# $$p$$-Adic Capped Relative Elements¶

$$p$$-Adic Capped Relative Elements

Elements of $$p$$-Adic Rings with Capped Relative Precision

AUTHORS:

• David Roe: initial version, rewriting to use templates (2012-3-1)
• Genya Zaytman: documentation
• David Harvey: doctests

Initialization.

EXAMPLES:

sage: a = Zp(5)(1/2,3); a
3 + 2*5 + 2*5^2 + O(5^3)
sage: type(a)
sage: TestSuite(a).run()


Returns a new element with absolute precision decreased to absprec.

INPUT:

• absprec – an integer or infinity

OUTPUT:

an equal element with precision set to the minimum of self's precision and absprec

EXAMPLE:

sage: R = Zp(7,4,'capped-rel','series'); a = R(8); a.add_bigoh(1)
1 + O(7)
O(7^3)
sage: R = Qp(7,4); a = R(8); a.add_bigoh(1)
1 + O(7)
O(7^3)

The precision never increases::

4 + O(7^2)

Another example that illustrates that the precision does
not increase::

sage: k = Qp(3,5)
sage: a = k(1234123412/3^70); a
2*3^-70 + 3^-69 + 3^-68 + 3^-67 + O(3^-65)
2*3^-70 + 3^-69 + 3^-68 + 3^-67 + O(3^-65)

sage: k = Qp(5,10)
sage: a = k(1/5^3 + 5^2); a
5^-3 + 5^2 + O(5^7)
5^-3 + O(5^2)
5^-3 + O(5^-1)

is_equal_to(_right, absprec=None)

Returns whether self is equal to right modulo $$\pi^{\mbox{absprec}}$$.

If absprec is None, returns True if self and right are equal to the minimum of their precisions.

INPUT:

• right – a $$p$$-adic element
• absprec – an integer, infinity, or None

EXAMPLES:

sage: R = Zp(5, 10); a = R(0); b = R(0, 3); c = R(75, 5)
sage: aa = a + 625; bb = b + 625; cc = c + 625
sage: a.is_equal_to(aa), a.is_equal_to(aa, 4), a.is_equal_to(aa, 5)
(False, True, False)
sage: a.is_equal_to(aa, 15)
Traceback (most recent call last):
...
PrecisionError: Elements not known to enough precision

sage: a.is_equal_to(a, 50000)
True

sage: a.is_equal_to(b), a.is_equal_to(b, 2)
(True, True)
sage: a.is_equal_to(b, 5)
Traceback (most recent call last):
...
PrecisionError: Elements not known to enough precision

sage: b.is_equal_to(b, 5)
Traceback (most recent call last):
...
PrecisionError: Elements not known to enough precision

sage: b.is_equal_to(bb, 3)
True
sage: b.is_equal_to(bb, 4)
Traceback (most recent call last):
...
PrecisionError: Elements not known to enough precision

sage: c.is_equal_to(b, 2), c.is_equal_to(b, 3)
(True, False)
sage: c.is_equal_to(b, 4)
Traceback (most recent call last):
...
PrecisionError: Elements not known to enough precision

sage: c.is_equal_to(cc, 2), c.is_equal_to(cc, 4), c.is_equal_to(cc, 5)
(True, True, False)


TESTS:

sage: aa.is_equal_to(a), aa.is_equal_to(a, 4), aa.is_equal_to(a, 5)
(False, True, False)
sage: aa.is_equal_to(a, 15)
Traceback (most recent call last):
...
PrecisionError: Elements not known to enough precision

sage: b.is_equal_to(a), b.is_equal_to(a, 2)
(True, True)
sage: b.is_equal_to(a, 5)
Traceback (most recent call last):
...
PrecisionError: Elements not known to enough precision

sage: bb.is_equal_to(b, 3)
True
sage: bb.is_equal_to(b, 4)
Traceback (most recent call last):
...
PrecisionError: Elements not known to enough precision

sage: b.is_equal_to(c, 2), b.is_equal_to(c, 3)
(True, False)
sage: b.is_equal_to(c, 4)
Traceback (most recent call last):
...
PrecisionError: Elements not known to enough precision

sage: cc.is_equal_to(c, 2), cc.is_equal_to(c, 4), cc.is_equal_to(c, 5)
(True, True, False)

is_zero(absprec=None)

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 = Zp(5); a = R(0); b = R(0,5); c = R(75)
sage: a.is_zero(), a.is_zero(6)
(True, True)
sage: b.is_zero(), b.is_zero(5)
(True, True)
sage: c.is_zero(), c.is_zero(2), c.is_zero(3)
(False, True, False)
sage: b.is_zero(6)
Traceback (most recent call last):
...
PrecisionError: Not enough precision to determine if element is zero

list(lift_mode='simple', start_val=None)

Returns a list of coefficients in a power series expansion of self in terms of $$\pi$$. If self is a field element, they start at $$\pi^{\mbox{valuation}}$$, if a ring element at $$\pi^0$$.

For each lift mode, this funciton returns a list of $$a_i$$ so that this element can be expressed as

$\pi^v \cdot \sum_{i=0}^\infty a_i \pi^i$

where $$v$$ is the valuation of this element when the parent is a field, and $$v = 0$$ otherwise.

Different lift modes affect the choice of $$a_i$$. When lift_mode is 'simple', the resulting $$a_i$$ will be non-negative: if the residue field is $$\mathbb{F}_p$$ then they will be integers with $$0 \le a_i < p$$; otherwise they will be a list of integers in the same range giving the coefficients of a polynomial in the indeterminant representing the maximal unramified subextension.

Choosing lift_mode as 'smallest' is similar to 'simple', but uses a balanced representation $$-p/2 < a_i \le p/2$$.

Finally, setting lift_mode = 'teichmuller' will yield Teichmuller representatives for the $$a_i$$: $$a_i^q = a_i$$. In this case the $$a_i$$ will also be $$p$$-adic elements.

INPUT:

• lift_mode'simple', 'smallest' or 'teichmuller' (default: 'simple')
• start_val – start at this valuation rather than the default ($$0$$ or the valuation of this element). If start_val is larger than the valuation of this element a ValueError is raised.

OUTPUT:

• the list of coefficients of this element. For base elements these will be integers if lift_mode is 'simple' or 'smallest', and elements of self.parent() if lift_mode is 'teichmuller'.

Note

Use slice operators to get a particular range.

EXAMPLES:

sage: R = Zp(7,6); a = R(12837162817); a
3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)
sage: L = a.list(); L
[3, 4, 4, 0, 4]
sage: sum([L[i] * 7^i for i in range(len(L))]) == a
True
sage: L = a.list('smallest'); L
[3, -3, -2, 1, -3, 1]
sage: sum([L[i] * 7^i for i in range(len(L))]) == a
True
sage: L = a.list('teichmuller'); L
[3 + 4*7 + 6*7^2 + 3*7^3 + 2*7^5 + O(7^6),
0,
5 + 2*7 + 3*7^3 + O(7^4),
1 + O(7^3),
3 + 4*7 + O(7^2),
5 + O(7)]
sage: sum([L[i] * 7^i for i in range(len(L))])
3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)

sage: R(0, 7).list()
[]

sage: R = Qp(7,4); a = R(6*7+7**2); a.list()
[6, 1]
sage: a.list('smallest')
[-1, 2]
sage: a.list('teichmuller')
[6 + 6*7 + 6*7^2 + 6*7^3 + O(7^4),
2 + 4*7 + 6*7^2 + O(7^3),
3 + 4*7 + O(7^2),
3 + O(7)]


TESTS:

Check to see that trac ticket #10292 is resolved:

sage: E = EllipticCurve('37a')
sage: len(R.list())
19

precision_absolute()

Returns the absolute precision of this element.

This is the power of the maximal ideal modulo which this element is defined.

EXAMPLES:

sage: R = Zp(7,3,'capped-rel'); a = R(7); a.precision_absolute()
4
sage: R = Qp(7,3); a = R(7); a.precision_absolute()
4
sage: R(7^-3).precision_absolute()
0

sage: R(0).precision_absolute()
+Infinity
sage: R(0,7).precision_absolute()
7

precision_relative()

Returns the relative precision of this element.

This is the power of the maximal ideal modulo which the unit part of self is defined.

EXAMPLES:

sage: R = Zp(7,3,'capped-rel'); a = R(7); a.precision_relative()
3
sage: R = Qp(7,3); a = R(7); a.precision_relative()
3
sage: a = R(7^-2, -1); a.precision_relative()
1
sage: a
7^-2 + O(7^-1)

sage: R(0).precision_relative()
0
sage: R(0,7).precision_relative()
0

teichmuller_list()

Returns a list [$$a_0$$, $$a_1$$,..., $$a_n$$] such that

• $$a_i^q = a_i$$, where $$q$$ is the cardinality of the residue field,
• self.unit_part() = $$\sum_{i = 0}^n a_i p^i$$, and
• if $$a_i \ne 0$$, the absolute precision of $$a_i$$ is self.precision_relative() - i

EXAMPLES:

sage: R = Qp(5,5); R(70).list('teichmuller') #indirect doctest
[4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5),
3 + 3*5 + 2*5^2 + 3*5^3 + O(5^4),
2 + 5 + 2*5^2 + O(5^3),
1 + O(5^2),
4 + O(5)]

unit_part()

Returns $$u$$, where this element is $$\pi^v u$$.

EXAMPLES:

sage: R = Zp(17,4,'capped-rel')
sage: a = R(18*17)
sage: a.unit_part()
1 + 17 + O(17^4)
sage: type(a)
sage: R = Qp(17,4,'capped-rel')
sage: a = R(18*17)
sage: a.unit_part()
1 + 17 + O(17^4)
sage: type(a)
sage: a = R(2*17^2); a
2*17^2 + O(17^6)
sage: a.unit_part()
2 + O(17^4)
sage: b=1/a; b
9*17^-2 + 8*17^-1 + 8 + 8*17 + O(17^2)
sage: b.unit_part()
9 + 8*17 + 8*17^2 + 8*17^3 + O(17^4)
sage: Zp(5)(75).unit_part()
3 + O(5^20)

sage: R(0).unit_part()
Traceback (most recent call last):
...
ValueError: unit part of 0 not defined
sage: R(0,7).unit_part()
O(17^0)

val_unit(p=None)

Returns a pair (self.valuation(), self.unit_part()).

INPUT:

• p – a prime (default: None). If specified, will make sure that p==self.parent().prime()

Note

The optional argument p is used for consistency with the valuation methods on integer and rational.

EXAMPLES:

sage: R = Zp(5); a = R(75, 20); a
3*5^2 + O(5^20)
sage: a.val_unit()
(2, 3 + O(5^18))
sage: R(0).val_unit()
Traceback (most recent call last):
...
ValueError: unit part of 0 not defined
sage: R(0, 10).val_unit()
(10, O(5^0))


Returns a base-$$p$$ list of digits of n.

INPUT:

• n – a positive Integer.

• pos – a boolean. If True, then returns the standard base $$p$$ expansion.

Otherwise, the digits lie in the range $$-p/2$$ to $$p/2$$.

• prime_pow – A PowComputer giving the prime.

EXAMPLES:

sage: from sage.rings.padics.padic_capped_relative_element import base_p_list
sage: base_p_list(192837, True, Zp(5).prime_pow)
[2, 2, 3, 2, 3, 1, 2, 2]
sage: 2 + 2*5 + 3*5^2 + 2*5^3 + 3*5^4 + 5^5 + 2*5^6 + 2*5^7
192837
sage: base_p_list(192837, False, Zp(5).prime_pow)
[2, 2, -2, -2, -1, 2, 2, 2]
sage: 2 + 2*5 - 2*5^2 - 2*5^3 - 5^4 + 2*5^5 + 2*5^6 + 2*5^7
192837


Constructs new element with given parent and value.

INPUT:

• x – value to coerce into a capped relative ring or field
• absprec – maximum number of digits of absolute precision
• relprec – maximum number of digits of relative precision

EXAMPLES:

sage: R = Zp(5, 10, 'capped-rel')


Construct from integers:

sage: R(3)
3 + O(5^10)
sage: R(75)
3*5^2 + O(5^12)
sage: R(0)
0
sage: R(-1)
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)
sage: R(-5)
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 + 4*5^10 + O(5^11)
sage: R(-7*25)
3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + 4*5^10 + 4*5^11 + O(5^12)


Construct from rationals:

sage: R(1/2)
3 + 2*5 + 2*5^2 + 2*5^3 + 2*5^4 + 2*5^5 + 2*5^6 + 2*5^7 + 2*5^8 + 2*5^9 + O(5^10)
sage: R(-7875/874)
3*5^3 + 2*5^4 + 2*5^5 + 5^6 + 3*5^7 + 2*5^8 + 3*5^10 + 3*5^11 + 3*5^12 + O(5^13)
sage: R(15/425)
Traceback (most recent call last):
...
ValueError: p divides the denominator


Construct from IntegerMod:

sage: R(Integers(125)(3))
3 + O(5^3)
sage: R(Integers(5)(3))
3 + O(5)
sage: R(Integers(5^30)(3))
3 + O(5^10)
sage: R(Integers(5^30)(1+5^23))
1 + O(5^10)
sage: R(Integers(49)(3))
Traceback (most recent call last):
...
TypeError: cannot coerce from the given integer mod ring (not a power of the same prime)

sage: R(Integers(48)(3))
Traceback (most recent call last):
...
TypeError: cannot coerce from the given integer mod ring (not a power of the same prime)


Some other conversions:

sage: R(R(5))
5 + O(5^11)


Construct from Pari objects:

sage: R = Zp(5)
sage: x = pari(123123) ; R(x)
3 + 4*5 + 4*5^2 + 4*5^3 + 5^4 + 4*5^5 + 2*5^6 + 5^7 + O(5^20)
sage: R(pari(R(5252)))
2 + 2*5^3 + 3*5^4 + 5^5 + O(5^20)
sage: R = Zp(5,prec=5)
sage: R(pari(-1))
4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5)
sage: pari(R(-1))
4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5)
sage: pari(R(0))
0
sage: R(pari(R(0,5)))
O(5^5)


# todo: doctests for converting from other types of p-adic rings

lift()

Return an integer or rational congruent to self modulo self‘s precision. If a rational is returned, its denominator will equal p^ordp(self).

EXAMPLES:

sage: R = Zp(7,4,'capped-rel'); a = R(8); a.lift()
8
sage: R = Qp(7,4); a = R(8); a.lift()
8
sage: R = Qp(7,4); a = R(8/7); a.lift()
8/7

residue(absprec=1)

Reduces this element modulo $$p^{\mbox{absprec}}$$.

INPUT:

• absprec - an integer (default: 1)

OUTPUT:

Element of $$\ZZ/(p^{\mbox{absprec}} \ZZ)$$ – the reduction modulo $$p^{\mbox{absprec}}$$

EXAMPLES:

sage: R = Zp(7,4,'capped-rel'); a = R(8); a.residue(1)
1
sage: R = Qp(7,4,'capped-rel'); a = R(8); a.residue(1)
1
sage: a.residue(6)
Traceback (most recent call last):
...
PrecisionError: Not enough precision known in order to compute residue.
sage: b = a/7
sage: b.residue(1)
Traceback (most recent call last):
...
ValueError: Element must have non-negative valuation in order to compute residue.


The canonical inclusion from the rationals to a capped relative field.

EXAMPLES:

sage: f = Qp(5).coerce_map_from(QQ); f
Ring Coercion morphism:
From: Rational Field
To:   5-adic Field with capped relative precision 20

section()

Returns a map back to the rationals that approximates an element by a rational number.

EXAMPLES:

sage: f = Qp(5).coerce_map_from(QQ).section()
sage: f(Qp(5)(1/4))
1/4
sage: f(Qp(5)(1/5))
1/5


The canonical inclusion from the integer ring to a capped relative ring.

EXAMPLES:

sage: f = Zp(5).coerce_map_from(ZZ); f
Ring Coercion morphism:
From: Integer Ring
To:   5-adic Ring with capped relative precision 20

section()

Returns a map back to the ring of integers that approximates an element by an integer.

EXAMPLES:

sage: f = Zp(5).coerce_map_from(ZZ).section()
sage: f(Zp(5)(-1)) - 5^20
-1


The map from the capped relative ring back to the rationals that returns a rational approximation of its input.

EXAMPLES:

sage: f = Qp(5).coerce_map_from(QQ).section(); f
Set-theoretic ring morphism:
From: 5-adic Field with capped relative precision 20
To:   Rational Field


The map from a capped relative ring back to the ring of integers that returns the 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 integers.

EXAMPLES:

sage: f = Zp(5).coerce_map_from(ZZ).section(); f
Set-theoretic ring morphism:
From: 5-adic Ring with capped relative precision 20
To:   Integer Ring


The inclusion map from the rationals to a capped relative ring that is defined on all elements with non-negative $$p$$-adic valuation.

EXAMPLES:

sage: f = Zp(5).convert_map_from(QQ); f
Generic morphism:
From: Rational Field
To:   5-adic Ring with capped relative precision 20

section()

Returns the map back to the rationals that returns the smallest non-negative integer approximation to its input which is accurate up to the precision.

EXAMPLES:

sage: f = Zp(5,4).convert_map_from(QQ).section()
sage: f(Zp(5,4)(-1))
-1


A class for common functionality among the $$p$$-adic template classes.

INPUT:

• parent – a local ring or field
• x – data defining this element. Various types are supported, including ints, Integers, Rationals, PARI p-adics, integers mod $$p^k$$ and other Sage p-adics.
• absprec – a cap on the absolute precision of this element
• relprec – a cap on the relative precision of this element

EXAMPLES:

sage: Zp(17)(17^3, 8, 4)
17^3 + O(17^7)

lift_to_precision(absprec=None)

Returns another element of the same parent with absolute precision at least absprec, congruent to this $$p$$-adic element modulo the precision of this element.

INPUT:

• absprec – an integer or None (default: None), the absolute precision of the result. If None, lifts to the maximum precision allowed.

Note

If setting absprec that high would violate the precision cap, raises a precision error. Note that the new digits will not necessarily be zero.

EXAMPLES:

sage: R = ZpCA(17)
sage: R(-1,2).lift_to_precision(10)
16 + 16*17 + O(17^10)
sage: R(1,15).lift_to_precision(10)
1 + O(17^15)
sage: R(1,15).lift_to_precision(30)
Traceback (most recent call last):
...
PrecisionError: Precision higher than allowed by the precision cap.
sage: R(-1,2).lift_to_precision().precision_absolute() == R.precision_cap()
True

sage: R = Zp(5); c = R(17,3); c.lift_to_precision(8)
2 + 3*5 + O(5^8)
sage: c.lift_to_precision().precision_relative() == R.precision_cap()
True


Fixed modulus elements don’t raise errors:

sage: R = ZpFM(5); a = R(5); a.lift_to_precision(7)
5 + O(5^20)
sage: a.lift_to_precision(10000)
5 + O(5^20)


Returns a list of coefficients of the uniformizer $$\pi$$ starting with $$\pi^0$$ up to $$\pi^n$$ exclusive (padded with zeros if needed).

For a field element of valuation $$v$$, starts at $$\pi^v$$ instead.

INPUT:

• n - an integer
• lift_mode - ‘simple’, ‘smallest’ or ‘teichmuller’

EXAMPLES:

sage: R = Zp(7,4,'capped-abs'); a = R(2*7+7**2); a.padded_list(5)
[0, 2, 1, 0, 0]
sage: R = Zp(7,4,'fixed-mod'); a = R(2*7+7**2); a.padded_list(5)
[0, 2, 1, 0, 0]


For elements with positive valuation, this function will return a list with leading 0s if the parent is not a field:

sage: R = Zp(7,3,'capped-rel'); a = R(2*7+7**2); a.padded_list(5)
[0, 2, 1, 0, 0]
sage: R = Qp(7,3); a = R(2*7+7**2); a.padded_list(5)
[2, 1, 0, 0]
[2, 1]

unit_part()

Returns the unit part of this element.

This is the $$p$$-adic element $$u$$ in the same ring so that this element is $$\pi^v u$$, where $$\pi$$ is a uniformizer and $$v$$ is the valuation of this element.

Unpickles a capped relative element.

EXAMPLES:

sage: from sage.rings.padics.padic_capped_relative_element import unpickle_cre_v2
sage: R = Zp(5); a = R(85,6)
sage: b = unpickle_cre_v2(a.__class__, R, 17, 1, 5)
sage: a == b
True
sage: a.precision_relative() == b.precision_relative()
True


Unpickles a capped relative element.

EXAMPLES:

sage: from sage.rings.padics.padic_capped_relative_element import unpickle_pcre_v1
sage: R = Zp(5)
sage: a = unpickle_pcre_v1(R, 17, 2, 5); a
2*5^2 + 3*5^3 + O(5^7)


#### Next topic

$$p$$-Adic Capped Absolute Elements