-Adic ZZ_pX CA Element
This file implements elements of eisenstein and unramified extensions of Zp with capped absolute 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:
EXAMPLES:
An eisenstein extension:
sage: R = ZpCA(5,5)
sage: S.<x> = ZZ[]
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 absolute precision 5 in w defined by (1 + O(5^5))*x^5 + (O(5^5))*x^4 + (3*5^2 + O(5^5))*x^3 + (2*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5))*x^2 + (5^3 + O(5^5))*x + (4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5))
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^12)
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)
sage: FFA = A.residue_field()
sage: a0 = FFA.gen(); A(a0^3)
(2*a + 2) + O(5)
Different printing modes:
sage: R = ZpCA(5, print_mode='digits'); S.<x> = ZZ[]; 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 = ZpCA(5, print_mode='bars'); S.<x> = ZZ[]; 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 = ZpCA(5, print_mode='terse'); S.<x> = ZZ[]; 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 = ZpCA(5, print_mode='val-unit'); S.<x> = ZZ[]; 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 representation:
sage: z._ntl_rep()
[6 95367431640505 25 95367431640560 5]
sage: y._ntl_rep()
[5 95367431640505 25 95367431640560 5]
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:
For pickling. Makes a pAdicZZpXCAElement with given parent, value, absprec.
EXAMPLES:
sage: from sage.rings.padics.padic_ZZ_pX_CA_element import make_ZZpXCAElement
sage: R = ZpCA(5,5)
sage: S.<x> = ZZ[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: make_ZZpXCAElement(W, ntl.ZZ_pX([3,2,4],5^3),13,0)
3 + 2*w + 4*w^2 + O(w^13)
Bases: sage.rings.padics.padic_ZZ_pX_element.pAdicZZpXElement
Creates an element of a capped absolute precision, unramified or eisenstein extension of Zp or Qp.
INPUT:
EXAMPLES:
sage: R = ZpCA(5,5)
sage: S.<x> = ZZ[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
sage: z = (1+w)^5; z # indirect doctest
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: W(R(3,3))
3 + O(w^15)
sage: W(pari('3 + O(5^3)'))
3 + O(w^15)
sage: W(w, 14)
w + O(w^14)
TESTS:
Check that trac ticket #13600 is fixed:
sage: K = W.fraction_field()
sage: W(K.zero())
O(w^25)
sage: W(K.one())
1 + O(w^25)
sage: W(K.zero().add_bigoh(3))
O(w^3)
Check that trac ticket #3865 is fixed:
sage: W(gp(‘5 + O(5^2)’)) w^5 + 2*w^7 + 4*w^9 + O(w^10)
Check that trac ticket #13612 has been fixed:
sage: R = ZpCA(3)
sage: S.<a> = R[]
sage: W.<a> = R.extension(a^2+1)
sage: W(W.residue_field().zero())
O(3)
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 = ZpCA(5,5)
sage: S.<x> = ZZ[]
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
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 = ZpCA(5,5)
sage: S.<x> = ZZ[]
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
Returns a pAdicZZpXCAElement congruent to self but with absolute precision at least absprec.
INPUT:
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(5,5)
sage: S.<x> = ZZ[]
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 # indirect doctest
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()
[345]
sage: b._ntl_rep()
[345]
sage: c._ntl_rep()
[345]
sage: a.lift_to_precision().precision_absolute() == W.precision_cap()
True
Returns a list giving a series representation of self.
EXAMPLES:
sage: R = ZpCA(5,5)
sage: S.<x> = ZZ[]
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(5^5)
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(5^5)
sage: W(0).list()
[0]
sage: A(0,4).list()
[[]]
Returns the matrix of right multiplication by the element on
the power basis for this
extension field. Thus the rows of this matrix give the
images of each of the
. The entries of the matrices are
IntegerMod elements, defined modulo p^(self.absprec() / e).
EXAMPLES:
sage: R = ZpCA(5,5)
sage: S.<x> = ZZ[]
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]
Returns the absolute precision of self, ie the power of the uniformizer modulo which this element is defined.
EXAMPLES:
sage: R = ZpCA(5,5)
sage: S.<x> = ZZ[]
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)
Returns the relative precision of self, ie the power of the uniformizer modulo which the unit part of self is defined.
EXAMPLES:
sage: R = ZpCA(5,5)
sage: S.<x> = ZZ[]
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)
Returns a list [,
,...,
] such that
EXAMPLES:
sage: R.<a> = Zq(5^4,4)
sage: L = a.teichmuller_list(); L
[a + (2*a^3 + 2*a^2 + 3*a + 4)*5 + (4*a^3 + 3*a^2 + 3*a + 2)*5^2 + (4*a^2 + 2*a + 2)*5^3 + O(5^4), (3*a^3 + 3*a^2 + 2*a + 1) + (a^3 + 4*a^2 + 1)*5 + (a^2 + 4*a + 4)*5^2 + O(5^3), (4*a^3 + 2*a^2 + a + 1) + (2*a^3 + 2*a^2 + 2*a + 4)*5 + O(5^2), (a^3 + a^2 + a + 4) + O(5)]
sage: sum([5^i*L[i] for i in range(4)])
a + O(5^4)
sage: all([L[i]^625 == L[i] for i in range(4)])
True
sage: S.<x> = ZZ[]
sage: f = x^3 - 98*x + 7
sage: W.<w> = ZpCA(7,3).ext(f)
sage: b = (1+w)^5; L = b.teichmuller_list(); L
[1 + O(w^9), 5 + 5*w^3 + w^6 + 4*w^7 + O(w^8), 3 + 3*w^3 + O(w^7), 3 + 3*w^3 + O(w^6), O(w^5), 4 + 5*w^3 + O(w^4), 3 + O(w^3), 6 + O(w^2), 6 + O(w)]
sage: sum([w^i*L[i] for i in range(9)]) == b
True
sage: all([L[i]^(7^3) == L[i] for i in range(9)])
True
sage: L = W(3).teichmuller_list(); L
[3 + 3*w^3 + w^7 + O(w^9), O(w^8), O(w^7), 4 + 5*w^3 + O(w^6), O(w^5), O(w^4), 3 + O(w^3), 6 + O(w^2)]
sage: sum([w^i*L[i] for i in range(len(L))])
3 + O(w^9)
Returns self cast into the fraction field of self.parent().
EXAMPLES:
sage: R = ZpCA(5,5)
sage: S.<x> = ZZ[]
sage: f = x^5 + 75*x^3 - 15*x^2 +125*x - 5
sage: W.<w> = R.ext(f)
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.to_fraction_field(); y #indirect doctest
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.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))
Returns the unit part of self, ie self / uniformizer^(self.valuation())
EXAMPLES:
sage: R = ZpCA(5,5)
sage: S.<x> = ZZ[]
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)