\(p\)-adic Capped Relative Dense Polynomials¶

class sage.rings.polynomial.padics.polynomial_padic_capped_relative_dense.Polynomial_padic_capped_relative_dense(parent, x=None, check=True, is_gen=False, construct=False, absprec=+Infinity, relprec=+Infinity)[source]¶

Bases: Polynomial_generic_cdv, Polynomial_padic

degree(secure=False)[source]¶

Return the degree of self.

INPUT:

secure – boolean (default: False)

If secure is True and the degree of this polynomial is not determined (because the leading coefficient is indistinguishable from 0), an error is raised.

If secure is False, the returned value is the largest \(n\) so that the coefficient of \(x^n\) does not compare equal to \(0\).

EXAMPLES:

Sage

sage: K = Qp(3,10)
sage: R.<T> = K[]
sage: f = T + 2; f
(1 + O(3^10))*T + 2 + O(3^10)
sage: f.degree()
1
sage: (f-T).degree()
0
sage: (f-T).degree(secure=True)
Traceback (most recent call last):
...
PrecisionError: the leading coefficient is indistinguishable from 0

sage: x = O(3^5)
sage: li = [3^i * x for i in range(0,5)]; li
[O(3^5), O(3^6), O(3^7), O(3^8), O(3^9)]
sage: f = R(li); f
O(3^9)*T^4 + O(3^8)*T^3 + O(3^7)*T^2 + O(3^6)*T + O(3^5)
sage: f.degree()
-1
sage: f.degree(secure=True)
Traceback (most recent call last):
...
PrecisionError: the leading coefficient is indistinguishable from 0

Python

>>> from sage.all import *
>>> K = Qp(Integer(3),Integer(10))
>>> R = K['T']; (T,) = R._first_ngens(1)
>>> f = T + Integer(2); f
(1 + O(3^10))*T + 2 + O(3^10)
>>> f.degree()
1
>>> (f-T).degree()
0
>>> (f-T).degree(secure=True)
Traceback (most recent call last):
...
PrecisionError: the leading coefficient is indistinguishable from 0

>>> x = O(Integer(3)**Integer(5))
>>> li = [Integer(3)**i * x for i in range(Integer(0),Integer(5))]; li
[O(3^5), O(3^6), O(3^7), O(3^8), O(3^9)]
>>> f = R(li); f
O(3^9)*T^4 + O(3^8)*T^3 + O(3^7)*T^2 + O(3^6)*T + O(3^5)
>>> f.degree()
-1
>>> f.degree(secure=True)
Traceback (most recent call last):
...
PrecisionError: the leading coefficient is indistinguishable from 0

Sage Live

K = Qp(3,10)
R.<T> = K[]
f = T + 2; f
f.degree()
(f-T).degree()
(f-T).degree(secure=True)
x = O(3^5)
li = [3^i * x for i in range(0,5)]; li
f = R(li); f
f.degree()
f.degree(secure=True)

disc()[source]¶

factor_mod()[source]¶: Return the factorization of self modulo \(p\).

is_eisenstein(secure=False)[source]¶

Return True if this polynomial is an Eisenstein polynomial.

EXAMPLES:

Sage

sage: K = Qp(5)
sage: R.<t> = K[]
sage: f = 5 + 5*t + t^4
sage: f.is_eisenstein()
True

Python

>>> from sage.all import *
>>> K = Qp(Integer(5))
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> f = Integer(5) + Integer(5)*t + t**Integer(4)
>>> f.is_eisenstein()
True

Sage Live

K = Qp(5)
R.<t> = K[]
f = 5 + 5*t + t^4
f.is_eisenstein()

AUTHOR:

Xavier Caruso (2013-03)

lift()[source]¶

Return an integer polynomial congruent to this one modulo the precision of each coefficient.

Note

The lift that is returned will not necessarily be the same for polynomials with the same coefficients (i.e. same values and precisions): it will depend on how the polynomials are created.

EXAMPLES:

Sage

sage: K = Qp(13,7)
sage: R.<t> = K[]
sage: a = 13^7*t^3 + K(169,4)*t - 13^4
sage: a.lift()
62748517*t^3 + 169*t - 28561

Python

>>> from sage.all import *
>>> K = Qp(Integer(13),Integer(7))
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> a = Integer(13)**Integer(7)*t**Integer(3) + K(Integer(169),Integer(4))*t - Integer(13)**Integer(4)
>>> a.lift()
62748517*t^3 + 169*t - 28561

Sage Live

K = Qp(13,7)
R.<t> = K[]
a = 13^7*t^3 + K(169,4)*t - 13^4
a.lift()

list(copy=True)[source]¶

Return a list of coefficients of self.

Note

The length of the list returned may be greater than expected since it includes any leading zeros that have finite absolute precision.

EXAMPLES:

Sage

sage: K = Qp(13,7)
sage: R.<t> = K[]
sage: a = 2*t^3 + 169*t - 1
sage: a
(2 + O(13^7))*t^3 + (13^2 + O(13^9))*t + 12 + 12*13 + 12*13^2 + 12*13^3 + 12*13^4 + 12*13^5 + 12*13^6 + O(13^7)
sage: a.list()
[12 + 12*13 + 12*13^2 + 12*13^3 + 12*13^4 + 12*13^5 + 12*13^6 + O(13^7),
 13^2 + O(13^9),
 0,
 2 + O(13^7)]

Python

>>> from sage.all import *
>>> K = Qp(Integer(13),Integer(7))
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> a = Integer(2)*t**Integer(3) + Integer(169)*t - Integer(1)
>>> a
(2 + O(13^7))*t^3 + (13^2 + O(13^9))*t + 12 + 12*13 + 12*13^2 + 12*13^3 + 12*13^4 + 12*13^5 + 12*13^6 + O(13^7)
>>> a.list()
[12 + 12*13 + 12*13^2 + 12*13^3 + 12*13^4 + 12*13^5 + 12*13^6 + O(13^7),
 13^2 + O(13^9),
 0,
 2 + O(13^7)]

Sage Live

K = Qp(13,7)
R.<t> = K[]
a = 2*t^3 + 169*t - 1
a
a.list()

lshift_coeffs(shift, no_list=False)[source]¶

Return a new polynomials whose coefficients are multiplied by p^shift.

EXAMPLES:

Sage

sage: K = Qp(13, 4)
sage: R.<t> = K[]
sage: a = t + 52
sage: a.lshift_coeffs(3)
(13^3 + O(13^7))*t + 4*13^4 + O(13^8)

Python

>>> from sage.all import *
>>> K = Qp(Integer(13), Integer(4))
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> a = t + Integer(52)
>>> a.lshift_coeffs(Integer(3))
(13^3 + O(13^7))*t + 4*13^4 + O(13^8)

Sage Live

K = Qp(13, 4)
R.<t> = K[]
a = t + 52
a.lshift_coeffs(3)

newton_polygon()[source]¶

Return the Newton polygon of this polynomial.

Note

If some coefficients have not enough precision an error is raised.

OUTPUT: a NewtonPolygon

EXAMPLES:

Sage

sage: K = Qp(2, prec=5)
sage: P.<x> = K[]
sage: f = x^4 + 2^3*x^3 + 2^13*x^2 + 2^21*x + 2^37
sage: f.newton_polygon()                                                    # needs sage.geometry.polyhedron
Finite Newton polygon with 4 vertices: (0, 37), (1, 21), (3, 3), (4, 0)

sage: K = Qp(5)
sage: R.<t> = K[]
sage: f = 5 + 3*t + t^4 + 25*t^10
sage: f.newton_polygon()                                                    # needs sage.geometry.polyhedron
Finite Newton polygon with 4 vertices: (0, 1), (1, 0), (4, 0), (10, 2)

Python

>>> from sage.all import *
>>> K = Qp(Integer(2), prec=Integer(5))
>>> P = K['x']; (x,) = P._first_ngens(1)
>>> f = x**Integer(4) + Integer(2)**Integer(3)*x**Integer(3) + Integer(2)**Integer(13)*x**Integer(2) + Integer(2)**Integer(21)*x + Integer(2)**Integer(37)
>>> f.newton_polygon()                                                    # needs sage.geometry.polyhedron
Finite Newton polygon with 4 vertices: (0, 37), (1, 21), (3, 3), (4, 0)

>>> K = Qp(Integer(5))
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> f = Integer(5) + Integer(3)*t + t**Integer(4) + Integer(25)*t**Integer(10)
>>> f.newton_polygon()                                                    # needs sage.geometry.polyhedron
Finite Newton polygon with 4 vertices: (0, 1), (1, 0), (4, 0), (10, 2)

Sage Live

K = Qp(2, prec=5)
P.<x> = K[]
f = x^4 + 2^3*x^3 + 2^13*x^2 + 2^21*x + 2^37
f.newton_polygon()                                                    # needs sage.geometry.polyhedron
K = Qp(5)
R.<t> = K[]
f = 5 + 3*t + t^4 + 25*t^10
f.newton_polygon()                                                    # needs sage.geometry.polyhedron

Here is an example where the computation fails because precision is not sufficient:

Sage

sage: g = f + K(0,0)*t^4; g
(5^2 + O(5^22))*t^10 + O(5^0)*t^4 + (3 + O(5^20))*t + 5 + O(5^21)
sage: g.newton_polygon()                                                    # needs sage.geometry.polyhedron
Traceback (most recent call last):
...
PrecisionError: The coefficient of t^4 has not enough precision

Python

>>> from sage.all import *
>>> g = f + K(Integer(0),Integer(0))*t**Integer(4); g
(5^2 + O(5^22))*t^10 + O(5^0)*t^4 + (3 + O(5^20))*t + 5 + O(5^21)
>>> g.newton_polygon()                                                    # needs sage.geometry.polyhedron
Traceback (most recent call last):
...
PrecisionError: The coefficient of t^4 has not enough precision

Sage Live

g = f + K(0,0)*t^4; g
g.newton_polygon()                                                    # needs sage.geometry.polyhedron

AUTHOR:

Xavier Caruso (2013-03-20)

newton_slopes(repetition=True)[source]¶

Return a list of the Newton slopes of this polynomial.

These are the valuations of the roots of this polynomial.

If repetition is True, each slope is repeated a number of times equal to its multiplicity. Otherwise it appears only one time.

INPUT:

repetition – boolean (default: True)

OUTPUT: list of rationals

EXAMPLES:

Sage

sage: K = Qp(5)
sage: R.<t> = K[]
sage: f = 5 + 3*t + t^4 + 25*t^10
sage: f.newton_polygon()                                                    # needs sage.geometry.polyhedron
Finite Newton polygon with 4 vertices: (0, 1), (1, 0), (4, 0),
(10, 2)
sage: f.newton_slopes()                                                     # needs sage.geometry.polyhedron
[1, 0, 0, 0, -1/3, -1/3, -1/3, -1/3, -1/3, -1/3]

sage: f.newton_slopes(repetition=False)                                     # needs sage.geometry.polyhedron
[1, 0, -1/3]

Python

>>> from sage.all import *
>>> K = Qp(Integer(5))
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> f = Integer(5) + Integer(3)*t + t**Integer(4) + Integer(25)*t**Integer(10)
>>> f.newton_polygon()                                                    # needs sage.geometry.polyhedron
Finite Newton polygon with 4 vertices: (0, 1), (1, 0), (4, 0),
(10, 2)
>>> f.newton_slopes()                                                     # needs sage.geometry.polyhedron
[1, 0, 0, 0, -1/3, -1/3, -1/3, -1/3, -1/3, -1/3]

>>> f.newton_slopes(repetition=False)                                     # needs sage.geometry.polyhedron
[1, 0, -1/3]

Sage Live

K = Qp(5)
R.<t> = K[]
f = 5 + 3*t + t^4 + 25*t^10
f.newton_polygon()                                                    # needs sage.geometry.polyhedron
f.newton_slopes()                                                     # needs sage.geometry.polyhedron
f.newton_slopes(repetition=False)                                     # needs sage.geometry.polyhedron

AUTHOR:

Xavier Caruso (2013-03-20)

prec_degree()[source]¶

Return the largest \(n\) so that precision information is stored about the coefficient of \(x^n\).

Always greater than or equal to degree.

EXAMPLES:

Sage

sage: K = Qp(3,10)
sage: R.<T> = K[]
sage: f = T + 2; f
(1 + O(3^10))*T + 2 + O(3^10)
sage: f.prec_degree()
1

Python

>>> from sage.all import *
>>> K = Qp(Integer(3),Integer(10))
>>> R = K['T']; (T,) = R._first_ngens(1)
>>> f = T + Integer(2); f
(1 + O(3^10))*T + 2 + O(3^10)
>>> f.prec_degree()
1

Sage Live

K = Qp(3,10)
R.<T> = K[]
f = T + 2; f
f.prec_degree()

precision_absolute(n=None)[source]¶

Return absolute precision information about self.

INPUT:

self – a \(p\)-adic polynomial
n – None or integer (default: None)

OUTPUT:

If n is None, returns a list of absolute precisions of coefficients. Otherwise, returns the absolute precision of the coefficient of \(x^n\).

EXAMPLES:

Sage

sage: K = Qp(3,10)
sage: R.<T> = K[]
sage: f = T + 2; f
(1 + O(3^10))*T + 2 + O(3^10)
sage: f.precision_absolute()
[10, 10]

Python

>>> from sage.all import *
>>> K = Qp(Integer(3),Integer(10))
>>> R = K['T']; (T,) = R._first_ngens(1)
>>> f = T + Integer(2); f
(1 + O(3^10))*T + 2 + O(3^10)
>>> f.precision_absolute()
[10, 10]

Sage Live

K = Qp(3,10)
R.<T> = K[]
f = T + 2; f
f.precision_absolute()

precision_relative(n=None)[source]¶

Return relative precision information about self.

INPUT:

self – a \(p\)-adic polynomial
n – None or integer (default: None)

OUTPUT:

If n is None, returns a list of relative precisions of coefficients. Otherwise, returns the relative precision of the coefficient of \(x^n\).

EXAMPLES:

Sage

sage: K = Qp(3,10)
sage: R.<T> = K[]
sage: f = T + 2; f
(1 + O(3^10))*T + 2 + O(3^10)
sage: f.precision_relative()
[10, 10]

Python

>>> from sage.all import *
>>> K = Qp(Integer(3),Integer(10))
>>> R = K['T']; (T,) = R._first_ngens(1)
>>> f = T + Integer(2); f
(1 + O(3^10))*T + 2 + O(3^10)
>>> f.precision_relative()
[10, 10]

Sage Live

K = Qp(3,10)
R.<T> = K[]
f = T + 2; f
f.precision_relative()

quo_rem(right, secure=False)[source]¶

Return the quotient and remainder in division of self by right.

EXAMPLES:

Sage

sage: K = Qp(3,10)
sage: R.<T> = K[]
sage: f = T + 2
sage: g = T**4 + 3*T+22
sage: g.quo_rem(f)
((1 + O(3^10))*T^3 + (1 + 2*3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + O(3^10))*T^2 + (1 + 3 + O(3^10))*T + 1 + 3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + O(3^10),
 2 + 3 + 3^3 + O(3^10))

Python

>>> from sage.all import *
>>> K = Qp(Integer(3),Integer(10))
>>> R = K['T']; (T,) = R._first_ngens(1)
>>> f = T + Integer(2)
>>> g = T**Integer(4) + Integer(3)*T+Integer(22)
>>> g.quo_rem(f)
((1 + O(3^10))*T^3 + (1 + 2*3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + O(3^10))*T^2 + (1 + 3 + O(3^10))*T + 1 + 3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + O(3^10),
 2 + 3 + 3^3 + O(3^10))

Sage Live

K = Qp(3,10)
R.<T> = K[]
f = T + 2
g = T**4 + 3*T+22
g.quo_rem(f)

rescale(a)[source]¶

Return \(f(a\cdot x)\).

Todo

Need to write this function for integer polynomials before this works.

EXAMPLES:

Sage

sage: K = Zp(13, 5)
sage: R.<t> = K[]
sage: f = t^3 + K(13, 3) * t
sage: f.rescale(2)  # not implemented

Python

>>> from sage.all import *
>>> K = Zp(Integer(13), Integer(5))
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> f = t**Integer(3) + K(Integer(13), Integer(3)) * t
>>> f.rescale(Integer(2))  # not implemented

Sage Live

K = Zp(13, 5)
R.<t> = K[]
f = t^3 + K(13, 3) * t
f.rescale(2)  # not implemented

reverse(degree=None)[source]¶

Return the reverse of the input polynomial, thought as a polynomial of degree degree.

If \(f\) is a degree-\(d\) polynomial, its reverse is \(x^d f(1/x)\).

INPUT:

degree – None or integer; if specified, truncate or zero pad the list of coefficients to this degree before reversing it

EXAMPLES:

Sage

sage: K = Qp(13,7)
sage: R.<t> = K[]
sage: f = t^3 + 4*t; f
(1 + O(13^7))*t^3 + (4 + O(13^7))*t
sage: f.reverse()
0*t^3 + (4 + O(13^7))*t^2 + 1 + O(13^7)
sage: f.reverse(3)
0*t^3 + (4 + O(13^7))*t^2 + 1 + O(13^7)
sage: f.reverse(2)
0*t^2 + (4 + O(13^7))*t
sage: f.reverse(4)
0*t^4 + (4 + O(13^7))*t^3 + (1 + O(13^7))*t
sage: f.reverse(6)
0*t^6 + (4 + O(13^7))*t^5 + (1 + O(13^7))*t^3

Python

>>> from sage.all import *
>>> K = Qp(Integer(13),Integer(7))
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> f = t**Integer(3) + Integer(4)*t; f
(1 + O(13^7))*t^3 + (4 + O(13^7))*t
>>> f.reverse()
0*t^3 + (4 + O(13^7))*t^2 + 1 + O(13^7)
>>> f.reverse(Integer(3))
0*t^3 + (4 + O(13^7))*t^2 + 1 + O(13^7)
>>> f.reverse(Integer(2))
0*t^2 + (4 + O(13^7))*t
>>> f.reverse(Integer(4))
0*t^4 + (4 + O(13^7))*t^3 + (1 + O(13^7))*t
>>> f.reverse(Integer(6))
0*t^6 + (4 + O(13^7))*t^5 + (1 + O(13^7))*t^3

Sage Live

K = Qp(13,7)
R.<t> = K[]
f = t^3 + 4*t; f
f.reverse()
f.reverse(3)
f.reverse(2)
f.reverse(4)
f.reverse(6)

rshift_coeffs(shift, no_list=False)[source]¶

Return a new polynomial whose coefficients are \(p\)-adically shifted to the right by shift.

Note

Type Qp(5)(0).__rshift__? for more information.

EXAMPLES:

Sage

sage: K = Zp(13, 4)
sage: R.<t> = K[]
sage: a = t^2 + K(13,3)*t + 169; a
(1 + O(13^4))*t^2 + (13 + O(13^3))*t + 13^2 + O(13^6)
sage: b = a.rshift_coeffs(1); b
O(13^3)*t^2 + (1 + O(13^2))*t + 13 + O(13^5)
sage: b.list()
[13 + O(13^5), 1 + O(13^2), O(13^3)]
sage: b = a.rshift_coeffs(2); b
O(13^2)*t^2 + O(13)*t + 1 + O(13^4)
sage: b.list()
[1 + O(13^4), O(13), O(13^2)]

Python

>>> from sage.all import *
>>> K = Zp(Integer(13), Integer(4))
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> a = t**Integer(2) + K(Integer(13),Integer(3))*t + Integer(169); a
(1 + O(13^4))*t^2 + (13 + O(13^3))*t + 13^2 + O(13^6)
>>> b = a.rshift_coeffs(Integer(1)); b
O(13^3)*t^2 + (1 + O(13^2))*t + 13 + O(13^5)
>>> b.list()
[13 + O(13^5), 1 + O(13^2), O(13^3)]
>>> b = a.rshift_coeffs(Integer(2)); b
O(13^2)*t^2 + O(13)*t + 1 + O(13^4)
>>> b.list()
[1 + O(13^4), O(13), O(13^2)]

Sage Live

K = Zp(13, 4)
R.<t> = K[]
a = t^2 + K(13,3)*t + 169; a
b = a.rshift_coeffs(1); b
b.list()
b = a.rshift_coeffs(2); b
b.list()

valuation(val_of_var=None)[source]¶

Return the valuation of self.

INPUT:

self – a \(p\)-adic polynomial
val_of_var – None or a rational (default: None)

OUTPUT:

If val_of_var is None, returns the largest power of the variable dividing self. Otherwise, returns the valuation of self where the variable is assigned valuation val_of_var

EXAMPLES:

Sage

sage: K = Qp(3,10)
sage: R.<T> = K[]
sage: f = T + 2; f
(1 + O(3^10))*T + 2 + O(3^10)
sage: f.valuation()
0

Python

>>> from sage.all import *
>>> K = Qp(Integer(3),Integer(10))
>>> R = K['T']; (T,) = R._first_ngens(1)
>>> f = T + Integer(2); f
(1 + O(3^10))*T + 2 + O(3^10)
>>> f.valuation()
0

Sage Live

K = Qp(3,10)
R.<T> = K[]
f = T + 2; f
f.valuation()

valuation_of_coefficient(n=None)[source]¶

Return valuation information about self’s coefficients.

INPUT:

self – a \(p\)-adic polynomial
n – None or integer (default: None)

OUTPUT:

If n is None, returns a list of valuations of coefficients. Otherwise, returns the valuation of the coefficient of \(x^n\).

EXAMPLES:

Sage

sage: K = Qp(3,10)
sage: R.<T> = K[]
sage: f = T + 2; f
(1 + O(3^10))*T + 2 + O(3^10)
sage: f.valuation_of_coefficient(1)
0

Python

>>> from sage.all import *
>>> K = Qp(Integer(3),Integer(10))
>>> R = K['T']; (T,) = R._first_ngens(1)
>>> f = T + Integer(2); f
(1 + O(3^10))*T + 2 + O(3^10)
>>> f.valuation_of_coefficient(Integer(1))
0

Sage Live

K = Qp(3,10)
R.<T> = K[]
f = T + 2; f
f.valuation_of_coefficient(1)

sage.rings.polynomial.padics.polynomial_padic_capped_relative_dense.make_padic_poly(parent, x, version)[source]¶