# $$p$$-adic Capped Absolute Elements#

Elements of $$p$$-adic Rings with Absolute Precision Cap

AUTHORS:

• David Roe

• Genya Zaytman: documentation

• David Harvey: doctests

Return 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)
2 + 3 + 3^2 + 3^3 + O(3^5)
2 + 3 + 3^2 + O(3^3)

is_equal_to(_right, absprec=None)#

Determine 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

is_zero(absprec=None)#

Determine 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

polynomial(var='x')#

Return 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)

precision_absolute()#

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

precision_relative()#

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

unit_part()#

Return 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)
sage: R(0).unit_part()
O(17^0)

val_unit()#

Return 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))


Bases: object

An iterator over a $$p$$-adic expansion.

This class should not be instantiated directly, but instead using expansion().

INPUT:

• elt – the $$p$$-adic element

• prec – the number of terms to be emitted

• mode – either simple_mode, smallest_mode or teichmuller_mode

EXAMPLES:

sage: E = Zp(5,4)(373).expansion()
sage: I = iter(E) # indirect doctest
sage: type(I)


Bases: object

An iterable storing a $$p$$-adic expansion of an element.

This class should not be instantiated directly, but instead using expansion().

INPUT:

• elt – the $$p$$-adic element

• prec – the number of terms to be emitted

• val_shift – how many zeros to add at the beginning of the expansion, or the number of initial terms to truncate (if negative)

• mode – one of the following:

• 'simple_mode'

• 'smallest_mode'

• 'teichmuller_mode'

EXAMPLES:

sage: E = Zp(5,4)(373).expansion() # indirect doctest
sage: type(E)


A PowComputer for a capped-absolute padic ring.

Unpickles a capped absolute element.

EXAMPLES:

sage: from sage.rings.padics.padic_capped_absolute_element import make_pAdicCappedAbsoluteElement
sage: R = ZpCA(5)
sage: a = make_pAdicCappedAbsoluteElement(R, 17*25, 5); a
2*5^2 + 3*5^3 + O(5^5)


Bases: CAElement

Constructs new element with given parent and value.

INPUT:

• x – value to coerce into a capped absolute ring

• absprec – maximum number of digits of absolute precision

• relprec – maximum number of digits of relative precision

EXAMPLES:

sage: R = ZpCA(3, 5)
sage: R(2)
2 + O(3^5)
sage: R(2, absprec=2)
2 + O(3^2)
sage: R(3, relprec=2)
3 + O(3^3)
sage: R(Qp(3)(10))
1 + 3^2 + O(3^5)
sage: R(pari(6))
2*3 + O(3^5)
sage: R(pari(1/2))
2 + 3 + 3^2 + 3^3 + 3^4 + O(3^5)
sage: R(1/2)
2 + 3 + 3^2 + 3^3 + 3^4 + O(3^5)
sage: R(mod(-1, 3^7))
2 + 2*3 + 2*3^2 + 2*3^3 + 2*3^4 + O(3^5)
sage: R(mod(-1, 3^2))
2 + 2*3 + O(3^2)
sage: R(3 + O(3^2))
3 + O(3^2)

lift()#

EXAMPLES:

sage: R = ZpCA(3)
sage: R(10).lift()
10
sage: R(-1).lift()
3486784400

multiplicative_order()#

Return the minimum possible multiplicative order of this element.

OUTPUT:

The multiplicative order of self. This is the minimum multiplicative order of all elements of $$\ZZ_p$$ lifting self to infinite precision.

EXAMPLES:

sage: R = ZpCA(7, 6)
sage: R(1/3)
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + 4*7^5 + O(7^6)
sage: R(1/3).multiplicative_order()
+Infinity
sage: R(7).multiplicative_order()
+Infinity
sage: R(1).multiplicative_order()
1
sage: R(-1).multiplicative_order()
2
sage: R.teichmuller(3).multiplicative_order()
6

residue(absprec=1, field=None, check_prec=True)#

Reduces self modulo $$p^\mathrm{absprec}$$.

INPUT:

• absprec – a non-negative integer (default: 1)

• field – boolean (default None). Whether to return an element of GF(p) or Zmod(p).

• check_prec – boolean (default True). Whether to raise an error if this element has insufficient precision to determine the reduction.

OUTPUT:

This element reduced modulo $$p^\mathrm{absprec}$$ as an element of $$\ZZ/p^\mathrm{absprec}\ZZ$$

EXAMPLES:

sage: R = Zp(7,10,'capped-abs')
sage: a = R(8)
sage: a.residue(1)
1


This is different from applying % p^n which returns an element in the same ring:

sage: b = a.residue(2); b
8
sage: b.parent()
Ring of integers modulo 49
sage: c = a % 7^2; c
1 + 7 + O(7^10)
sage: c.parent()
7-adic Ring with capped absolute precision 10


Note that reduction of c dropped to the precision of the unit part of 7^2, see _mod_():

sage: R(7^2).unit_part()
1 + O(7^8)


_mod_()

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: 3-adic Unramified Extension Ring in a defined by x^3 + 2*x + 1
To:   3-adic Unramified Extension Field in a defined by x^3 + 2*x + 1

is_injective()#

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

is_surjective()#

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

section()#

Return 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)
sage: f.section()
Generic morphism:
From: 3-adic Unramified Extension Field in a defined by x^3 + 2*x + 1
To:   3-adic Unramified Extension Ring in a defined by x^3 + 2*x + 1


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

section()#

Return 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


Bases: 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


Bases: 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: 3-adic Unramified Extension Field in a defined by x^3 + 2*x + 1
To:   3-adic Unramified Extension Ring in a defined by x^3 + 2*x + 1


Bases: 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


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)

expansion(n=None, lift_mode='simple', start_val=None)#

Return the coefficients in a $$\pi$$-adic expansion. If this is a field element, start at $$\pi^{\mbox{valuation}}$$, if a ring element at $$\pi^0$$.

For each lift mode, this function 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 $$\GF{p}$$ 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 lie in the ring of integers of the maximal unramified subextension of the parent of this element.

INPUT:

• n – integer (default None). If given, returns the corresponding entry in the expansion. Can also accept a slice (see slice())

• 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).

OUTPUT:

• If n is None, an iterable giving a $$\pi$$-adic expansion of this element. For base elements the contents will be integers if lift_mode is 'simple' or 'smallest', and elements of self.parent() if lift_mode is 'teichmuller'.

• If n is an integer, the coefficient of $$\pi^n$$ in the $$\pi$$-adic expansion of this element.

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: E = a.expansion(); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)
sage: list(E)
[3, 4, 4, 0, 4, 0]
sage: sum([c * 7^i for i, c in enumerate(E)]) == a
True
sage: E = a.expansion(lift_mode='smallest'); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6) (balanced)
sage: list(E)
[3, -3, -2, 1, -3, 1]
sage: sum([c * 7^i for i, c in enumerate(E)]) == a
True
sage: E = a.expansion(lift_mode='teichmuller'); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6) (teichmuller)
sage: list(E)
[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(c * 7^i for i, c in enumerate(E))
3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)


If the element has positive valuation then the list will start with some zeros:

sage: a = R(7^3 * 17)
sage: E = a.expansion(); E
7-adic expansion of 3*7^3 + 2*7^4 + O(7^9)
sage: list(E)
[0, 0, 0, 3, 2, 0, 0, 0, 0]


The expansion of 0 is truncated:

sage: E = R(0, 7).expansion(); E
sage: len(E)
0
sage: list(E)
[]


In fields, on the other hand, the expansion starts at the valuation:

sage: R = Qp(7,4); a = R(6*7+7**2); E = a.expansion(); E
7-adic expansion of 6*7 + 7^2 + O(7^5)
sage: list(E)
[6, 1, 0, 0]
sage: list(a.expansion(lift_mode='smallest'))
[-1, 2, 0, 0]
sage: list(a.expansion(lift_mode='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)]


You can ask for a specific entry in the expansion:

sage: a.expansion(1)
6
sage: a.expansion(1, lift_mode='smallest')
-1
sage: a.expansion(2, lift_mode='teichmuller')
2 + 4*7 + 6*7^2 + O(7^3)

lift_to_precision(absprec=None)#

Return 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
sage: a.lift_to_precision(10000)
5

residue(absprec=1, field=None, check_prec=True)#

Reduce this element modulo $$p^\mathrm{absprec}$$.

INPUT:

• absprec0 or 1.

• field – boolean (default None). For precision 1, whether to return an element of the residue field or a residue ring. Currently unused.

• check_prec – boolean (default True). Whether to raise an error if this element has insufficient precision to determine the reduction. Errors are never raised for fixed-mod or floating-point types.

OUTPUT:

This element reduced modulo $$p^\mathrm{absprec}$$ as an element of the residue field or the null ring.

EXAMPLES:

sage: R.<a> = Zq(27, 4)
sage: (3 + 3*a).residue()
0
sage: (a + 1).residue()
a0 + 1

teichmuller_expansion(n=None)#

Returns an iterator over coefficients $$a_0, a_1, \dots, a_n$$ such that

• $$a_i^q = a_i$$, where $$q$$ is the cardinality of the residue field,

• 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.

• if $$a_i \ne 0$$, the precision of $$a_i$$ is $$i$$ less than the precision of this element (relative in the case that the parent is a field, absolute otherwise)

Note

The coefficients will lie in the ring of integers of the maximal unramified subextension.

INPUT:

• n – integer (default None). If given, returns the coefficient of $$\pi^n$$ in the expansion.

EXAMPLES:

For fields, the expansion starts at the valuation:

sage: R = Qp(5,5); list(R(70).teichmuller_expansion())
[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)]


But if you specify n, you get the coefficient of $$\pi^n$$:

sage: R(70).teichmuller_expansion(2)
3 + 3*5 + 2*5^2 + 3*5^3 + O(5^4)

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.

EXAMPLES:

sage: R.<a> = Zq(125)
sage: (5*a).unit_part()
a + O(5^20)


Unpickle capped absolute elements.

INPUT:

• cls – the class of the capped absolute element

• 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