Module padic_ZZ_pX_CA_element
File: sage/rings/padics/padic_ZZ_pX_CA_element.pyx (starting at line 1)
This file implements elements of eisenstein and unramified extensions of Zp and Qp 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:
absprec (long) -- An integer giving the precision to which this element is defined. This is the power of the uniformizer
modulo which the element is well defined.
value (ZZ_pX_c) -- An ntl ZZ_pX storing the value. The varible x is the uniformizer in the case of eisenstein extensions.
This ZZ_pX is created with global ntl modulus determined by absprec. Let a be absprec and e be the
ramification index over Qp or Zp. Then the modulus is given by p^ceil(a/e). Note that all kinds
of problems arise if you try to mix moduli. ZZ_pX_conv_modulus gives a semi-safe way to convert
between different moduli without having to pass through ZZX (see sage/libs/ntl/decl.pxi and c_lib/src/ntl_wrap.cpp)
prime_pow (some subclass of PowComputer_ZZ_pX) -- a class, identical among all elements with the same parent, holding
common data.
prime_pow.deg -- The degree of the extension
prime_pow.e -- The ramification index
prime_pow.f -- The inertia degree
prime_pow.prec_cap -- the unramified precision cap. For eisenstein extensions this is the smallest power of p that is zero.
prime_pow.ram_prec_cap -- the ramified precision cap. For eisenstein extensions this will be the smallest power of x that is
indistinugishable from zero.
prime_pow.pow_ZZ_tmp, prime_pow.pow_mpz_t_tmp, prime_pow.pow_Integer -- functions for accessing powers of p.
The first two return pointers. See sage/rings/padics/pow_computer_ext for examples and important warnings.
prime_pow.get_context, prime_pow.get_context_capdiv, prime_pow.get_top_context -- obtain an ntl_ZZ_pContext_class corresponding to p^n.
The capdiv version divides by prime_pow.e as appropriate. top_context corresponds to prec_cap.
prime_pow.restore_context, prime_pow.restore_context_capdiv, prime_pow.restore_top_context -- restores the given context.
prime_pow.get_modulus, get_modulus_capdiv, get_top_modulus -- Returns a ZZ_pX_Modulus_c* pointing to a polynomial modulus defined modulo
p^n (appropriately divided by prime_pow.e in the capdiv case).
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 + (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(5^5))*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))
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)
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:
-- David Roe (2008-01-01) initial version
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make_ZZpXCAElement(...)
File: sage/rings/padics/padic_ZZ_pX_CA_element.pyx (starting at line 1825)
For pickling. |
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File: sage/rings/padics/padic_ZZ_pX_CA_element.pyx (starting at line 1825)
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)
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