Univariate Polynomials over domains and fields¶
AUTHORS:
William Stein: first version
Martin Albrecht: Added singular coercion.
David Harvey: split off polynomial_integer_dense_ntl.pyx (2007-09)
Robert Bradshaw: split off polynomial_modn_dense_ntl.pyx (2007-09)
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_cdv(parent, is_gen=False, construct=False)[source]¶
Bases:
Polynomial_generic_domain
A generic class for polynomials over complete discrete valuation domains and fields.
AUTHOR:
Xavier Caruso (2013-03)
- factor_of_slope(slope=None)[source]¶
INPUT:
slope
– a rational number (default: the first slope in the Newton polygon ofself
)
OUTPUT:
The factor of
self
corresponding to the slopeslope
(i.e. the unique monic divisor ofself
whose slope isslope
and degree is the length ofslope
in the Newton polygon).EXAMPLES:
sage: # needs sage.geometry.polyhedron sage.rings.padics sage: K = Qp(5) sage: R.<x> = K[] sage: K = Qp(5) sage: R.<t> = K[] sage: f = 5 + 3*t + t^4 + 25*t^10 sage: f.newton_slopes() [1, 0, 0, 0, -1/3, -1/3, -1/3, -1/3, -1/3, -1/3] sage: g = f.factor_of_slope(0) sage: g.newton_slopes() [0, 0, 0] sage: (f % g).is_zero() True sage: h = f.factor_of_slope() sage: h.newton_slopes() [1] sage: (f % h).is_zero() True
>>> from sage.all import * >>> # needs sage.geometry.polyhedron sage.rings.padics >>> K = Qp(Integer(5)) >>> R = K['x']; (x,) = R._first_ngens(1) >>> 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_slopes() [1, 0, 0, 0, -1/3, -1/3, -1/3, -1/3, -1/3, -1/3] >>> g = f.factor_of_slope(Integer(0)) >>> g.newton_slopes() [0, 0, 0] >>> (f % g).is_zero() True >>> h = f.factor_of_slope() >>> h.newton_slopes() [1] >>> (f % h).is_zero() True
If
slope
is not a slope ofself
, the corresponding factor is \(1\):sage: f.factor_of_slope(-1) # needs sage.geometry.polyhedron sage.rings.padics 1 + O(5^20)
>>> from sage.all import * >>> f.factor_of_slope(-Integer(1)) # needs sage.geometry.polyhedron sage.rings.padics 1 + O(5^20)
AUTHOR:
Xavier Caruso (2013-03-20)
- hensel_lift(a)[source]¶
Lift \(a\) to a root of this polynomial (using Newton iteration).
If \(a\) is not close enough to a root (so that Newton iteration does not converge), an error is raised.
EXAMPLES:
sage: # needs sage.rings.padics sage: K = Qp(5, 10) sage: P.<x> = PolynomialRing(K) sage: f = x^2 + 1 sage: root = f.hensel_lift(2); root 2 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + O(5^10) sage: f(root) O(5^10) sage: g = (x^2 + 1) * (x - 7) # needs sage.rings.padics sage: g.hensel_lift(2) # here, 2 is a multiple root modulo p # needs sage.rings.padics Traceback (most recent call last): ... ValueError: a is not close enough to a root of this polynomial
>>> from sage.all import * >>> # needs sage.rings.padics >>> K = Qp(Integer(5), Integer(10)) >>> P = PolynomialRing(K, names=('x',)); (x,) = P._first_ngens(1) >>> f = x**Integer(2) + Integer(1) >>> root = f.hensel_lift(Integer(2)); root 2 + 5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + O(5^10) >>> f(root) O(5^10) >>> g = (x**Integer(2) + Integer(1)) * (x - Integer(7)) # needs sage.rings.padics >>> g.hensel_lift(Integer(2)) # here, 2 is a multiple root modulo p # needs sage.rings.padics Traceback (most recent call last): ... ValueError: a is not close enough to a root of this polynomial
AUTHOR:
Xavier Caruso (2013-03-23)
- newton_polygon()[source]¶
Return a list of vertices of the Newton polygon of this polynomial.
Note
If some coefficients have not enough precision an error is raised.
EXAMPLES:
sage: # needs sage.geometry.polyhedron sage.rings.padics sage: K = Qp(5) sage: R.<t> = K[] sage: f = 5 + 3*t + t^4 + 25*t^10 sage: f.newton_polygon() Finite Newton polygon with 4 vertices: (0, 1), (1, 0), (4, 0), (10, 2) 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() Traceback (most recent call last): ... PrecisionError: The coefficient of t^4 has not enough precision
>>> from sage.all import * >>> # needs sage.geometry.polyhedron sage.rings.padics >>> 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() Finite Newton polygon with 4 vertices: (0, 1), (1, 0), (4, 0), (10, 2) >>> 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() 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
isTrue
, each slope is repeated a number of times equal to its multiplicity. Otherwise it appears only one time.EXAMPLES:
sage: # needs sage.geometry.polyhedron sage.rings.padics sage: K = Qp(5) sage: R.<t> = K[] sage: f = 5 + 3*t + t^4 + 25*t^10 sage: f.newton_polygon() Finite Newton polygon with 4 vertices: (0, 1), (1, 0), (4, 0), (10, 2) sage: f.newton_slopes() [1, 0, 0, 0, -1/3, -1/3, -1/3, -1/3, -1/3, -1/3] sage: f.newton_slopes(repetition=False) [1, 0, -1/3]
>>> from sage.all import * >>> # needs sage.geometry.polyhedron sage.rings.padics >>> 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() Finite Newton polygon with 4 vertices: (0, 1), (1, 0), (4, 0), (10, 2) >>> f.newton_slopes() [1, 0, 0, 0, -1/3, -1/3, -1/3, -1/3, -1/3, -1/3] >>> f.newton_slopes(repetition=False) [1, 0, -1/3]
AUTHOR:
Xavier Caruso (2013-03-20)
- slope_factorization()[source]¶
Return a factorization of
self
into a product of factors corresponding to each slope in the Newton polygon.EXAMPLES:
sage: # needs sage.geometry.polyhedron sage.rings.padics sage: K = Qp(5) sage: R.<x> = K[] sage: K = Qp(5) sage: R.<t> = K[] sage: f = 5 + 3*t + t^4 + 25*t^10 sage: f.newton_slopes() [1, 0, 0, 0, -1/3, -1/3, -1/3, -1/3, -1/3, -1/3] sage: F = f.slope_factorization() sage: F.prod() == f True sage: for (f,_) in F: ....: print(f.newton_slopes()) [-1/3, -1/3, -1/3, -1/3, -1/3, -1/3] [0, 0, 0] [1]
>>> from sage.all import * >>> # needs sage.geometry.polyhedron sage.rings.padics >>> K = Qp(Integer(5)) >>> R = K['x']; (x,) = R._first_ngens(1) >>> 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_slopes() [1, 0, 0, 0, -1/3, -1/3, -1/3, -1/3, -1/3, -1/3] >>> F = f.slope_factorization() >>> F.prod() == f True >>> for (f,_) in F: ... print(f.newton_slopes()) [-1/3, -1/3, -1/3, -1/3, -1/3, -1/3] [0, 0, 0] [1]
AUTHOR:
Xavier Caruso (2013-03-20)
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_cdvf(parent, is_gen=False, construct=False)[source]¶
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_cdvr(parent, is_gen=False, construct=False)[source]¶
Bases:
Polynomial_generic_cdv
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_cdv[source]¶
Bases:
Polynomial_generic_dense_inexact
,Polynomial_generic_cdv
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_cdvf[source]¶
Bases:
Polynomial_generic_dense_cdv
,Polynomial_generic_cdvf
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_cdvr[source]¶
Bases:
Polynomial_generic_dense_cdv
,Polynomial_generic_cdvr
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_field(parent, x=None, check=True, is_gen=False, construct=False)[source]¶
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_domain(parent, is_gen=False, construct=False)[source]¶
Bases:
Polynomial
,IntegralDomainElement
- is_unit()[source]¶
Return
True
if this polynomial is a unit.EXERCISE (Atiyah-McDonald, Ch 1): Let \(A[x]\) be a polynomial ring in one variable. Then \(f=\sum a_i x^i \in A[x]\) is a unit if and only if \(a_0\) is a unit and \(a_1,\ldots, a_n\) are nilpotent.
EXAMPLES:
sage: R.<z> = PolynomialRing(ZZ, sparse=True) sage: (2 + z^3).is_unit() False sage: f = -1 + 3*z^3; f 3*z^3 - 1 sage: f.is_unit() False sage: R(-3).is_unit() False sage: R(-1).is_unit() True sage: R(0).is_unit() False
>>> from sage.all import * >>> R = PolynomialRing(ZZ, sparse=True, names=('z',)); (z,) = R._first_ngens(1) >>> (Integer(2) + z**Integer(3)).is_unit() False >>> f = -Integer(1) + Integer(3)*z**Integer(3); f 3*z^3 - 1 >>> f.is_unit() False >>> R(-Integer(3)).is_unit() False >>> R(-Integer(1)).is_unit() True >>> R(Integer(0)).is_unit() False
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_field(parent, is_gen=False, construct=False)[source]¶
Bases:
Polynomial_singular_repr
,Polynomial_generic_domain
,EuclideanDomainElement
- quo_rem(other)[source]¶
Return a tuple
(quotient, remainder)
whereself = quotient * other + remainder
.EXAMPLES:
sage: # needs sage.rings.number_field sage: R.<y> = PolynomialRing(QQ) sage: K.<t> = NumberField(y^2 - 2) sage: P.<x> = PolynomialRing(K) sage: x.quo_rem(K(1)) (x, 0) sage: x.xgcd(K(1)) (1, 0, 1)
>>> from sage.all import * >>> # needs sage.rings.number_field >>> R = PolynomialRing(QQ, names=('y',)); (y,) = R._first_ngens(1) >>> K = NumberField(y**Integer(2) - Integer(2), names=('t',)); (t,) = K._first_ngens(1) >>> P = PolynomialRing(K, names=('x',)); (x,) = P._first_ngens(1) >>> x.quo_rem(K(Integer(1))) (x, 0) >>> x.xgcd(K(Integer(1))) (1, 0, 1)
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_sparse(parent, x=None, check=True, is_gen=False, construct=False)[source]¶
Bases:
Polynomial
A generic sparse polynomial.
The
Polynomial_generic_sparse
class defines functionality for sparse polynomials over any base ring. A sparse polynomial is represented using a dictionary which maps each exponent to the corresponding coefficient. The coefficients must never be zero.EXAMPLES:
sage: R.<x> = PolynomialRing(PolynomialRing(QQ, 'y'), sparse=True) sage: f = x^3 - x + 17 sage: type(f) <class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain_with_category.element_class'> sage: loads(f.dumps()) == f True
>>> from sage.all import * >>> R = PolynomialRing(PolynomialRing(QQ, 'y'), sparse=True, names=('x',)); (x,) = R._first_ngens(1) >>> f = x**Integer(3) - x + Integer(17) >>> type(f) <class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain_with_category.element_class'> >>> loads(f.dumps()) == f True
A more extensive example:
sage: # needs sage.libs.pari sage: A.<T> = PolynomialRing(Integers(5), sparse=True) sage: f = T^2 + 1; B = A.quo(f) sage: C.<s> = PolynomialRing(B) sage: C Univariate Polynomial Ring in s over Univariate Quotient Polynomial Ring in Tbar over Ring of integers modulo 5 with modulus T^2 + 1 sage: s + T s + Tbar sage: (s + T)**2 s^2 + 2*Tbar*s + 4
>>> from sage.all import * >>> # needs sage.libs.pari >>> A = PolynomialRing(Integers(Integer(5)), sparse=True, names=('T',)); (T,) = A._first_ngens(1) >>> f = T**Integer(2) + Integer(1); B = A.quo(f) >>> C = PolynomialRing(B, names=('s',)); (s,) = C._first_ngens(1) >>> C Univariate Polynomial Ring in s over Univariate Quotient Polynomial Ring in Tbar over Ring of integers modulo 5 with modulus T^2 + 1 >>> s + T s + Tbar >>> (s + T)**Integer(2) s^2 + 2*Tbar*s + 4
- coefficients(sparse=True)[source]¶
Return the coefficients of the monomials appearing in
self
.EXAMPLES:
sage: R.<w> = PolynomialRing(Integers(8), sparse=True) sage: f = 5 + w^1997 - w^10000; f 7*w^10000 + w^1997 + 5 sage: f.coefficients() [5, 1, 7]
>>> from sage.all import * >>> R = PolynomialRing(Integers(Integer(8)), sparse=True, names=('w',)); (w,) = R._first_ngens(1) >>> f = Integer(5) + w**Integer(1997) - w**Integer(10000); f 7*w^10000 + w^1997 + 5 >>> f.coefficients() [5, 1, 7]
- degree(gen=None)[source]¶
Return the degree of this sparse polynomial.
EXAMPLES:
sage: R.<z> = PolynomialRing(ZZ, sparse=True) sage: f = 13*z^50000 + 15*z^2 + 17*z sage: f.degree() 50000
>>> from sage.all import * >>> R = PolynomialRing(ZZ, sparse=True, names=('z',)); (z,) = R._first_ngens(1) >>> f = Integer(13)*z**Integer(50000) + Integer(15)*z**Integer(2) + Integer(17)*z >>> f.degree() 50000
- dict()[source]¶
Return a new copy of the dict of the underlying elements of
self
.EXAMPLES:
sage: R.<w> = PolynomialRing(Integers(8), sparse=True) sage: f = 5 + w^1997 - w^10000; f 7*w^10000 + w^1997 + 5 sage: d = f.monomial_coefficients(); d {0: 5, 1997: 1, 10000: 7} sage: d[0] = 10 sage: f.monomial_coefficients() {0: 5, 1997: 1, 10000: 7}
>>> from sage.all import * >>> R = PolynomialRing(Integers(Integer(8)), sparse=True, names=('w',)); (w,) = R._first_ngens(1) >>> f = Integer(5) + w**Integer(1997) - w**Integer(10000); f 7*w^10000 + w^1997 + 5 >>> d = f.monomial_coefficients(); d {0: 5, 1997: 1, 10000: 7} >>> d[Integer(0)] = Integer(10) >>> f.monomial_coefficients() {0: 5, 1997: 1, 10000: 7}
dict
is an alias:sage: f.dict() {0: 5, 1997: 1, 10000: 7}
>>> from sage.all import * >>> f.dict() {0: 5, 1997: 1, 10000: 7}
- exponents()[source]¶
Return the exponents of the monomials appearing in
self
.EXAMPLES:
sage: R.<w> = PolynomialRing(Integers(8), sparse=True) sage: f = 5 + w^1997 - w^10000; f 7*w^10000 + w^1997 + 5 sage: f.exponents() [0, 1997, 10000]
>>> from sage.all import * >>> R = PolynomialRing(Integers(Integer(8)), sparse=True, names=('w',)); (w,) = R._first_ngens(1) >>> f = Integer(5) + w**Integer(1997) - w**Integer(10000); f 7*w^10000 + w^1997 + 5 >>> f.exponents() [0, 1997, 10000]
- gcd(other, algorithm=None)[source]¶
Return the gcd of this polynomial and
other
.INPUT:
other
– a polynomial defined over the same ring as this polynomial
ALGORITHM:
Two algorithms are provided:
'generic'
– uses the generic implementation, which depends on the base ring being a UFD or a field'dense'
– the polynomials are converted to the dense representation, their gcd is computed and is converted back to the sparse representation
Default is
'dense'
for polynomials over \(\ZZ\) and'generic'
in the other cases.EXAMPLES:
sage: R.<x> = PolynomialRing(ZZ, sparse=True) sage: p = x^6 + 7*x^5 + 8*x^4 + 6*x^3 + 2*x^2 + x + 2 sage: q = 2*x^4 - x^3 - 2*x^2 - 4*x - 1 sage: gcd(p, q) x^2 + x + 1 sage: gcd(p, q, algorithm='dense') x^2 + x + 1 sage: gcd(p, q, algorithm='generic') x^2 + x + 1 sage: gcd(p, q, algorithm='foobar') Traceback (most recent call last): ... ValueError: Unknown algorithm 'foobar'
>>> from sage.all import * >>> R = PolynomialRing(ZZ, sparse=True, names=('x',)); (x,) = R._first_ngens(1) >>> p = x**Integer(6) + Integer(7)*x**Integer(5) + Integer(8)*x**Integer(4) + Integer(6)*x**Integer(3) + Integer(2)*x**Integer(2) + x + Integer(2) >>> q = Integer(2)*x**Integer(4) - x**Integer(3) - Integer(2)*x**Integer(2) - Integer(4)*x - Integer(1) >>> gcd(p, q) x^2 + x + 1 >>> gcd(p, q, algorithm='dense') x^2 + x + 1 >>> gcd(p, q, algorithm='generic') x^2 + x + 1 >>> gcd(p, q, algorithm='foobar') Traceback (most recent call last): ... ValueError: Unknown algorithm 'foobar'
- integral(var=None)[source]¶
Return the integral of this polynomial.
By default, the integration variable is the variable of the polynomial.
Otherwise, the integration variable is the optional parameter
var
Note
The integral is always chosen so that the constant term is 0.
EXAMPLES:
sage: R.<x> = PolynomialRing(ZZ, sparse=True) sage: (1 + 3*x^10 - 2*x^100).integral() -2/101*x^101 + 3/11*x^11 + x
>>> from sage.all import * >>> R = PolynomialRing(ZZ, sparse=True, names=('x',)); (x,) = R._first_ngens(1) >>> (Integer(1) + Integer(3)*x**Integer(10) - Integer(2)*x**Integer(100)).integral() -2/101*x^101 + 3/11*x^11 + x
- list(copy=True)[source]¶
Return a new copy of the list of the underlying elements of
self
.EXAMPLES:
sage: R.<z> = PolynomialRing(Integers(100), sparse=True) sage: f = 13*z^5 + 15*z^2 + 17*z sage: f.list() [0, 17, 15, 0, 0, 13]
>>> from sage.all import * >>> R = PolynomialRing(Integers(Integer(100)), sparse=True, names=('z',)); (z,) = R._first_ngens(1) >>> f = Integer(13)*z**Integer(5) + Integer(15)*z**Integer(2) + Integer(17)*z >>> f.list() [0, 17, 15, 0, 0, 13]
- monomial_coefficients()[source]¶
Return a new copy of the dict of the underlying elements of
self
.EXAMPLES:
sage: R.<w> = PolynomialRing(Integers(8), sparse=True) sage: f = 5 + w^1997 - w^10000; f 7*w^10000 + w^1997 + 5 sage: d = f.monomial_coefficients(); d {0: 5, 1997: 1, 10000: 7} sage: d[0] = 10 sage: f.monomial_coefficients() {0: 5, 1997: 1, 10000: 7}
>>> from sage.all import * >>> R = PolynomialRing(Integers(Integer(8)), sparse=True, names=('w',)); (w,) = R._first_ngens(1) >>> f = Integer(5) + w**Integer(1997) - w**Integer(10000); f 7*w^10000 + w^1997 + 5 >>> d = f.monomial_coefficients(); d {0: 5, 1997: 1, 10000: 7} >>> d[Integer(0)] = Integer(10) >>> f.monomial_coefficients() {0: 5, 1997: 1, 10000: 7}
dict
is an alias:sage: f.dict() {0: 5, 1997: 1, 10000: 7}
>>> from sage.all import * >>> f.dict() {0: 5, 1997: 1, 10000: 7}
- number_of_terms()[source]¶
Return the number of nonzero terms.
EXAMPLES:
sage: R.<x> = PolynomialRing(ZZ, sparse=True) sage: p = x^100 - 3*x^10 + 12 sage: p.number_of_terms() 3
>>> from sage.all import * >>> R = PolynomialRing(ZZ, sparse=True, names=('x',)); (x,) = R._first_ngens(1) >>> p = x**Integer(100) - Integer(3)*x**Integer(10) + Integer(12) >>> p.number_of_terms() 3
- quo_rem(other)[source]¶
Return the quotient and remainder of the Euclidean division of
self
andother
.Raises
ZeroDivisionError
ifother
is zero.Raises
ArithmeticError
ifother
has a nonunit leading coefficient and this causes the Euclidean division to fail.EXAMPLES:
sage: P.<x> = PolynomialRing(ZZ, sparse=True) sage: R.<y> = PolynomialRing(P, sparse=True) sage: f = R.random_element(10) sage: while x.divides(f.leading_coefficient()): ....: f = R.random_element(10) sage: g = y^5 + R.random_element(4) sage: q, r = f.quo_rem(g) sage: f == q*g + r and r.degree() < g.degree() True sage: g = x*y^5 sage: f.quo_rem(g) Traceback (most recent call last): ... ArithmeticError: Division non exact (consider coercing to polynomials over the fraction field) sage: g = 0 sage: f.quo_rem(g) Traceback (most recent call last): ... ZeroDivisionError: Division by zero polynomial
>>> from sage.all import * >>> P = PolynomialRing(ZZ, sparse=True, names=('x',)); (x,) = P._first_ngens(1) >>> R = PolynomialRing(P, sparse=True, names=('y',)); (y,) = R._first_ngens(1) >>> f = R.random_element(Integer(10)) >>> while x.divides(f.leading_coefficient()): ... f = R.random_element(Integer(10)) >>> g = y**Integer(5) + R.random_element(Integer(4)) >>> q, r = f.quo_rem(g) >>> f == q*g + r and r.degree() < g.degree() True >>> g = x*y**Integer(5) >>> f.quo_rem(g) Traceback (most recent call last): ... ArithmeticError: Division non exact (consider coercing to polynomials over the fraction field) >>> g = Integer(0) >>> f.quo_rem(g) Traceback (most recent call last): ... ZeroDivisionError: Division by zero polynomial
If the leading coefficient of
other
is not a unit, Euclidean division may still work:sage: f = -x*y^10 + 2*x*y^7 + y^3 - 2*x^2*y^2 - y sage: g = x*y^5 sage: f.quo_rem(g) (-y^5 + 2*y^2, y^3 - 2*x^2*y^2 - y)
>>> from sage.all import * >>> f = -x*y**Integer(10) + Integer(2)*x*y**Integer(7) + y**Integer(3) - Integer(2)*x**Integer(2)*y**Integer(2) - y >>> g = x*y**Integer(5) >>> f.quo_rem(g) (-y^5 + 2*y^2, y^3 - 2*x^2*y^2 - y)
Polynomials over noncommutative rings are also allowed:
sage: # needs sage.combinat sage.modules sage: HH = QuaternionAlgebra(QQ, -1, -1) sage: P.<x> = PolynomialRing(HH, sparse=True) sage: f = P.random_element(5) sage: g = P.random_element((0, 5)) sage: q, r = f.quo_rem(g) sage: f == q*g + r True
>>> from sage.all import * >>> # needs sage.combinat sage.modules >>> HH = QuaternionAlgebra(QQ, -Integer(1), -Integer(1)) >>> P = PolynomialRing(HH, sparse=True, names=('x',)); (x,) = P._first_ngens(1) >>> f = P.random_element(Integer(5)) >>> g = P.random_element((Integer(0), Integer(5))) >>> q, r = f.quo_rem(g) >>> f == q*g + r True
AUTHORS:
Bruno Grenet (2014-07-09)
- reverse(degree=None)[source]¶
Return this polynomial but with the coefficients reversed.
If an optional degree argument is given, the coefficient list will be truncated or zero padded as necessary and the reverse polynomial will have the specified degree.
EXAMPLES:
sage: R.<x> = PolynomialRing(ZZ, sparse=True) sage: p = x^4 + 2*x^2^100 sage: p.reverse() x^1267650600228229401496703205372 + 2 sage: p.reverse(10) x^6
>>> from sage.all import * >>> R = PolynomialRing(ZZ, sparse=True, names=('x',)); (x,) = R._first_ngens(1) >>> p = x**Integer(4) + Integer(2)*x**Integer(2)**Integer(100) >>> p.reverse() x^1267650600228229401496703205372 + 2 >>> p.reverse(Integer(10)) x^6
- shift(n)[source]¶
Return this polynomial multiplied by the power \(x^n\).
If \(n\) is negative, terms below \(x^n\) will be discarded. Does not change this polynomial.
EXAMPLES:
sage: R.<x> = PolynomialRing(ZZ, sparse=True) sage: p = x^100000 + 2*x + 4 sage: type(p) <class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain_with_category.element_class'> sage: p.shift(0) x^100000 + 2*x + 4 sage: p.shift(-1) x^99999 + 2 sage: p.shift(-100002) 0 sage: p.shift(2) x^100002 + 2*x^3 + 4*x^2
>>> from sage.all import * >>> R = PolynomialRing(ZZ, sparse=True, names=('x',)); (x,) = R._first_ngens(1) >>> p = x**Integer(100000) + Integer(2)*x + Integer(4) >>> type(p) <class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain_with_category.element_class'> >>> p.shift(Integer(0)) x^100000 + 2*x + 4 >>> p.shift(-Integer(1)) x^99999 + 2 >>> p.shift(-Integer(100002)) 0 >>> p.shift(Integer(2)) x^100002 + 2*x^3 + 4*x^2
AUTHOR: - David Harvey (2006-08-06)
- truncate(n)[source]¶
Return the polynomial of degree \(< n\) equal to
self
modulo \(x^n\).EXAMPLES:
sage: R.<x> = PolynomialRing(ZZ, sparse=True) sage: (x^11 + x^10 + 1).truncate(11) x^10 + 1 sage: (x^2^500 + x^2^100 + 1).truncate(2^101) x^1267650600228229401496703205376 + 1
>>> from sage.all import * >>> R = PolynomialRing(ZZ, sparse=True, names=('x',)); (x,) = R._first_ngens(1) >>> (x**Integer(11) + x**Integer(10) + Integer(1)).truncate(Integer(11)) x^10 + 1 >>> (x**Integer(2)**Integer(500) + x**Integer(2)**Integer(100) + Integer(1)).truncate(Integer(2)**Integer(101)) x^1267650600228229401496703205376 + 1
- valuation(p=None)[source]¶
Return the valuation of
self
.EXAMPLES:
sage: # needs sage.rings.finite_rings sage: R.<w> = PolynomialRing(GF(9, 'a'), sparse=True) sage: f = w^1997 - w^10000 sage: f.valuation() 1997 sage: R(19).valuation() 0 sage: R(0).valuation() +Infinity
>>> from sage.all import * >>> # needs sage.rings.finite_rings >>> R = PolynomialRing(GF(Integer(9), 'a'), sparse=True, names=('w',)); (w,) = R._first_ngens(1) >>> f = w**Integer(1997) - w**Integer(10000) >>> f.valuation() 1997 >>> R(Integer(19)).valuation() 0 >>> R(Integer(0)).valuation() +Infinity
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_sparse_cdv(parent, x=None, check=True, is_gen=False, construct=False)[source]¶
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_sparse_cdvf(parent, x=None, check=True, is_gen=False, construct=False)[source]¶
Bases:
Polynomial_generic_sparse_cdv
,Polynomial_generic_cdvf
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_sparse_cdvr(parent, x=None, check=True, is_gen=False, construct=False)[source]¶
Bases:
Polynomial_generic_sparse_cdv
,Polynomial_generic_cdvr
- class sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_sparse_field(parent, x=None, check=True, is_gen=False, construct=False)[source]¶
Bases:
Polynomial_generic_sparse
,Polynomial_generic_field
EXAMPLES:
sage: R.<x> = PolynomialRing(Frac(RR['t']), sparse=True) sage: f = x^3 - x + 17 sage: type(f) <class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_field_with_category.element_class'> sage: loads(f.dumps()) == f True
>>> from sage.all import * >>> R = PolynomialRing(Frac(RR['t']), sparse=True, names=('x',)); (x,) = R._first_ngens(1) >>> f = x**Integer(3) - x + Integer(17) >>> type(f) <class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_field_with_category.element_class'> >>> loads(f.dumps()) == f True