\(p\)-adic ZZ_pX CR Element¶
This file implements elements of Eisenstein and unramified extensions of \(\ZZ_p\) and \(\QQ_p\) with capped relative precision.
For the parent class see sage.rings.padics.padic_extension_leaves.
The underlying implementation is through NTL’s ZZ_pX class. Each
element contains the following data:
ordp–long; a power of the uniformizer to scale the unit by. For unramified extensions this uniformizer is \(p\), for Eisenstein extensions it is not. A value equal to the maximum value of alongindicates that the element is an exact zero.relprec–long; a signed integer giving the precision to which this element is defined. For nonzerorelprec, the absolute value gives the power of the uniformizer modulo which the unit is defined. A positive value indicates that the element is normalized (ieunitis actually a unit: in the case of Eisenstein extensions the constant term is not divisible by \(p\), in the case of unramified extensions that there is at least one coefficient that is not divisible by \(p\)). A negative value indicates that the element may or may not be normalized. A zero value indicates that the element is zero to some precision. If so,ordpgives the absolute precision of the element. Ifordpis greater thanmaxordp, then the element is an exact zero.unit–ZZ_pX_c; an ntlZZ_pXstoring the unit part The variable \(x\) is the uniformizer in the case of Eisenstein extensions. If the element is not normalized, theunitmay or may not actually be a unit. ThisZZ_pXis created with global ntl modulus determined by the absolute value ofrelprec. Ifrelprecis 0,unitis not initialized, or destructed if normalized and found to be zero. Otherwise, let \(r\) berelprecand \(e\) be the ramification index over \(\QQ_p\) or \(\ZZ_p\). Then the modulus of unit is given by \(p^{ceil(r/e)}\). Note that all kinds of problems arise if you try to mix moduli.ZZ_pX_conv_modulusgives a semi-safe way to convert between different moduli without having to pass throughZZX.prime_pow(some subclass ofPowComputer_ZZ_pX) – a class, identical among all elements with the same parent, holding common data.prime_pow.deg– the degree of the extensionprime_pow.e– the ramification indexprime_pow.f– the inertia degreeprime_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 indistinguishable 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. Seesage.rings.padics.pow_computer_extfor examples and important warnings.prime_pow.get_context,prime_pow.get_context_capdiv,prime_pow.get_top_context– obtain anntl_ZZ_pContext_classcorresponding to \(p^n\). The capdiv version divides byprime_pow.eas appropriate.top_contextcorresponds to \(p^{\texttt{prec\_cap}}\).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 aZZ_pX_Modulus_c*pointing to a polynomial modulus defined modulo \(p^n\) (appropriately divided byprime_pow.ein the capdiv case).
EXAMPLES:
An Eisenstein extension:
sage: R = Zp(5,5)
sage: S.<x> = R[]
sage: f = x^5 + 75*x^3 - 15*x^2 + 125*x - 5
sage: W.<w> = R.ext(f); W
5-adic Eisenstein Extension Ring in w defined by x^5 + 75*x^3 - 15*x^2 + 125*x - 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^13)
sage: (1/w).parent()
5-adic Eisenstein Extension Field in w defined by x^5 + 75*x^3 - 15*x^2 + 125*x - 5
>>> from sage.all import *
>>> R = Zp(Integer(5),Integer(5))
>>> S = R['x']; (x,) = S._first_ngens(1)
>>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x - Integer(5)
>>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1); W
5-adic Eisenstein Extension Ring in w defined by x^5 + 75*x^3 - 15*x^2 + 125*x - 5
>>> z = (Integer(1)+w)**Integer(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)
>>> y = z >> Integer(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)
>>> y.valuation()
4
>>> y.precision_relative()
20
>>> y.precision_absolute()
24
>>> z - (y << Integer(1))
1 + O(w^25)
>>> (Integer(1)/w)**Integer(12)+w
w^-12 + w + O(w^13)
>>> (Integer(1)/w).parent()
5-adic Eisenstein Extension Field in w defined by x^5 + 75*x^3 - 15*x^2 + 125*x - 5
Unramified extensions:
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: FFp = R.residue_field()
sage: R(FFp(3))
3 + O(5)
sage: QQq.<zz> = Qq(25,4)
sage: QQq(FFp(3))
3 + O(5)
sage: FFq = QQq.residue_field(); QQq(FFq(3))
3 + O(5)
sage: zz0 = FFq.gen(); QQq(zz0^2)
(zz + 3) + O(5)
>>> from sage.all import *
>>> g = x**Integer(3) + Integer(3)*x + Integer(3)
>>> A = R.ext(g, names=('a',)); (a,) = A._first_ngens(1)
>>> z = (Integer(1)+a)**Integer(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)
>>> z - Integer(1) - Integer(5)*a - Integer(10)*a**Integer(2) - Integer(10)*a**Integer(3) - Integer(5)*a**Integer(4) - a**Integer(5)
O(5^5)
>>> y = z >> Integer(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)
>>> Integer(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)
>>> FFp = R.residue_field()
>>> R(FFp(Integer(3)))
3 + O(5)
>>> QQq = Qq(Integer(25),Integer(4), names=('zz',)); (zz,) = QQq._first_ngens(1)
>>> QQq(FFp(Integer(3)))
3 + O(5)
>>> FFq = QQq.residue_field(); QQq(FFq(Integer(3)))
3 + O(5)
>>> zz0 = FFq.gen(); QQq(zz0**Integer(2))
(zz + 3) + O(5)
Different printing modes:
sage: R = Zp(5, print_mode='digits'); S.<x> = R[]; 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 = Zp(5, print_mode='bars'); S.<x> = R[]; 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 = Zp(5, print_mode='terse'); S.<x> = R[]; 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 = Zp(5, print_mode='val-unit'); S.<x> = R[]; 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)
>>> from sage.all import *
>>> R = Zp(Integer(5), print_mode='digits'); S = R['x']; (x,) = S._first_ngens(1); f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x -Integer(5); W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1)
>>> z = (Integer(1)+w)**Integer(5); repr(z)
'...4110403113210310442221311242000111011201102002023303214332011214403232013144001400444441030421100001'
>>> R = Zp(Integer(5), print_mode='bars'); S = R['x']; (x,) = S._first_ngens(1); g = x**Integer(3) + Integer(3)*x + Integer(3); A = R.ext(g, names=('a',)); (a,) = A._first_ngens(1)
>>> z = (Integer(1)+a)**Integer(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]'
>>> R = Zp(Integer(5), print_mode='terse'); S = R['x']; (x,) = S._first_ngens(1); f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x -Integer(5); W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1)
>>> z = (Integer(1)+w)**Integer(5); z
6 + 95367431640505*w + 25*w^2 + 95367431640560*w^3 + 5*w^4 + O(w^100)
>>> R = Zp(Integer(5), print_mode='val-unit'); S = R['x']; (x,) = S._first_ngens(1); f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x -Integer(5); W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1)
>>> y = (Integer(1)+w)**Integer(5) - Integer(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 unit:
sage: z._ntl_rep()
[6 95367431640505 25 95367431640560 5]
sage: y._ntl_rep()
[2090041 19073486126901 1258902 674 16785]
sage: y._ntl_rep_abs()
([5 95367431640505 25 95367431640560 5], 0)
>>> from sage.all import *
>>> z._ntl_rep()
[6 95367431640505 25 95367431640560 5]
>>> y._ntl_rep()
[2090041 19073486126901 1258902 674 16785]
>>> y._ntl_rep_abs()
([5 95367431640505 25 95367431640560 5], 0)
Note
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
Robert Harron (2011-09): fixes/enhancements
Julian Rueth (2014-05-09): enable caching through
_cache_key
- sage.rings.padics.padic_ZZ_pX_CR_element.make_ZZpXCRElement(parent, unit, ordp, relprec, version)[source]¶
Unpickling.
EXAMPLES:
sage: R = Zp(5,5) sage: S.<x> = R[] 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: loads(dumps(y)) # indirect doctest w^10 + 4*w^12 + 2*w^14 + w^15 + 2*w^16 + 4*w^17 + w^18 + O(w^19) sage: from sage.rings.padics.padic_ZZ_pX_CR_element import make_ZZpXCRElement sage: make_ZZpXCRElement(W, y._ntl_rep(), 3, 9, 0) w^3 + 4*w^5 + 2*w^7 + w^8 + 2*w^9 + 4*w^10 + w^11 + O(w^12)
>>> from sage.all import * >>> R = Zp(Integer(5),Integer(5)) >>> S = R['x']; (x,) = S._first_ngens(1) >>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x - Integer(5) >>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1) >>> y = W(Integer(775), Integer(19)); y w^10 + 4*w^12 + 2*w^14 + w^15 + 2*w^16 + 4*w^17 + w^18 + O(w^19) >>> loads(dumps(y)) # indirect doctest w^10 + 4*w^12 + 2*w^14 + w^15 + 2*w^16 + 4*w^17 + w^18 + O(w^19) >>> from sage.rings.padics.padic_ZZ_pX_CR_element import make_ZZpXCRElement >>> make_ZZpXCRElement(W, y._ntl_rep(), Integer(3), Integer(9), Integer(0)) w^3 + 4*w^5 + 2*w^7 + w^8 + 2*w^9 + 4*w^10 + w^11 + O(w^12)
- class sage.rings.padics.padic_ZZ_pX_CR_element.pAdicZZpXCRElement[source]¶
Bases:
pAdicZZpXElementCreate an element of a capped relative precision, unramified or Eisenstein extension of \(\ZZ_p\) or \(\QQ_p\).
INPUT:
parent– either anEisensteinRingCappedRelativeorUnramifiedRingCappedRelativex– integer; rational, \(p\)-adic element, polynomial, list, integer_mod, pari int/frac/poly_t/pol_mod, anntl_ZZ_pX, anntl_ZZ, anntl_ZZ_p, anntl_ZZX, or something convertible into parent.residue_field()absprec– an upper bound on the absolute precision of the element createdrelprec– an upper bound on the relative precision of the element createdempty– whether to return after initializing to zero (without setting the valuation)
EXAMPLES:
sage: R = Zp(5,5) sage: S.<x> = R[] 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(pari('3 + O(5^3)')) 3 + O(w^15) sage: W(R(3,3)) 3 + O(w^15) sage: W.<w> = R.ext(x^625 + 915*x^17 - 95) sage: W(3) 3 + O(w^3125) sage: W(w, 14) w + O(w^14)
>>> from sage.all import * >>> R = Zp(Integer(5),Integer(5)) >>> S = R['x']; (x,) = S._first_ngens(1) >>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x - Integer(5) >>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1) >>> z = (Integer(1)+w)**Integer(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) >>> W(pari('3 + O(5^3)')) 3 + O(w^15) >>> W(R(Integer(3),Integer(3))) 3 + O(w^15) >>> W = R.ext(x**Integer(625) + Integer(915)*x**Integer(17) - Integer(95), names=('w',)); (w,) = W._first_ngens(1) >>> W(Integer(3)) 3 + O(w^3125) >>> W(w, Integer(14)) w + O(w^14)
- expansion(n=None, lift_mode='simple')[source]¶
Return a list giving a series representation of
self.If
lift_mode == 'simple'or'smallest', the returned list will consist of integers (in the Eisenstein case) or a list of lists of integers (in the unramified case).selfcan be reconstructed as a sum of elements of the list times powers of the uniformiser (in the Eisenstein case), or as a sum of powers of the \(p\) times polynomials in the generator (in the unramified case).If
lift_mode == 'simple', all integers will be in the interval \([0,p-1]\).If
lift_mode == 'smallest'they will be in the interval \([(1-p)/2, p/2]\).
If
lift_mode == 'teichmuller', returns a list ofpAdicZZpXCRElements, all of which are Teichmuller representatives and such thatselfis the sum of that list times powers of the uniformizer.
Note that zeros are truncated from the returned list if
self.parent()is a field, so you must use thevaluationfunction to fully reconstructself.INPUT:
n– integer (default:None); if given, returns the corresponding entry in the expansion
EXAMPLES:
sage: R = Zp(5,5) sage: S.<x> = R[] 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).expansion() [0, 1, 0, 4, 0, 2, 1, 2, 4, 1] sage: (y>>9).expansion(lift_mode='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(5^6) sage: E = y.expansion(); E 5-adic expansion of 4*a*5 + (3*a^2 + a + 3)*5^2 + 4*a^2*5^3 + a^2*5^4 + O(5^6) sage: list(E) [[], [0, 4], [3, 1, 3], [0, 0, 4], [0, 0, 1], []] sage: list(y.expansion(lift_mode='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(5^6) sage: list(W(0).expansion()) [] sage: list(W(0,4).expansion()) [] sage: list(A(0,4).expansion()) []
>>> from sage.all import * >>> R = Zp(Integer(5),Integer(5)) >>> S = R['x']; (x,) = S._first_ngens(1) >>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x - Integer(5) >>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1) >>> y = W(Integer(775), Integer(19)); y w^10 + 4*w^12 + 2*w^14 + w^15 + 2*w^16 + 4*w^17 + w^18 + O(w^19) >>> (y>>Integer(9)).expansion() [0, 1, 0, 4, 0, 2, 1, 2, 4, 1] >>> (y>>Integer(9)).expansion(lift_mode='smallest') [0, 1, 0, -1, 0, 2, 1, 2, 0, 1] >>> w**Integer(10) - w**Integer(12) + Integer(2)*w**Integer(14) + w**Integer(15) + Integer(2)*w**Integer(16) + w**Integer(18) + O(w**Integer(19)) w^10 + 4*w^12 + 2*w^14 + w^15 + 2*w^16 + 4*w^17 + w^18 + O(w^19) >>> g = x**Integer(3) + Integer(3)*x + Integer(3) >>> A = R.ext(g, names=('a',)); (a,) = A._first_ngens(1) >>> y = Integer(75) + Integer(45)*a + Integer(1200)*a**Integer(2); y 4*a*5 + (3*a^2 + a + 3)*5^2 + 4*a^2*5^3 + a^2*5^4 + O(5^6) >>> E = y.expansion(); E 5-adic expansion of 4*a*5 + (3*a^2 + a + 3)*5^2 + 4*a^2*5^3 + a^2*5^4 + O(5^6) >>> list(E) [[], [0, 4], [3, 1, 3], [0, 0, 4], [0, 0, 1], []] >>> list(y.expansion(lift_mode='smallest')) [[], [0, -1], [-2, 2, -2], [1], [0, 0, 2], []] >>> Integer(5)*((-Integer(2)*Integer(5) + Integer(25)) + (-Integer(1) + Integer(2)*Integer(5))*a + (-Integer(2)*Integer(5) + Integer(2)*Integer(125))*a**Integer(2)) 4*a*5 + (3*a^2 + a + 3)*5^2 + 4*a^2*5^3 + a^2*5^4 + O(5^6) >>> list(W(Integer(0)).expansion()) [] >>> list(W(Integer(0),Integer(4)).expansion()) [] >>> list(A(Integer(0),Integer(4)).expansion()) []
- is_equal_to(right, absprec=None)[source]¶
Return whether this element is equal to
rightmoduloself.uniformizer()^absprec.If
absprecisNone, checks whether this element is equal torightmodulo the lower of their two precisions.EXAMPLES:
sage: R = Zp(5,5) sage: S.<x> = R[] 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
>>> from sage.all import * >>> R = Zp(Integer(5),Integer(5)) >>> S = R['x']; (x,) = S._first_ngens(1) >>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x - Integer(5) >>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1) >>> a = W(Integer(47)); b = W(Integer(47) + Integer(25)) >>> a.is_equal_to(b) False >>> a.is_equal_to(b, Integer(7)) True
- is_zero(absprec=None)[source]¶
Return whether the valuation of this element is at least
absprec. IfabsprecisNone, checks if this element is indistinguishable from zero.If this element is an inexact zero of valuation less than
absprec, raises aPrecisionError.EXAMPLES:
sage: R = Zp(5,5) sage: S.<x> = R[] 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
>>> from sage.all import * >>> R = Zp(Integer(5),Integer(5)) >>> S = R['x']; (x,) = S._first_ngens(1) >>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x - Integer(5) >>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1) >>> O(w**Integer(189)).is_zero() True >>> W(Integer(0)).is_zero() True >>> a = W(Integer(675)) >>> a.is_zero() False >>> a.is_zero(Integer(7)) True >>> a.is_zero(Integer(21)) False
- lift_to_precision(absprec=None)[source]¶
Return a
pAdicZZpXCRElementcongruent to this element but with absolute precision at leastabsprec.INPUT:
absprec– (default:None) the absolute precision of the result. IfNone, lifts to the maximum precision allowed.
Note
If setting
absprecthat high would violate the precision cap, raises a precision error. Ifselfis an inexact zero andabsprecis greater than the maximum allowed valuation, raises an error.Note that the new digits will not necessarily be zero.
EXAMPLES:
sage: R = Zp(5,5) sage: S.<x> = R[] 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 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() [19 35 118 60 121] sage: b._ntl_rep() [19 35 118 60 121] sage: c._ntl_rep() [19 35 118 60 121] sage: a.lift_to_precision().precision_relative() == W.precision_cap() True
>>> from sage.all import * >>> R = Zp(Integer(5),Integer(5)) >>> S = R['x']; (x,) = S._first_ngens(1) >>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x - Integer(5) >>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1) >>> a = W(Integer(345), Integer(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) >>> b = a.lift_to_precision(Integer(19)); b 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) >>> c = a.lift_to_precision(Integer(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) >>> a._ntl_rep() [19 35 118 60 121] >>> b._ntl_rep() [19 35 118 60 121] >>> c._ntl_rep() [19 35 118 60 121] >>> a.lift_to_precision().precision_relative() == W.precision_cap() True
- matrix_mod_pn()[source]¶
Return the matrix of right multiplication by the element on the power basis \(1, x, x^2, \ldots, x^{d-1}\) for this extension field. Thus the rows of this matrix give the images of each of the \(x^i\). The entries of the matrices are
IntegerModelements, defined modulo \(p^{N / e}\) where \(N\) is the absolute precision of this element (unless this element is zero to arbitrary precision; in that case the entries are integer zeros.)Raises an error if this element has negative valuation.
EXAMPLES:
sage: R = ZpCR(5,5) sage: S.<x> = R[] 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]
>>> from sage.all import * >>> R = ZpCR(Integer(5),Integer(5)) >>> S = R['x']; (x,) = S._first_ngens(1) >>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x - Integer(5) >>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1) >>> a = (Integer(3)+w)**Integer(7) >>> 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]
- polynomial(var='x')[source]¶
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: S.<x> = ZZ[] sage: W.<w> = Zp(5).extension(x^2 - 5) sage: (w + W(5, 7)).polynomial() (1 + O(5^3))*x + 5 + O(5^4)
>>> from sage.all import * >>> S = ZZ['x']; (x,) = S._first_ngens(1) >>> W = Zp(Integer(5)).extension(x**Integer(2) - Integer(5), names=('w',)); (w,) = W._first_ngens(1) >>> (w + W(Integer(5), Integer(7))).polynomial() (1 + O(5^3))*x + 5 + O(5^4)
- precision_absolute()[source]¶
Return the absolute precision of this element, i.e., the power of the uniformizer modulo which this element is defined.
EXAMPLES:
sage: R = Zp(5,5) sage: S.<x> = R[] 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) sage: (a.unit_part() - 3).precision_absolute() 9
>>> from sage.all import * >>> R = Zp(Integer(5),Integer(5)) >>> S = R['x']; (x,) = S._first_ngens(1) >>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x - Integer(5) >>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1) >>> a = W(Integer(75), Integer(19)); a 3*w^10 + 2*w^12 + w^14 + w^16 + w^17 + 3*w^18 + O(w^19) >>> a.valuation() 10 >>> a.precision_absolute() 19 >>> a.precision_relative() 9 >>> a.unit_part() 3 + 2*w^2 + w^4 + w^6 + w^7 + 3*w^8 + O(w^9) >>> (a.unit_part() - Integer(3)).precision_absolute() 9
- precision_relative()[source]¶
Return the relative precision of this element, i.e., the power of the uniformizer modulo which the unit part of
selfis defined.EXAMPLES:
sage: R = Zp(5,5) sage: S.<x> = R[] 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)
>>> from sage.all import * >>> R = Zp(Integer(5),Integer(5)) >>> S = R['x']; (x,) = S._first_ngens(1) >>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x - Integer(5) >>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1) >>> a = W(Integer(75), Integer(19)); a 3*w^10 + 2*w^12 + w^14 + w^16 + w^17 + 3*w^18 + O(w^19) >>> a.valuation() 10 >>> a.precision_absolute() 19 >>> a.precision_relative() 9 >>> a.unit_part() 3 + 2*w^2 + w^4 + w^6 + w^7 + 3*w^8 + O(w^9)
- teichmuller_expansion(n=None)[source]¶
Return a list [\(a_0\), \(a_1\),…, \(a_n\)] such that:
\(a_i^q = a_i\)
self.unit_part()= \(\sum_{i = 0}^n a_i \pi^i\), where \(\pi\) is a uniformizer ofself.parent()if \(a_i \ne 0\), the absolute precision of \(a_i\) is
self.precision_relative() - i
INPUT:
n– integer (default:None); if given, returns the corresponding entry in the expansion
EXAMPLES:
sage: R.<a> = ZqCR(5^4,4) sage: E = a.teichmuller_expansion(); E 5-adic expansion of a + O(5^4) (teichmuller) sage: list(E) [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([c * 5^i for i, c in enumerate(E)]) a + O(5^4) sage: all(c^625 == c for c in E) True sage: S.<x> = ZZ[] sage: f = x^3 - 98*x + 7 sage: W.<w> = ZpCR(7,3).ext(f) sage: b = (1+w)^5; L = b.teichmuller_expansion(); 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_expansion(); 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)
>>> from sage.all import * >>> R = ZqCR(Integer(5)**Integer(4),Integer(4), names=('a',)); (a,) = R._first_ngens(1) >>> E = a.teichmuller_expansion(); E 5-adic expansion of a + O(5^4) (teichmuller) >>> list(E) [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)] >>> sum([c * Integer(5)**i for i, c in enumerate(E)]) a + O(5^4) >>> all(c**Integer(625) == c for c in E) True >>> S = ZZ['x']; (x,) = S._first_ngens(1) >>> f = x**Integer(3) - Integer(98)*x + Integer(7) >>> W = ZpCR(Integer(7),Integer(3)).ext(f, names=('w',)); (w,) = W._first_ngens(1) >>> b = (Integer(1)+w)**Integer(5); L = b.teichmuller_expansion(); 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)] >>> sum([w**i*L[i] for i in range(Integer(9))]) == b True >>> all(L[i]**(Integer(7)**Integer(3)) == L[i] for i in range(Integer(9))) True >>> L = W(Integer(3)).teichmuller_expansion(); 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)] >>> sum([w**i*L[i] for i in range(len(L))]) 3 + O(w^9)
- unit_part()[source]¶
Return the unit part of this element, ie
self / uniformizer^(self.valuation()).EXAMPLES:
sage: R = Zp(5,5) sage: S.<x> = R[] 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)
>>> from sage.all import * >>> R = Zp(Integer(5),Integer(5)) >>> S = R['x']; (x,) = S._first_ngens(1) >>> f = x**Integer(5) + Integer(75)*x**Integer(3) - Integer(15)*x**Integer(2) + Integer(125)*x - Integer(5) >>> W = R.ext(f, names=('w',)); (w,) = W._first_ngens(1) >>> a = W(Integer(75), Integer(19)); a 3*w^10 + 2*w^12 + w^14 + w^16 + w^17 + 3*w^18 + O(w^19) >>> a.valuation() 10 >>> a.precision_absolute() 19 >>> a.precision_relative() 9 >>> a.unit_part() 3 + 2*w^2 + w^4 + w^6 + w^7 + 3*w^8 + O(w^9)