# p-adic Capped Relative Dense Polynomials#

degree(secure=False)[source]#

Return the degree of self.

INPUT:

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

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: 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
>>> 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
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: 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)]
>>> 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)]
lshift_coeffs(shift, no_list=False)[source]#

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

EXAMPLES:

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)
>>> 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)
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: 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)
>>> 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)

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

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
>>> 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

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:

• a list of rationals

EXAMPLES:

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]
>>> 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]

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: 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
>>> 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
precision_absolute(n=None)[source]#

Return absolute precision information about self.

INPUT:

• self – a p-adic polynomial

• nNone or an 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: 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]
>>> 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]
precision_relative(n=None)[source]#

Return relative precision information about self.

INPUT:

• self – a p-adic polynomial

• nNone or an 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: 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]
>>> 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]
quo_rem(right, secure=False)[source]#

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

EXAMPLES:

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))
>>> 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))
rescale(a)[source]#

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

Todo

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

EXAMPLES:

sage: K = Zp(13, 5)
sage: R.<t> = K[]
sage: f = t^3 + K(13, 3) * t
sage: f.rescale(2)  # not implemented
>>> 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
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 an integer) – if specified, truncate or zero pad the list of coefficients to this degree before reversing it.

EXAMPLES:

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
>>> 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
rshift_coeffs(shift, no_list=False)[source]#

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

Note

EXAMPLES:

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)]
>>> 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)]
valuation(val_of_var=None)[source]#

Return the valuation of self.

INPUT:

• self – a p-adic polynomial

• val_of_varNone 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: 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
>>> 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
valuation_of_coefficient(n=None)[source]#

Return valuation information about self’s coefficients.

INPUT:

• self – a p-adic polynomial

• nNone or an 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: 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
>>> 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