\(p\)-adic Generic Element#

Elements of \(p\)-adic Rings and Fields

AUTHORS:

David Roe
Genya Zaytman: documentation
David Harvey: doctests
Julian Rueth: fixes for exp() and log(), implemented gcd, xgcd

sage.rings.padics.padic_generic_element.dwork_mahler_coeffs(R, bd=20)#

Compute Dwork’s formula for Mahler coefficients of \(p\)-adic Gamma.

This is called internally when one computes Gamma for a \(p\)-adic integer. Normally there is no need to call it directly.

INPUT:

R – \(p\)-adic ring in which to compute
bd – integer. Number of terms in the expansion to use

OUTPUT:

A list of \(p\)-adic integers.

EXAMPLES:

sage: from sage.rings.padics.padic_generic_element import dwork_mahler_coeffs, evaluate_dwork_mahler
sage: R = Zp(3)
sage: v = dwork_mahler_coeffs(R)
sage: x = R(1/7)
sage: evaluate_dwork_mahler(v, x, 3, 20, 1)
2 + 2*3 + 3^2 + 3^3 + 3^4 + 3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^11
 + 2*3^12 + 3^13 + 3^14 + 2*3^16 + 3^17 + 3^19 + O(3^20)
sage: x.dwork_expansion(a=1)  # Same result
2 + 2*3 + 3^2 + 3^3 + 3^4 + 3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^11
 + 2*3^12 + 3^13 + 3^14 + 2*3^16 + 3^17 + 3^19 + O(3^20)

sage.rings.padics.padic_generic_element.evaluate_dwork_mahler(v, x, p, bd, a)#

Evaluate Dwork’s Mahler series for \(p\)-adic Gamma.

EXAMPLES:

sage: from sage.rings.padics.padic_generic_element import dwork_mahler_coeffs, evaluate_dwork_mahler
sage: R = Zp(3)
sage: v = dwork_mahler_coeffs(R)
sage: x = R(1/7)
sage: evaluate_dwork_mahler(v, x, 3, 20, 1)
2 + 2*3 + 3^2 + 3^3 + 3^4 + 3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^11
 + 2*3^12 + 3^13 + 3^14 + 2*3^16 + 3^17 + 3^19 + O(3^20)
sage: x.dwork_expansion(a=1)  # Same result
2 + 2*3 + 3^2 + 3^3 + 3^4 + 3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^11
 + 2*3^12 + 3^13 + 3^14 + 2*3^16 + 3^17 + 3^19 + O(3^20)

sage.rings.padics.padic_generic_element.gauss_table(p, f, prec, use_longs)#

Compute a table of Gauss sums using the Gross-Koblitz formula.

This is used in the computation of L-functions of hypergeometric motives. The Gross-Koblitz formula is used as in \(sage.rings.padics.misc.gauss_sum\), but further unpacked for efficiency.

INPUT:

p - prime
f, prec - positive integers
use_longs - boolean; if True, computations are done in C long long
integers rather than Sage \(p\)-adics, and the results are returned as a Python array rather than a list.

OUTPUT:

A list of length \(q-1=p^f-1\). The entries are \(p\)-adic units created with absolute precision prec.

EXAMPLES:

sage: from sage.rings.padics.padic_generic_element import gauss_table
sage: gauss_table(2,2,4,False)
[1 + 2 + 2^2 + 2^3, 1 + 2 + 2^2 + 2^3, 1 + 2 + 2^2 + 2^3]
sage: gauss_table(3,2,4,False)[3]
2 + 3 + 2*3^2

class sage.rings.padics.padic_generic_element.pAdicGenericElement#

Bases: LocalGenericElement

abs(prec=None)#

Return the \(p\)-adic absolute value of self.

This is normalized so that the absolute value of \(p\) is \(1/p\).

INPUT:

prec – Integer. The precision of the real field in which the answer is returned. If None, returns a rational for absolutely unramified fields, or a real with 53 bits of precision for ramified fields.

EXAMPLES:

sage: a = Qp(5)(15); a.abs()
1/5
sage: a.abs(53)                                                             # needs sage.rings.real_mpfr
0.200000000000000
sage: Qp(7)(0).abs()
0
sage: Qp(7)(0).abs(prec=20)                                                 # needs sage.rings.real_mpfr
0.00000

An unramified extension:

sage: # needs sage.libs.ntl
sage: R = Zp(5,5)
sage: P.<x> = PolynomialRing(R)
sage: Z25.<u> = R.ext(x^2 - 3)
sage: u.abs()
1
sage: (u^24-1).abs()
1/5

A ramified extension:

sage: # needs sage.libs.ntl
sage: W.<w> = R.ext(x^5 + 75*x^3 - 15*x^2 + 125*x - 5)
sage: w.abs()
0.724779663677696
sage: W(0).abs()
0.000000000000000

additive_order(prec=None)#

Return the additive order of this element truncated at precision prec.

INPUT:

prec – an integer or None (default: None)

OUTPUT:

The additive order of this element

EXAMPLES:

sage: R = Zp(7, 4, 'capped-rel', 'series'); a = R(7^3); a.additive_order(3)
1
sage: a.additive_order(4)
+Infinity
sage: R = Zp(7, 4, 'fixed-mod', 'series'); a = R(7^5); a.additive_order(6)
1

algdep(n)#

Returns a polynomial of degree at most \(n\) which is approximately satisfied by this number. Note that the returned polynomial need not be irreducible, and indeed usually won’t be if this number is a good approximation to an algebraic number of degree less than \(n\).

ALGORITHM: Uses the PARI C-library pari:algdep command.

INPUT:

self – a \(p\)-adic element
n – an integer

OUTPUT:

polynomial – degree \(n\) polynomial approximately satisfied by self

EXAMPLES:

sage: K = Qp(3,20,'capped-rel','series'); R = Zp(3,20,'capped-rel','series')
sage: a = K(7/19); a
1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12
  + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20)
sage: a.algdep(1)
19*x - 7
sage: K2 = Qp(7,20,'capped-rel')
sage: b = K2.zeta(); b.algdep(2)
x^2 - x + 1
sage: K2 = Qp(11,20,'capped-rel')
sage: b = K2.zeta(); b.algdep(4)
x^4 - x^3 + x^2 - x + 1
sage: a = R(7/19); a
1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12
  + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20)
sage: a.algdep(1)
19*x - 7
sage: R2 = Zp(7,20,'capped-rel')
sage: b = R2.zeta(); b.algdep(2)
x^2 - x + 1
sage: R2 = Zp(11,20,'capped-rel')
sage: b = R2.zeta(); b.algdep(4)
x^4 - x^3 + x^2 - x + 1

algebraic_dependency(n)#

ALGORITHM: Uses the PARI C-library pari:algdep command.

INPUT:

self – a p-adic element
n – an integer

OUTPUT:

polynomial – degree \(n\) polynomial approximately satisfied by self

EXAMPLES:

sage: K = Qp(3,20,'capped-rel','series'); R = Zp(3,20,'capped-rel','series')
sage: a = K(7/19); a
1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12
  + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20)
sage: a.algebraic_dependency(1)
19*x - 7
sage: K2 = Qp(7,20,'capped-rel')
sage: b = K2.zeta(); b.algebraic_dependency(2)
x^2 - x + 1
sage: K2 = Qp(11,20,'capped-rel')
sage: b = K2.zeta(); b.algebraic_dependency(4)
x^4 - x^3 + x^2 - x + 1
sage: a = R(7/19); a
1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12
  + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20)
sage: a.algebraic_dependency(1)
19*x - 7
sage: R2 = Zp(7,20,'capped-rel')
sage: b = R2.zeta(); b.algebraic_dependency(2)
x^2 - x + 1
sage: R2 = Zp(11,20,'capped-rel')
sage: b = R2.zeta(); b.algebraic_dependency(4)
x^4 - x^3 + x^2 - x + 1

artin_hasse_exp(prec=None, algorithm=None)#

Return the Artin-Hasse exponential of this element.

INPUT:

prec – an integer or None (default: None) the desired precision on the result; if None, the precision is derived from the precision on the input
algorithm – 'direct', 'series', 'newton' or None (default)

The direct algorithm computes the Artin-Hasse exponential of \(x\), namely AH(x) as

\[AH(x) = \exp(x + \frac{x^p}{p} + \frac{x^{p^2}}{p^2} + \dots\]

It runs roughly as fast as the computation of the exponential (since the computation of the argument is not that costly).

The series algorithm computes the series defining the Artin-Hasse exponential and evaluates it.

The 'newton' algorithm solves the equation

\[\log(AH(x)) = x + \frac{x^p}{p} + \frac{x^{p^2}}{p^2} + \dots\]

using a Newton scheme. It runs roughly as fast as the computation of the logarithm.

By default, we use the 'direct' algorithm if a fast algorithm for computing the exponential is available. If not, we use the 'newton' algorithm if a fast algorithm for computing the logarithm is available. Otherwise we switch to the 'series' algorithm.

OUTPUT:

The Artin-Hasse exponential of this element.

See Wikipedia article Artin-Hasse_exponential for more information.

EXAMPLES:

sage: x = Zp(5)(45/7)
sage: y = x.artin_hasse_exp(); y
1 + 2*5 + 4*5^2 + 3*5^3 + 5^7 + 2*5^8 + 3*5^10 + 2*5^11 + 2*5^12 +
2*5^13 + 5^14 + 3*5^17 + 2*5^18 + 2*5^19 + O(5^20)

sage: y * (-x).artin_hasse_exp()
1 + O(5^20)

The function respects your precision:

sage: x = Zp(3,30)(45/7)
sage: x.artin_hasse_exp()
1 + 2*3^2 + 3^4 + 2*3^5 + 3^6 + 2*3^7 + 2*3^8 + 3^9 + 2*3^10 + 3^11 +
3^13 + 2*3^15 + 2*3^16 + 2*3^17 + 3^19 + 3^20 + 2*3^21 + 3^23 + 3^24 +
3^26 + 3^27 + 2*3^28 + O(3^30)

Unless you tell it not to:

sage: x = Zp(3,30)(45/7)
sage: x.artin_hasse_exp()
1 + 2*3^2 + 3^4 + 2*3^5 + 3^6 + 2*3^7 + 2*3^8 + 3^9 + 2*3^10 + 3^11 +
3^13 + 2*3^15 + 2*3^16 + 2*3^17 + 3^19 + 3^20 + 2*3^21 + 3^23 + 3^24 +
3^26 + 3^27 + 2*3^28 + O(3^30)
sage: x.artin_hasse_exp(10)
1 + 2*3^2 + 3^4 + 2*3^5 + 3^6 + 2*3^7 + 2*3^8 + 3^9 + O(3^10)

For precision 1 the function just returns 1 since the exponential is always a 1-unit:

sage: x = Zp(3).random_element()
sage: while x.dist(0) >= 1:
....:     x = Zp(3).random_element()
sage: x.artin_hasse_exp(1)
1 + O(3)

AUTHORS:

Mitchell Owen, Sebastian Pancrantz (2012-02): initial version.
Xavier Caruso (2018-08): extend to any p-adic rings and fields and implement several algorithms.

dwork_expansion(bd=20, a=0)#

Return the value of a function defined by Dwork.

Used to compute the \(p\)-adic Gamma function, see gamma().

INPUT:

bd – integer. Precision bound, defaults to 20
a – integer. Offset parameter, defaults to 0

OUTPUT:

A \(p\)-adic integer.

Note

This is based on GP code written by Fernando Rodriguez Villegas (http://www.ma.utexas.edu/cnt/cnt-frames.html). William Stein sped it up for GP (http://sage.math.washington.edu/home/wstein/www/home/wbhart/pari-2.4.2.alpha/src/basemath/trans2.c). The output is a \(p\)-adic integer from Dwork’s expansion, used to compute the \(p\)-adic gamma function as in [RV2007] section 6.2. The coefficients of the expansion are now cached to speed up multiple evaluation, as in the trace formula for hypergeometric motives.

EXAMPLES:

sage: R = Zp(17)
sage: x = R(5+3*17+13*17^2+6*17^3+12*17^5+10*17^(14)+5*17^(17)+O(17^(19)))
sage: x.dwork_expansion(18)
16 + 7*17 + 11*17^2 + 4*17^3 + 8*17^4 + 10*17^5 + 11*17^6 + 6*17^7
+ 17^8 + 8*17^10 + 13*17^11 + 9*17^12 + 15*17^13  + 2*17^14 + 6*17^15
+ 7*17^16 + 6*17^17 + O(17^18)

sage: R = Zp(5)
sage: x = R(3*5^2+4*5^3+1*5^4+2*5^5+1*5^(10)+O(5^(20)))
sage: x.dwork_expansion()
4 + 4*5 + 4*5^2 + 4*5^3 + 2*5^4 + 4*5^5 + 5^7 + 3*5^9 + 4*5^10 + 3*5^11
+ 5^13 + 4*5^14 + 2*5^15 + 2*5^16 + 2*5^17 + 3*5^18 + O(5^20)

exp(aprec=None, algorithm=None)#

Compute the \(p\)-adic exponential of this element if the exponential series converges.

INPUT:

aprec – an integer or None (default: None); if specified, computes only up to the indicated precision
algorithm – 'generic', 'binary_splitting', 'newton' or None (default)

The 'generic' algorithm evaluates naively the series defining the exponential, namely

\[\exp(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + \cdots.\]

Its binary complexity is quadratic with respect to the precision.

The 'binary_splitting' algorithm is faster, it has a quasi-linear complexity.

The 'newton' algorithms solve the equation \(\log(x) =\) self using a Newton scheme. It runs roughly as fast as the computation of the logarithm.

By default, we use the 'binary_splitting' if it is available. If it is not, we use the 'newton' algorithm if a fast algorithm for computing the logarithm is available. Otherwise we switch to the 'generic' algorithm.

EXAMPLES:

log() and exp() are inverse to each other:

sage: Z13 = Zp(13, 10)
sage: a = Z13(14); a
1 + 13 + O(13^10)
sage: a.log().exp()
1 + 13 + O(13^10)

An error occurs if this is called with an element for which the exponential series does not converge:

sage: Z13.one().exp()
Traceback (most recent call last):
...
ValueError: Exponential does not converge for that input.

The next few examples illustrate precision when computing \(p\)-adic exponentials:

sage: R = Zp(5,10)
sage: e = R(2*5 + 2*5**2 + 4*5**3 + 3*5**4
....:        + 5**5 + 3*5**7 + 2*5**8 + 4*5**9).add_bigoh(10); e
2*5 + 2*5^2 + 4*5^3 + 3*5^4 + 5^5 + 3*5^7 + 2*5^8 + 4*5^9 + O(5^10)
sage: e.exp()*R.teichmuller(4)
4 + 2*5 + 3*5^3 + O(5^10)

sage: K = Qp(5,10)
sage: e = K(2*5 + 2*5**2 + 4*5**3 + 3*5**4
....:        + 5**5 + 3*5**7 + 2*5**8 + 4*5**9).add_bigoh(10); e
2*5 + 2*5^2 + 4*5^3 + 3*5^4 + 5^5 + 3*5^7 + 2*5^8 + 4*5^9 + O(5^10)
sage: e.exp()*K.teichmuller(4)
4 + 2*5 + 3*5^3 + O(5^10)

Logarithms and exponentials in extension fields. First, in an Eisenstein extension:

sage: # needs sage.libs.ntl
sage: R = Zp(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)
sage: z.log().exp()
1 + w^2 + 4*w^7 + O(w^20)

Now an unramified example:

sage: # needs sage.libs.ntl
sage: R = Zp(5,5)
sage: S.<x> = R[]
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); b
1 + (a^2 + 1)*5 + (2*a + 3)*5^3 + O(5^5)
sage: b.log().exp()
1 + (a^2 + 1)*5 + (2*a + 3)*5^3 + O(5^5)

AUTHORS:

Genya Zaytman (2007-02-15)
Amnon Besser, Marc Masdeu (2012-02-23): Complete rewrite
Julian Rueth (2013-02-14): Added doctests, fixed some corner cases
Xavier Caruso (2017-06): Added binary splitting and Newton algorithms

gamma(algorithm='pari')#

Return the value of the \(p\)-adic Gamma function.

INPUT:

algorithm – string. Can be set to 'pari' to call the PARI function, or 'sage' to call the function implemented in Sage. The default is 'pari' since PARI is about 10 times faster than Sage.

OUTPUT:

a \(p\)-adic integer

Note

This is based on GP code written by Fernando Rodriguez Villegas (http://www.ma.utexas.edu/cnt/cnt-frames.html). William Stein sped it up for GP (http://sage.math.washington.edu/home/wstein/www/home/wbhart/pari-2.4.2.alpha/src/basemath/trans2.c). The 'sage' version uses dwork_expansion() to compute the \(p\)-adic gamma function of self as in [RV2007] section 6.2.

EXAMPLES:

This example illustrates x.gamma() for \(x\) a \(p\)-adic unit:

sage: R = Zp(7)
sage: x = R(2+3*7^2+4*7^3+O(7^20))
sage: x.gamma('pari')
1 + 2*7^2 + 4*7^3 + 5*7^4 + 3*7^5 + 7^8 + 7^9 + 4*7^10 + 3*7^12
+ 7^13 + 5*7^14 + 3*7^15 + 2*7^16 + 2*7^17 + 5*7^18 + 4*7^19 + O(7^20)
sage: x.gamma('sage')
1 + 2*7^2 + 4*7^3 + 5*7^4 + 3*7^5 + 7^8 + 7^9 + 4*7^10 + 3*7^12
+ 7^13 + 5*7^14 + 3*7^15 + 2*7^16 + 2*7^17 + 5*7^18 + 4*7^19 + O(7^20)
sage: x.gamma('pari') == x.gamma('sage')
True

Now x.gamma() for \(x\) a \(p\)-adic integer but not a unit:

sage: R = Zp(17)
sage: x = R(17+17^2+3*17^3+12*17^8+O(17^13))
sage: x.gamma('pari')
1 + 12*17 + 13*17^2 + 13*17^3 + 10*17^4 + 7*17^5 + 16*17^7
+ 13*17^9 + 4*17^10 + 9*17^11 + 17^12 + O(17^13)
sage: x.gamma('sage')
1 + 12*17 + 13*17^2 + 13*17^3 + 10*17^4 + 7*17^5 + 16*17^7
+ 13*17^9 + 4*17^10 + 9*17^11 + 17^12 + O(17^13)
sage: x.gamma('pari') == x.gamma('sage')
True

Finally, this function is not defined if \(x\) is not a \(p\)-adic integer:

sage: K = Qp(7)
sage: x = K(7^-5 + 2*7^-4 + 5*7^-3 + 2*7^-2 + 3*7^-1 + 3 + 3*7
....:       + 7^3 + 4*7^4 + 5*7^5 + 6*7^8 + 3*7^9 + 6*7^10 + 5*7^11 + 6*7^12
....:       + 3*7^13 + 5*7^14 + O(7^15))
sage: x.gamma()
Traceback (most recent call last):
...
ValueError: The p-adic gamma function only works on elements of Zp

gcd(other)#

Return a greatest common divisor of self and other.

INPUT:

other – an element in the same ring as self

AUTHORS:

Julian Rueth (2012-10-19): initial version

Note

Since the elements are only given with finite precision, their greatest common divisor is in general not unique (not even up to units). For example \(O(3)\) is a representative for the elements 0 and 3 in the 3-adic ring \(\ZZ_3\). The greatest common divisor of \(O(3)\) and \(O(3)\) could be (among others) 3 or 0 which have different valuation. The algorithm implemented here, will return an element of minimal valuation among the possible greatest common divisors.

EXAMPLES:

The greatest common divisor is either zero or a power of the uniformizing parameter:

sage: R = Zp(3)
sage: R.zero().gcd(R.zero())
0
sage: R(3).gcd(9)
3 + O(3^21)

A non-zero result is always lifted to the maximal precision possible in the ring:

sage: a = R(3,2); a
3 + O(3^2)
sage: b = R(9,3); b
3^2 + O(3^3)
sage: a.gcd(b)
3 + O(3^21)
sage: a.gcd(0)
3 + O(3^21)

If both elements are zero, then the result is zero with the precision set to the smallest of their precisions:

sage: a = R.zero(); a
0
sage: b = R(0,2); b
O(3^2)
sage: a.gcd(b)
O(3^2)

One could argue that it is mathematically correct to return \(9 + O(3^{22})\) instead. However, this would lead to some confusing behaviour:

sage: alternative_gcd = R(9,22); alternative_gcd
3^2 + O(3^22)
sage: a.is_zero()
True
sage: b.is_zero()
True
sage: alternative_gcd.is_zero()
False

If exactly one element is zero, then the result depends on the valuation of the other element:

sage: R(0,3).gcd(3^4)
O(3^3)
sage: R(0,4).gcd(3^4)
O(3^4)
sage: R(0,5).gcd(3^4)
3^4 + O(3^24)

Over a field, the greatest common divisor is either zero (possibly with finite precision) or one:

sage: K = Qp(3)
sage: K(3).gcd(0)
1 + O(3^20)
sage: K.zero().gcd(0)
0
sage: K.zero().gcd(K(0,2))
O(3^2)
sage: K(3).gcd(4)
1 + O(3^20)

is_prime()#

Return whether this element is prime in its parent.

EXAMPLES:

sage: A = Zp(2)
sage: A(1).is_prime()
False
sage: A(2).is_prime()
True

sage: K = A.fraction_field()
sage: K(2).is_prime()
False

sage: # needs sage.libs.ntl
sage: x = polygen(ZZ, 'x')
sage: B.<pi> = A.extension(x^5 - 2)
sage: pi.is_prime()                                                         # needs sage.symbolic
True
sage: B(2).is_prime()
False

is_square()#

Returns whether this element is a square

INPUT:

self – a \(p\)-adic element

EXAMPLES:

sage: R = Zp(3,20,'capped-rel')
sage: R(0).is_square()
True
sage: R(1).is_square()
True
sage: R(2).is_square()
False

is_squarefree()#

Return whether this element is squarefree, i.e., whether there exists no non-unit \(g\) such that \(g^2\) divides this element.

EXAMPLES:

The zero element is never squarefree:

sage: K = Qp(2)
sage: K.zero().is_squarefree()
False

In \(p\)-adic rings, only elements of valuation at most 1 are squarefree:

sage: R = Zp(2)
sage: R(1).is_squarefree()
True
sage: R(2).is_squarefree()
True
sage: R(4).is_squarefree()
False

This works only if the precision is known sufficiently well:

sage: R(0,1).is_squarefree()
Traceback (most recent call last):
...
PrecisionError: element not known to sufficient precision to decide squarefreeness
sage: R(0,2).is_squarefree()
False
sage: R(1,1).is_squarefree()
True

For fields we are not so strict about the precision and treat inexact zeros as the zero element:

K(0,0).is_squarefree()
False

log(p_branch=None, pi_branch=None, aprec=None, change_frac=False, algorithm=None)#

Compute the \(p\)-adic logarithm of this element.

The usual power series for the logarithm with values in the additive group of a \(p\)-adic ring only converges for 1-units (units congruent to 1 modulo \(p\)). However, there is a unique extension of the logarithm to a homomorphism defined on all the units: If \(u = a \cdot v\) is a unit with \(v \equiv 1 \pmod{p}\) and \(a\) a Teichmuller representative, then we define \(log(u) = log(v)\). This is the correct extension because the units \(U\) split as a product \(U = V \times \langle w \rangle\), where \(V\) is the subgroup of 1-units and \(w\) is a fundamental root of unity. The \(\langle w \rangle\) factor is torsion, so must go to 0 under any homomorphism to the fraction field, which is a torsion free group.

INPUT:

p_branch – an element in the base ring or its fraction field; the implementation will choose the branch of the logarithm which sends \(p\) to branch
pi_branch – an element in the base ring or its fraction field; the implementation will choose the branch of the logarithm which sends the uniformizer to branch; you may specify at most one of p_branch and pi_branch, and must specify one of them if this element is not a unit
aprec – an integer or None (default: None); if not None, then the result will only be correct to precision aprec
change_frac – In general the codomain of the logarithm should be in the \(p\)-adic field, however, for most neighborhoods of 1, it lies in the ring of integers. This flag decides if the codomain should be the same as the input (default) or if it should change to the fraction field of the input.
algorithm – 'generic', 'binary_splitting' or None (default) The generic algorithm evaluates naively the series defining the log, namely

\[\log(1-x) = -x - 1/2 x^2 - 1/3 x^3 - 1/4 x^4 - 1/5 x^5 - \cdots.\]

Its binary complexity is quadratic with respect to the precision.

The 'binary_splitting' algorithm is faster, it has a quasi-linear complexity. By default, we use 'binary_splitting' if it is available. Otherwise we switch to the 'generic' algorithm.

Note

What some other systems do:

PARI: Seems to define the logarithm for units not congruent to 1 as we do.
MAGMA: Only implements logarithm for 1-units (version 2.19-2)

Todo

There is a soft-linear time algorithm for logarithm described by Dan Bernstein at http://cr.yp.to/lineartime/multapps-20041007.pdf

EXAMPLES:

sage: Z13 = Zp(13, 10)
sage: a = Z13(14); a
1 + 13 + O(13^10)
sage: a.log()
13 + 6*13^2 + 2*13^3 + 5*13^4 + 10*13^6 + 13^7 + 11*13^8 + 8*13^9 + O(13^10)

sage: Q13 = Qp(13, 10)
sage: a = Q13(14); a
1 + 13 + O(13^10)
sage: a.log()
13 + 6*13^2 + 2*13^3 + 5*13^4 + 10*13^6 + 13^7 + 11*13^8 + 8*13^9 + O(13^10)

Note that the relative precision decreases when we take log. Precisely the absolute precision on log(a) agrees with the relative precision on a thanks to the relation \(d\log(a) = da/a\).

The call log(a) works as well:

sage: log(a)
13 + 6*13^2 + 2*13^3 + 5*13^4 + 10*13^6 + 13^7 + 11*13^8 + 8*13^9 + O(13^10)
sage: log(a) == a.log()
True

The logarithm is not only defined for 1-units:

sage: R = Zp(5,10)
sage: a = R(2)
sage: a.log()
2*5 + 3*5^2 + 2*5^3 + 4*5^4 + 2*5^6 + 2*5^7 + 4*5^8 + 2*5^9 + O(5^10)

If you want to take the logarithm of a non-unit you must specify either p_branch or pi_branch:

sage: b = R(5)
sage: b.log()
Traceback (most recent call last):
...
ValueError: you must specify a branch of the logarithm for non-units
sage: b.log(p_branch=4)
4 + O(5^10)
sage: c = R(10)
sage: c.log(p_branch=4)
4 + 2*5 + 3*5^2 + 2*5^3 + 4*5^4 + 2*5^6 + 2*5^7 + 4*5^8 + 2*5^9 + O(5^10)

The branch parameters are only relevant for elements of non-zero valuation:

sage: a.log(p_branch=0)
2*5 + 3*5^2 + 2*5^3 + 4*5^4 + 2*5^6 + 2*5^7 + 4*5^8 + 2*5^9 + O(5^10)
sage: a.log(p_branch=1)
2*5 + 3*5^2 + 2*5^3 + 4*5^4 + 2*5^6 + 2*5^7 + 4*5^8 + 2*5^9 + O(5^10)

Logarithms can also be computed in extension fields. First, in an Eisenstein extension:

sage: R = Zp(5,5)
sage: S.<x> = ZZ[]
sage: f = x^4 + 15*x^2 + 625*x - 5
sage: W.<w> = R.ext(f)                                                      # needs sage.libs.ntl
sage: z = 1 + w^2 + 4*w^7; z                                                # needs sage.libs.ntl
1 + w^2 + 4*w^7 + O(w^20)
sage: z.log()                                                               # needs sage.libs.ntl
w^2 + 2*w^4 + 3*w^6 + 4*w^7 + w^9 + 4*w^10 + 4*w^11 + 4*w^12
 + 3*w^14 + w^15 + w^17 + 3*w^18 + 3*w^19 + O(w^20)

In an extension, there will usually be a difference between specifying p_branch and pi_branch:

sage: # needs sage.libs.ntl
sage: b = W(5)
sage: b.log()
Traceback (most recent call last):
...
ValueError: you must specify a branch of the logarithm for non-units
sage: b.log(p_branch=0)
O(w^20)
sage: b.log(p_branch=w)
w + O(w^20)
sage: b.log(pi_branch=0)
3*w^2 + 2*w^4 + 2*w^6 + 3*w^8 + 4*w^10 + w^13 + w^14 + 2*w^15
 + 2*w^16 + w^18 + 4*w^19 + O(w^20)
sage: b.unit_part().log()
3*w^2 + 2*w^4 + 2*w^6 + 3*w^8 + 4*w^10 + w^13 + w^14 + 2*w^15
 + 2*w^16 + w^18 + 4*w^19 + O(w^20)
sage: y = w^2 * 4*w^7; y
4*w^9 + O(w^29)
sage: y.log(p_branch=0)
2*w^2 + 2*w^4 + 2*w^6 + 2*w^8 + w^10 + w^12 + 4*w^13 + 4*w^14 + 3*w^15
 + 4*w^16 + 4*w^17 + w^18 + 4*w^19 + O(w^20)
sage: y.log(p_branch=w)
w + 2*w^2 + 2*w^4 + 4*w^5 + 2*w^6 + 2*w^7 + 2*w^8 + 4*w^9 + w^10
 + 3*w^11 + w^12 + 4*w^14 + 4*w^16 + 2*w^17 + w^19 + O(w^20)

Check that log is multiplicative:

sage: y.log(p_branch=0) + z.log() - (y*z).log(p_branch=0)                   # needs sage.libs.ntl
O(w^20)

Now an unramified example:

sage: # needs sage.libs.ntl
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()
(a^2 + 1)*5 + (3*a^2 + 4*a + 2)*5^2 + (3*a^2 + 2*a)*5^3
 + (3*a^2 + 2*a + 2)*5^4 + O(5^5)

Check that log is multiplicative:

sage: # needs sage.libs.ntl
sage: c = 3 + 5^2*(2 + 4*a)
sage: b.log() + c.log() - (b*c).log()
O(5^5)

We illustrate the effect of the precision argument:

sage: R = ZpCA(7,10)
sage: x = R(41152263); x
5 + 3*7^2 + 4*7^3 + 3*7^4 + 5*7^5 + 6*7^6 + 7^9 + O(7^10)
sage: x.log(aprec = 5)
7 + 3*7^2 + 4*7^3 + 3*7^4 + O(7^5)
sage: x.log(aprec = 7)
7 + 3*7^2 + 4*7^3 + 3*7^4 + 7^5 + 3*7^6 + O(7^7)
sage: x.log()
7 + 3*7^2 + 4*7^3 + 3*7^4 + 7^5 + 3*7^6 + 7^7 + 3*7^8 + 4*7^9 + O(7^10)

The logarithm is not defined for zero:

sage: R.zero().log()
Traceback (most recent call last):
...
ValueError: logarithm is not defined at zero

For elements in a \(p\)-adic ring, the logarithm will be returned in the same ring:

sage: x = R(2)
sage: x.log().parent()
7-adic Ring with capped absolute precision 10
sage: x = R(14)
sage: x.log(p_branch=0).parent()
7-adic Ring with capped absolute precision 10

This is not possible if the logarithm has negative valuation:

sage: R = ZpCA(3,10)
sage: S.<x> = R[]
sage: f = x^3 - 3
sage: W.<w> = R.ext(f)                                                      # needs sage.libs.ntl sage.rings.padics
sage: w.log(p_branch=2)                                                     # needs sage.libs.ntl sage.rings.padics
Traceback (most recent call last):
...
ValueError: logarithm is not integral, use change_frac=True
to obtain a result in the fraction field
sage: w.log(p_branch=2, change_frac=True)                                   # needs sage.libs.ntl sage.rings.padics
2*w^-3 + O(w^24)

AUTHORS:

William Stein: initial version
David Harvey (2006-09-13): corrected subtle precision bug (need to take denominators into account! – see github issue #53)
Genya Zaytman (2007-02-14): adapted to new \(p\)-adic class
Amnon Besser, Marc Masdeu (2012-02-21): complete rewrite, valid for generic \(p\)-adic rings.
Soroosh Yazdani (2013-02-1): Fixed a precision issue in _log_generic(). This should really fix the issue with divisions.
Julian Rueth (2013-02-14): Added doctests, some changes for capped-absolute implementations.
Xavier Caruso (2017-06): Added binary splitting type algorithms over Qp

minimal_polynomial(name='x', base=None)#

Returns the minimal polynomial of this element over base

INPUT:

name – string (default: 'x'): the name of the variable
base – a ring (default: the base ring of the parent): the base ring over which the minimal polynomial is computed

EXAMPLES:

sage: Zp(5,5)(1/3).minimal_polynomial('x')                                  # needs sage.libs.ntl
(1 + O(5^5))*x + 3 + 5 + 3*5^2 + 5^3 + 3*5^4 + O(5^5)

sage: Zp(5,5)(1/3).minimal_polynomial('foo')                                # needs sage.libs.ntl
(1 + O(5^5))*foo + 3 + 5 + 3*5^2 + 5^3 + 3*5^4 + O(5^5)

sage: # needs sage.libs.ntl
sage: K.<a> = QqCR(2^3,5)
sage: S.<x> = K[]
sage: L.<pi> = K.extension(x^4 - 2*a)
sage: pi.minimal_polynomial()                                               # needs sage.symbolic
(1 + O(2^5))*x^4 + a*2 + a*2^2 + a*2^3 + a*2^4 + a*2^5 + O(2^6)
sage: (pi^2).minimal_polynomial()                                           # needs sage.symbolic
(1 + O(2^5))*x^2 + a*2 + a*2^2 + a*2^3 + a*2^4 + a*2^5 + O(2^6)
sage: (1/pi).minimal_polynomial()                                           # needs sage.symbolic
(1 + O(2^5))*x^4 + (a^2 + 1)*2^-1 + O(2^4)
sage: elt = L.random_element()
sage: P = elt.minimal_polynomial()  # not tested, known bug (see :issue:`32111`)
sage: P(elt) == 0  # not tested
True

multiplicative_order(prec=None)#

Returns the multiplicative order of self, where self is considered to be one if it is one modulo \(p^{\mbox{prec}}\).

INPUT:

self – a \(p\)-adic element
prec – an integer

OUTPUT:

integer – the multiplicative order of self

EXAMPLES:

sage: K = Qp(5,20,'capped-rel')
sage: K(-1).multiplicative_order(20)
2
sage: K(1).multiplicative_order(20)
1
sage: K(2).multiplicative_order(20)
+Infinity
sage: K(5).multiplicative_order(20)
+Infinity
sage: K(1/5).multiplicative_order(20)
+Infinity
sage: K.zeta().multiplicative_order(20)
4

Over unramified extensions:

sage: # needs sage.libs.ntl
sage: L1.<a> = Qq(5^3)
sage: c = L1.teichmuller(a)
sage: c.multiplicative_order()
124
sage: c^124
1 + O(5^20)

Over totally ramified extensions:

sage: # needs sage.libs.ntl
sage: x = polygen(ZZ, 'x')
sage: L2.<pi> = Qp(5).extension(x^4 + 5*x^3 + 10*x^2 + 10*x + 5)
sage: u = 1 + pi
sage: u.multiplicative_order()
5
sage: v = L2.teichmuller(2)
sage: v.multiplicative_order()
4
sage: (u*v).multiplicative_order()
20

norm(base=None)#

Return the norm of this \(p\)-adic element over base.

Warning

This is not the \(p\)-adic absolute value. This is a field theoretic norm down to a base ring. If you want the \(p\)-adic absolute value, use the method abs() instead.

INPUT:

base – a subring of the parent (default: base ring)

OUTPUT:

The norm of this \(p\)-adic element over the given base.

EXAMPLES:

sage: Zp(5)(5).norm()                                                       # needs sage.libs.ntl
5 + O(5^21)

sage: # needs sage.libs.ntl
sage: K.<a> = QqCR(2^3,5)
sage: S.<x> = K[]
sage: L.<pi> = K.extension(x^4 - 2*a)
sage: pi.norm()  # norm over K                                              # needs sage.symbolic
a*2 + a*2^2 + a*2^3 + a*2^4 + a*2^5 + O(2^6)
sage: (pi^2).norm()                                                         # needs sage.symbolic
a^2*2^2 + O(2^7)
sage: pi.norm()^2                                                           # needs sage.symbolic
a^2*2^2 + O(2^7)

nth_root(n, all=False)#

Return the \(n\)-th root of this element.

INPUT:

n – an integer
all – a boolean (default: False): if True, return all \(n\)-th roots of this element, instead of just one.

EXAMPLES:

sage: A = Zp(5,10)
sage: x = A(61376); x
1 + 5^3 + 3*5^4 + 4*5^5 + 3*5^6 + O(5^10)
sage: y = x.nth_root(4); y
2 + 5 + 2*5^2 + 4*5^3 + 3*5^4 + 5^6 + O(5^10)
sage: y^4 == x
True

sage: x.nth_root(4, all=True)
[2 + 5 + 2*5^2 + 4*5^3 + 3*5^4 + 5^6 + O(5^10),
 4 + 4*5 + 4*5^2 + 4*5^4 + 3*5^5 + 5^6 + 3*5^7 + 5^8 + 5^9 + O(5^10),
 3 + 3*5 + 2*5^2 + 5^4 + 4*5^5 + 3*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10),
 1 + 4*5^3 + 5^5 + 3*5^6 + 5^7 + 3*5^8 + 3*5^9 + O(5^10)]

When \(n\) is divisible by the underlying prime \(p\), we are losing precision (which is consistent with the fact that raising to the \(p\)-th power increases precision):

sage: z = x.nth_root(5); z
1 + 5^2 + 3*5^3 + 2*5^4 + 5^5 + 3*5^7 + 2*5^8 + O(5^9)
sage: z^5
1 + 5^3 + 3*5^4 + 4*5^5 + 3*5^6 + O(5^10)

Everything works over extensions as well:

sage: # needs sage.libs.ntl
sage: W.<a> = Zq(5^3)
sage: S.<x> = W[]
sage: R.<pi> = W.extension(x^7 - 5)
sage: R(5).nth_root(7)
pi + O(pi^141)
sage: R(5).nth_root(7, all=True)
[pi + O(pi^141)]

An error is raised if the given element is not an \(n\)-th power in the ring:

sage: R(5).nth_root(11)                                                     # needs sage.libs.ntl
Traceback (most recent call last):
...
ValueError: this element is not a nth power

Similarly, when precision on the input is too small, an error is raised:

sage: # needs sage.libs.ntl
sage: x = R(1,6); x
1 + O(pi^6)
sage: x.nth_root(5)
Traceback (most recent call last):
...
PrecisionError: not enough precision to be sure that this element is a nth power

Check that github issue #30314 is fixed:

sage: # needs sage.libs.ntl
sage: K = Qp(29)
sage: x = polygen(K)
sage: L.<a> = K.extension(x^2 - 29)
sage: L(4).nth_root(2)
2 + O(a^40)

ordp(p=None)#

Return the valuation of self, normalized so that the valuation of \(p\) is 1.

INPUT:

self – a \(p\)-adic element
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 integers and rationals.

OUTPUT:

integer – the valuation of self, normalized so that the valuation of \(p\) is 1

EXAMPLES:

sage: R = Zp(5,20,'capped-rel')
sage: R(0).ordp()
+Infinity
sage: R(1).ordp()
0
sage: R(2).ordp()
0
sage: R(5).ordp()
1
sage: R(10).ordp()
1
sage: R(25).ordp()
2
sage: R(50).ordp()
2
sage: R(1/2).ordp()
0

polylog(n, p_branch=0)#

Return \(Li_n(\mathrm{self})\), the \(n\)-th \(p\)-adic polylogarithm of this element.

INPUT:

n – a non-negative integer
p_branch – an element in the base ring or its fraction field; the implementation will choose the branch of the logarithm which sends \(p\) to p_branch

EXAMPLES:

The \(n\)-th polylogarithm of \(-1\) is \(0\) for even \(n\):

sage: Qp(13)(-1).polylog(6) == 0                                            # needs sage.rings.real_mpfr sage.symbolic
True

We can check some identities, for example those mentioned in [DCW2016]:

sage: x = Qp(7, prec=30)(1/3)
sage: (x^2).polylog(4) - 8*x.polylog(4) - 8*(-x).polylog(4) == 0            # needs sage.symbolic
True

sage: x = Qp(5, prec=30)(4)
sage: x.polylog(2) + (1/x).polylog(2) + x.log(0)**2/2 == 0                  # needs sage.symbolic
True

sage: x = Qp(11, prec=30)(2)
sage: x.polylog(2) + (1-x).polylog(2) + x.log(0)**2*(1-x).log(0) == 0       # needs sage.symbolic
True

\(Li_1(z) = -\log(1-z)\) for \(|z| < 1\):

sage: Qp(5)(10).polylog(1) == -Qp(5)(1-10).log(0)
True

The dilogarithm of 1 is zero:

sage: Qp(5)(1).polylog(2)                                                   # needs sage.rings.real_mpfr sage.symbolic
O(5^20)

The cubing relation holds for the trilogarithm at 1:

sage: K = Qp(7)
sage: z = K.zeta(3)
sage: -8*K(1).polylog(3) == 9*(K(z).polylog(3) + K(z^2).polylog(3))         # needs sage.rings.padics sage.rings.real_mpfr sage.symbolic
True

The polylogarithm of 0 is 0:

sage: Qp(11)(0).polylog(7)
0

Only polylogarithms for positive \(n\) are defined:

sage: Qp(11)(2).polylog(-1)
Traceback (most recent call last):
...
ValueError: polylogarithm only implemented for n at least 0

Check that github issue #29222 is fixed:

sage: K = Qp(7)
sage: print(K(1 + 7^11).polylog(4))                                         # needs sage.symbolic
6*7^14 + 3*7^15 + 7^16 + 7^17 + O(7^18)

ALGORITHM:

The algorithm of Besser-de Jeu, as described in [BdJ2008] is used.

AUTHORS:

Jennifer Balakrishnan - Initial implementation
Alex J. Best (2017-07-21) - Extended to other residue disks

Todo

Implement for extensions.
Use the change method to create K from self.parent().

rational_reconstruction()#

Returns a rational approximation to this \(p\)-adic number.

This will raise an ArithmeticError if there are no valid approximations to the unit part with numerator and denominator bounded by sqrt(p^absprec / 2).