Raises an error if self has negative valuation.

EXAMPLES:

sage: R = ZpCR(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: a = (3+w)^7
sage: a.matrix_mod_pn()
[2757  333 1068  725 2510]
[  50 1507  483  318  725]
[ 500   50 3007 2358  318]
[1590 1375 1695 1032 2358]
[2415  590 2370 2970 1032]

precision_absolute()¶

Returns the absolute precision of self, ie the power of the uniformizer modulo which this element is defined.

EXAMPLES:

sage: R = Zp(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: a = W(75, 19); a
3*w^10 + 2*w^12 + w^14 + w^16 + w^17 + 3*w^18 + O(w^19)
sage: a.valuation()
10
sage: a.precision_absolute()
19
sage: a.precision_relative()
9
sage: a.unit_part()
3 + 2*w^2 + w^4 + w^6 + w^7 + 3*w^8 + O(w^9)
sage: (a.unit_part() - 3).precision_absolute()
9

precision_relative()¶

Returns the relative precision of self, ie the power of the uniformizer modulo which the unit part of self is defined.

EXAMPLES:

sage: R = Zp(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: a = W(75, 19); a
3*w^10 + 2*w^12 + w^14 + w^16 + w^17 + 3*w^18 + O(w^19)
sage: a.valuation()
10
sage: a.precision_absolute()
19
sage: a.precision_relative()
9
sage: a.unit_part()
3 + 2*w^2 + w^4 + w^6 + w^7 + 3*w^8 + O(w^9)

unit_part()¶

Returns the unit part of self, ie self / uniformizer^(self.valuation())

EXAMPLES:

sage: R = Zp(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: a = W(75, 19); a
3*w^10 + 2*w^12 + w^14 + w^16 + w^17 + 3*w^18 + O(w^19)
sage: a.valuation()
10
sage: a.precision_absolute()
19
sage: a.precision_relative()
9
sage: a.unit_part()
3 + 2*w^2 + w^4 + w^6 + w^7 + 3*w^8 + O(w^9)

$p$ -Adic `ZZ_pX` CR Element.¶

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$p$ -Adic `ZZ_pX` CR Element.¶