p-Adic Fixed-Mod Element

p-Adic Fixed-Mod Element

Elements of p-Adic Rings with Fixed Modulus

AUTHORS:

  • David Roe
  • Genya Zaytman: documentation
  • David Harvey: doctests
sage.rings.padics.padic_fixed_mod_element.make_pAdicFixedModElement(parent, value)

Unpickles a capped relative element.

EXAMPLES:

sage: from sage.rings.padics.padic_fixed_mod_element import make_pAdicFixedModElement
sage: R = ZpFM(5)
sage: a = make_pAdicFixedModElement(R, 17*25); a
2*5^2 + 3*5^3 + O(5^20)
class sage.rings.padics.padic_fixed_mod_element.pAdicFixedModElement

Bases: sage.rings.padics.padic_base_generic_element.pAdicBaseGenericElement

INPUT:
parent – a pAdicRingFixedMod object.
Types currently supported:
Integers Rationals – denominator must be relatively prime to p FixedMod p-adics
Types that should be supported:
Finite precision p-adics Lazy p-adics Elements of local extensions of THIS p-adic ring that actually lie in Zp Elements of IntegerModRing(p^k) for k less than or equal to the modulus
EXAMPLES:
sage: R = Zp(5, 20, ‘fixed-mod’, ‘terse’)
Construct from integers:

sage: R(3) 3 + O(5^20) sage: R(75) 75 + O(5^20) sage: R(0) 0 + O(5^20)

sage: R(-1) 95367431640624 + O(5^20) sage: R(-5) 95367431640620 + O(5^20)

Construct from rationals:
sage: R(1/2) 47683715820313 + O(5^20) sage: R(-7875/874) 9493096742250 + O(5^20) sage: R(15/425) Traceback (most recent call last): ... ValueError: p divides denominator

# todo: the above error message does not agree with the error message # in the corresponding capped-relative constructor

Construct from IntegerMod:

sage: R(Integers(125)(3)) 3 + O(5^20) sage: R(Integers(5)(3)) 3 + O(5^20) sage: R(Integers(5^30)(3)) 3 + O(5^20) sage: R(Integers(5^30)(1+5^23)) 1 + O(5^20) 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(5^20)

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

add_bigoh(absprec)

Returns a new element truncated modulo p^absprec.

INPUT:

- self -- a p-adic element
- absprec -- an integer

OUTPUT:

- element -- a new element truncated modulo p^absprec.

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod','series'); a = R(8); a.add_bigoh(1)
1 + O(7^4)
is_equal_to(right, absprec=None)

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

If absprec is None, returns if self == 0.

INPUT:

- self -- a p-adic element
- right -- a p-addic element with the same parent
- absprec -- a positive integer (or None)

OUTPUT:

boolean -- whether self is equal to right

EXAMPLES:

sage: R = ZpFM(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
is_zero(absprec=None)

Returns whether self is zero modulo p^{\mbox{absprec}}.

INPUT:

- self -- a p-adic element
- absprec -- an integer

OUTPUT:

boolean -- whether self is zero

EXAMPLES:

sage: R = ZpFM(17, 6)
sage: R(0).is_zero()
True
sage: R(17^6).is_zero()
True
sage: R(17^2).is_zero(absprec=2)
True
lift()

Return an integer congruent to self modulo self’s precision.

INPUT:

- self -- a p-adic element

OUTPUT:

- integer -- a integer congruent to self mod $p^{\mbox{prec}}$

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod'); a = R(8); a.lift()
8
sage: type(a.lift())
<type 'sage.rings.integer.Integer'>
lift_to_precision(absprec=None)

Returns self.

For compatibility with other p-adic types.

EXAMPLES:

sage: R = ZpFM(5); a = R(5); a.lift_to_precision(7)
5 + O(5^20)
list(lift_mode='simple')

Returns a list of coefficients of p starting with p^0.

INPUT:

- self -- a p-adic element
- lift_mode -- 'simple', 'smallest' or 'teichmuller' (default 'simple')

OUTPUT:

- list -- the list of coefficients of self

NOTES:

Returns a list [a_0, a_1, \ldots, a_n] so that each a_i is an integer
and \sum_{i = 0}^n a_i * p^i = self, modulo the precision cap.
If lift_mode = 'simple', 0 <= a_i < p.
If lift_mode = 'smallest', -p/2 < a_i <= p/2.
If lift_mode = 'teichmuller', a_i^p = a_i, modulo the precision cap.

EXAMPLES:

sage: R = ZpFM(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),
O(7^6),
5 + 2*7 + 3*7^3 + 6*7^4 + 4*7^5 + O(7^6),
1 + O(7^6),
3 + 4*7 + 6*7^2 + 3*7^3 + 2*7^5 + O(7^6),
5 + 2*7 + 3*7^3 + 6*7^4 + 4*7^5 + O(7^6)]
sage: sum([L[i] * 7^i for i in range(len(L))])
3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)
multiplicative_order()

Returns the minimum possible multiplicative order of self.

INPUT:

- self -- a p-adic element

OUTPUT:

- integer -- the multiplicative order of self.  This is
  the minimum multiplicative order of all elements of Z_p
  lifting self to infinite precision.

EXAMPLES:

sage: R = ZpFM(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
padded_list(n, list_mode='simple')

Returns a list of coefficients of p starting with p^0 up to p^n exclusive (padded with zeros if needed)

INPUT:

- self -- a p-adic element
- n -- an integer

OUTPUT:

- list -- the list of coefficients of self

EXAMPLES:

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

NOTE:

For elements with positive valuation, this function will
return a list with leading 0s, unlike for field elements.

The slice operators throw an error if asked for a slice
above the precision, while this function works
precision_absolute()

Returns the absolute precision of self.

INPUT:

- self -- a p-adic element

OUTPUT:

- integer -- the absolute precision of self

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod'); a = R(7); a.precision_absolute()
4
precision_relative()

Returns the relative precision of self

INPUT:

- self -- a p-adic element

OUTPUT:

- integer -- the relative precision of self

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod'); a = R(7); a.precision_relative()
3
sage: a = R(0); a.precision_relative()
0
residue(absprec=1)

Reduces this mod p^{\mbox{prec}}

INPUT:

- self -- a p-adic element
- absprec - an integer (default 1)

OUTPUT:

- element of Z/(p^prec Z) -- self reduced mod p^prec

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod'); a = R(8); a.residue(1)
1
unit_part()

Returns the unit part of self.

If the valuation of self is positive, then the high digits of the result will be zero.

INPUT:

- self -- a p-adic element

OUTPUT:

- p-adic element -- the unit part of self

EXAMPLES:

sage: R = Zp(17, 4, 'fixed-mod')
sage: R(5).unit_part()
5 + O(17^4)
sage: R(18*17).unit_part()
1 + 17 + O(17^4)
sage: R(0).unit_part()
O(17^4)
sage: type(R(5).unit_part())
<type 'sage.rings.padics.padic_fixed_mod_element.pAdicFixedModElement'>
sage: R = ZpFM(5, 5); a = R(75); a.unit_part()
3 + O(5^5)
val_unit()

Returns a 2-tuple, the first element set to the valuation of self, and the second to the unit part of self.

If self == 0, then the unit part is O(p^self.parent().precision_cap()).

EXAMPLES:

sage: R = ZpFM(5,5)
sage: a = R(75); b = a - a
sage: a.val_unit()
(2, 3 + O(5^5))
sage: b.val_unit()
(5, O(5^5))
valuation()

Returns the valuation of self.

If self is zero, the valuation returned is the precision of the ring.

INPUT:

- self -- a p-adic element

OUTPUT:

- integer -- the valuation of self.

EXAMPLES:

sage: R = Zp(17, 4,'fixed-mod')
sage: a = R(2*17^2)
sage: a.valuation()
2
sage: R = Zp(5, 4,'fixed-mod')
sage: R(0).valuation()
4
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            

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