28.2 Univariate Polynomial Base Class

Module: sage.rings.polynomial.polynomial_element

Univariate Polynomial Base Class

Author Log:

TESTS:

sage: R.<x> = ZZ[]
sage: f = x^5 + 2*x^2 + (-1)
sage: f == loads(dumps(f))
True

Module-level Functions

is_Polynomial( )

Return True if f is of type univariate polynomial.

Input:

f
- an object

sage: R.<x> = ZZ[]
sage: is_Polynomial(x^3 + x + 1)
True
sage: S.<y> = R[]
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: is_Polynomial(f)
True

However this function does not return True for genuine multivariate polynomial type objects or symbolic polynomials, since those are not of the same data type as univariate polynomials:

sage: R.<x,y> = QQ[]
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: is_Polynomial(f)
False
sage: var('x,y')
(x, y)
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: is_Polynomial(f)
False

make_generic_polynomial( )

Class: Polynomial

class Polynomial
A polynomial.

sage: R.<y> = QQ['y']
sage: S.<x> = R['x']  
sage: f = x*y; f
y*x
sage: type(f)
<type 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
sage: p = (y+1)^10; p(1)
1024

Functions: args,$ \,$ base_extend,$ \,$ base_ring,$ \,$ change_ring,$ \,$ change_variable_name,$ \,$ coefficients,$ \,$ coeffs,$ \,$ complex_roots,$ \,$ constant_coefficient,$ \,$ degree,$ \,$ denominator,$ \,$ derivative,$ \,$ dict,$ \,$ discriminant,$ \,$ exponents,$ \,$ factor,$ \,$ hamming_weight,$ \,$ integral,$ \,$ inverse_mod,$ \,$ inverse_of_unit,$ \,$ is_constant,$ \,$ is_gen,$ \,$ is_irreducible,$ \,$ is_monic,$ \,$ is_nilpotent,$ \,$ is_squarefree,$ \,$ is_unit,$ \,$ leading_coefficient,$ \,$ list,$ \,$ monic,$ \,$ name,$ \,$ newton_raphson,$ \,$ newton_slopes,$ \,$ norm,$ \,$ ord,$ \,$ padded_list,$ \,$ plot,$ \,$ polynomial,$ \,$ prec,$ \,$ radical,$ \,$ real_roots,$ \,$ resultant,$ \,$ reverse,$ \,$ root_field,$ \,$ roots,$ \,$ shift,$ \,$ square,$ \,$ squarefree_decomposition,$ \,$ subs,$ \,$ substitute,$ \,$ truncate,$ \,$ valuation,$ \,$ variable_name,$ \,$ variables

args( )

Returns the generator of this polynomial ring, which is the (only) argument used when calling self.

sage: R.<x> = QQ[]
sage: x.args()
(x,)

A constant polynomial has no variables, but still takes a single argument.

sage: R(2).args()
(x,)

base_extend( )

Return a copy of this polynomial but with coefficients in R, if there is a natural map from coefficient ring of self to R.

sage: R.<x> = QQ[]
sage: f = x^3 - 17*x + 3
sage: f.base_extend(GF(7))
Traceback (most recent call last):
...
TypeError: no such base extension
sage: f.change_ring(GF(7))
x^3 + 4*x + 3

base_ring( )

Return the base ring of the parent of self.

sage: R.<x> = ZZ[]
sage: x.base_ring()
Integer Ring
sage: (2*x+3).base_ring()
Integer Ring

change_ring( )

Return a copy of this polynomial but with coefficients in R, if at all possible.

sage: K.<z> = CyclotomicField(3)
sage: f = K.defining_polynomial()
sage: f.change_ring(GF(7))
x^2 + x + 1

change_variable_name( )

Return a new polynomial over the same base ring but in a different variable.

sage: x = polygen(QQ,'x')
sage: f = -2/7*x^3 + (2/3)*x - 19/993; f
-2/7*x^3 + 2/3*x - 19/993
sage: f.change_variable_name('theta')
-2/7*theta^3 + 2/3*theta - 19/993

coefficients( )

Return the coefficients of the monomials appearing in self.

sage: _.<x> = PolynomialRing(ZZ)
sage: f = x^4+2*x^2+1
sage: f.coefficients()
[1, 2, 1]

coeffs( )

Returns self.list().

(It potentially slightly faster better to use self.list() directly.)

sage: x = QQ['x'].0
sage: f = 10*x^3 + 5*x + 2/17
sage: f.coeffs()
[2/17, 5, 0, 10]

complex_roots( )

Return the complex roots of this polynomial, without multiplicities.

Calls self.roots(ring=CC), unless this is a polynomial with floating-point coefficients, in which case it is uses the appropriate precision from the input coefficients.

sage: x = polygen(ZZ)
sage: (x^3 - 1).complex_roots()   # note: low order bits slightly different on ppc.
[1.00000000000000, -0.500000000000000 + 0.86602540378443...*I,
-0.500000000000000 - 0.86602540378443...*I]

TESTS:

sage: x = polygen(RR)
sage: (x^3 - 1).complex_roots()[0].parent()
Complex Field with 53 bits of precision
sage: x = polygen(RDF)
sage: (x^3 - 1).complex_roots()[0].parent()
Complex Double Field
sage: x = polygen(RealField(200))
sage: (x^3 - 1).complex_roots()[0].parent()
Complex Field with 200 bits of precision
sage: x = polygen(CDF)
sage: (x^3 - 1).complex_roots()[0].parent()
Complex Double Field
sage: x = polygen(ComplexField(200))
sage: (x^3 - 1).complex_roots()[0].parent()
Complex Field with 200 bits of precision

constant_coefficient( )

Return the constant coefficient of this polynomial.

Output: elemenet of base ring

sage: R.<x> = QQ[]
sage: f = -2*x^3 + 2*x - 1/3
sage: f.constant_coefficient()
-1/3

degree( )

Return the degree of this polynomial. The zero polynomial has degree -1.

sage: x = ZZ['x'].0
sage: f = x^93 + 2*x + 1
sage: f.degree()
93
sage: x = PolynomialRing(QQ, 'x', sparse=True).0
sage: f = x^100000 
sage: f.degree()
100000

sage: x = QQ['x'].0
sage: f = 2006*x^2006 - x^2 + 3
sage: f.degree()
2006
sage: f = 0*x
sage: f.degree()
-1
sage: f = x + 33
sage: f.degree()
1

Author: Naqi Jaffery (2006-01-24): examples

denominator( )

Return the least common multiple of the denominators of the entries of self, when this makes sense, i.e., when the coefficients have a denominator function.

WARNING: This is not the denominator of the rational function defined by self, which would always be 1 since self is a polynomial.

First we compute the denominator of a polynomial with integer coefficients, which is of course 1.

sage: R.<x> = ZZ[]
sage: f = x^3 + 17*x + 1
sage: f.denominator()
1

Next we compute the denominator of a polynomial with rational coefficients.

sage: R.<x> = PolynomialRing(QQ)
sage: f = (1/17)*x^19 - (2/3)*x + 1/3; f
1/17*x^19 - 2/3*x + 1/3
sage: f.denominator()
51

Finally, we try to compute the denominator of a polynomial with coefficients in the real numbers, which is a ring whose elements do not have a denominator method.

sage: R.<x> = RR[]
sage: f = x + RR('0.3'); f
1.00000000000000*x + 0.300000000000000
sage: f.denominator()
Traceback (most recent call last):
...
AttributeError: 'sage.rings.real_mpfr.RealNumber' object has no attribute
'denominator'

derivative( )

The formal derivative of this polynomial, with respect to variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

SEE ALSO: self._derivative()

sage: R.<x> = PolynomialRing(QQ)
sage: g = -x^4 + x^2/2 - x
sage: g.derivative()
-4*x^3 + x - 1
sage: g.derivative(x)
-4*x^3 + x - 1
sage: g.derivative(x, x)
-12*x^2 + 1
sage: g.derivative(x, 2)
-12*x^2 + 1

sage: R.<t> = PolynomialRing(ZZ)
sage: S.<x> = PolynomialRing(R)
sage: f = t^3*x^2 + t^4*x^3
sage: f.derivative()
3*t^4*x^2 + 2*t^3*x
sage: f.derivative(x)
3*t^4*x^2 + 2*t^3*x
sage: f.derivative(t)
4*t^3*x^3 + 3*t^2*x^2

dict( )

Return a sparse dictionary representation of this univariate polynomial.

sage: R.<x> = QQ[]
sage: f = x^3 + -1/7*x + 13
sage: f.dict()
{0: 13, 1: -1/7, 3: 1}

discriminant( )

Returns the discrimant of self.

The discriminant is

$\displaystyle R_n := a_n^{2 n-2} \prod_{1<i<j<n} (r_i-r_j)^2,$

where $ n$ is the degree of self, $ a_n$ is the leading coefficient of self and the roots of self are $ r_1, \ldots, r_n$ .

Output: An element of the base ring of the polynomial ring.

NOTES: Uses the identity $ R_n(f) := (-1)^(n (n-1)/2) R(f, f') a_n^(n-k-2)$ , where $ n$ is the degree of self, $ a_n$ is the leading coefficient of self, $ f'$ is the derivative of $ f$ , and $ k$ is the degree of $ f'$ . Calls self.resultant.

In the case of elliptic curves in special form, the discriminant is easy to calculate:

sage: R.<x> = QQ[]
sage: f = x^3 + x + 1
sage: d = f.discriminant(); d
-31
sage: d.parent() is QQ
True
sage: EllipticCurve([1, 1]).discriminant()/16
-31

sage: R.<x> = QQ[]
sage: f = 2*x^3 + x + 1
sage: d = f.discriminant(); d
-116

We can also compute discriminants over univariate and multivariate polynomial rings, provided that PARI's variable ordering requirements are respected. Usually, your discriminants will work if you always ask for them in the variable x:

sage: R.<a> = QQ[]
sage: S.<x> = R[]
sage: f = a*x + x + a + 1
sage: d = f.discriminant(); d
1
sage: d.parent() is R
True

sage: R.<a, b> = QQ[]
sage: S.<x> = R[]
sage: f = x^2 + a + b
sage: d = f.discriminant(); d
-4*a - 4*b
sage: d.parent() is R
True

Unfortunately SAGE does not handle PARI's variable ordering requirements gracefully, so the following fails:

sage: R.<x, y> = QQ[]
sage: S.<a> = R[]
sage: f = x^2 + a
sage: f.discriminant()
Traceback (most recent call last):
...
PariError: (8)

exponents( )

Return the exponents of the monomials appearing in self.

sage: _.<x> = PolynomialRing(ZZ)
sage: f = x^4+2*x^2+1
sage: f.exponents()
[0, 2, 4]

factor( )

Return the factorization of self over the base ring of this polynomial. Factoring polynomials over $ \mathbf{Z}/n\mathbf{Z}$ for $ n$ composite is at the moment not implemented.

Input: a polynomial

Output:

Factorization
- the factorization of self, which is a product of a unit with a product of powers of irreducible factors.

Over a field the irreducible factors are all monic.

We factor some polynomials over $ \mathbf{Q}$ .

sage: x = QQ['x'].0
sage: f = (x^3 - 1)^2
sage: f.factor()
(x - 1)^2 * (x^2 + x + 1)^2

Notice that over the field $ \mathbf{Q}$ the irreducible factors are monic.

sage: f = 10*x^5 - 1
sage: f.factor()
(10) * (x^5 - 1/10)
sage: f = 10*x^5 - 10
sage: f.factor()
(10) * (x - 1) * (x^4 + x^3 + x^2 + x + 1)

Over $ \mathbf{Z}$ the irreducible factors need not be monic:

sage: x = ZZ['x'].0
sage: f = 10*x^5 - 1
sage: f.factor()
10*x^5 - 1

We factor a non-monic polynomial over the finite field $ F_{25}$ .

sage: k.<a> = GF(25)
sage: R.<x> = k[]
sage: f = 2*x^10 + 2*x + 2*a
sage: F = f.factor(); F
(2) * (x + a + 2) * (x^2 + 3*x + 4*a + 4) * (x^2 + (a + 1)*x + a + 2) *
(x^5 + (3*a + 4)*x^4 + (3*a + 3)*x^3 + 2*a*x^2 + (3*a + 1)*x + 3*a + 1)

Notice that the unit factor is included when we multiply $ F$ back out.

sage: expand(F)
2*x^10 + 2*x + 2*a

Factorization also works even if the variable of the finite field is nefariously labeled "x".

sage: x = GF(3^2, 'a')['x'].0
sage: f = x^10 +7*x -13
sage: G = f.factor(); G
(x + a) * (x + 2*a + 1) * (x^4 + (a + 2)*x^3 + (2*a + 2)*x + 2) * (x^4 +
2*a*x^3 + (a + 1)*x + 2)
sage: prod(G) == f
True

sage: f.parent().base_ring()._assign_names(['a'])
sage: f.factor()
(x + a) * (x + 2*a + 1) * (x^4 + (a + 2)*x^3 + (2*a + 2)*x + 2) * (x^4 +
2*a*x^3 + (a + 1)*x + 2)

sage: k = GF(9,'x')    # purposely calling it x to test robustness
sage: x = PolynomialRing(k,'x0').gen()
sage: f = x^3 + x + 1
sage: f.factor()
(x0 + 2) * (x0 + x) * (x0 + 2*x + 1)
sage: f = 0*x
sage: f.factor()
Traceback (most recent call last):
...
ValueError: factorization of 0 not defined

sage: f = x^0
sage: f.factor()
1

Arbitrary precision real and complex factorization:

sage: R.<x> = RealField(100)[]
sage: F = factor(x^2-3); F
(1.0000000000000000000000000000*x - 1.7320508075688772935274463415) *
(1.0000000000000000000000000000*x + 1.7320508075688772935274463415)
sage: expand(F)
1.0000000000000000000000000000*x^2 - 3.0000000000000000000000000000
sage: factor(x^2 + 1)
1.0000000000000000000000000000*x^2 + 1.0000000000000000000000000000
sage: C = ComplexField(100)
sage: R.<x> = C[]
sage: F = factor(x^2+3); F
(1.0000000000000000000000000000*x - 1.7320508075688772935274463415*I) *
(1.0000000000000000000000000000*x + 1.7320508075688772935274463415*I)
sage: expand(F)
1.0000000000000000000000000000*x^2 + 3.0000000000000000000000000000
sage: factor(x^2+1)
(1.0000000000000000000000000000*x - 1.0000000000000000000000000000*I) *
(1.0000000000000000000000000000*x + 1.0000000000000000000000000000*I)
sage: f = C.0 * (x^2 + 1) ; f
1.0000000000000000000000000000*I*x^2 + 1.0000000000000000000000000000*I
sage: F = factor(f); F
(1.0000000000000000000000000000*I) * (1.0000000000000000000000000000*x -
1.0000000000000000000000000000*I) * (1.0000000000000000000000000000*x +
1.0000000000000000000000000000*I)
sage: expand(F)
1.0000000000000000000000000000*I*x^2 + 1.0000000000000000000000000000*I

Over a complicated number field:

sage: x = polygen(QQ, 'x')
sage: f = x^6 + 10/7*x^5 - 867/49*x^4 - 76/245*x^3 + 3148/35*x^2 - 25944/245*x + 48771/1225
sage: K.<a> = NumberField(f)
sage: S.<T> = K[]
sage: ff = S(f); ff
T^6 + 10/7*T^5 + (-867/49)*T^4 + (-76/245)*T^3 + 3148/35*T^2 +
(-25944/245)*T + 48771/1225
sage: F = ff.factor()
sage: len(F)
4
sage: F[:2]
[(T - a, 1), (T - 40085763200/924556084127*a^5 -
145475769880/924556084127*a^4 + 527617096480/924556084127*a^3 +
1289745809920/924556084127*a^2 - 3227142391585/924556084127*a -
401502691578/924556084127, 1)]
sage: expand(F)
T^6 + 10/7*T^5 + (-867/49)*T^4 + (-76/245)*T^3 + 3148/35*T^2 +
(-25944/245)*T + 48771/1225

sage: f = x^2 - 1/3 ; K.<a> = NumberField(f) ; A.<T> = K[] ; g = A(x^2-1)
sage: g.factor()
(T - 1) * (T + 1)

sage: h = A(3*x^2-1) ; h.factor()
(3) * (T - a) * (T + a)

sage: h = A(x^2-1/3) ; h.factor()
(T - a) * (T + a)

Over the real double field:

sage: x = polygen(RDF)
sage: f = (x-1)^3
sage: f.factor() # random output (unfortunately)
(1.0*x - 1.00000859959) * (1.0*x^2 - 1.99999140041*x + 0.999991400484)
sage: (-2*x^2 - 1).factor() 
(-2.0) * (1.0*x^2 + 0.5) 
sage: (-2*x^2 - 1).factor().expand() 
-2.0*x^2 - 1.0

Note that this factorization suffers from the roots function:

sage: f.roots() # random output (unfortunately)
[1.00000859959, 0.999995700205 + 7.44736245561e-06*I, 0.999995700205 -
7.44736245561e-06*I]

Over the complex double field. Because this approximate, all factors will occur with multiplicity 1.

sage: x = CDF['x'].0; i = CDF.0
sage: f = (x^2 + 2*i)^3
sage: f.factor()    # random low order bits
(1.0*x + -0.999994409957 + 1.00001040378*I) * (1.0*x + -0.999993785062 +
0.999989956987*I) * (1.0*x + -1.00001180498 + 0.999999639235*I) * (1.0*x +
0.999995530902 - 0.999987780431*I) * (1.0*x + 1.00001281704 -
1.00000223945*I) * (1.0*x + 0.999991652054 - 1.00000998012*I)
sage: f(-f.factor()[0][0][0])   # random low order bits
-2.38358052913e-14 - 2.57571741713e-14*I

Over a relative number field:

sage: x = QQ['x'].0
sage: L.<a> = CyclotomicField(3).extension(x^3 - 2)
sage: x = L['x'].0
sage: f = (x^3 + x + a)*(x^5 + x + L.1); f
x^8 + x^6 + a*x^5 + x^4 + zeta3*x^3 + x^2 + (a + zeta3)*x + zeta3*a
sage: f.factor()
(x^3 + x + a) * (x^5 + x + zeta3)

Factoring polynomials over $ \mathbf{Z}/n\mathbf{Z}$ for composite $ n$ is not implemented:

sage: R.<x> = PolynomialRing(Integers(35))
sage: f = (x^2+2*x+2)*(x^2+3*x+9)
sage: f.factor()
Traceback (most recent call last):
...
NotImplementedError: factorization of polynomials over rings with composite
characteristic is not implemented

hamming_weight( )

Returns the number of non-zero coefficients of self.

sage: R.<x> = ZZ[]
sage: f = x^3 - x
sage: f.hamming_weight()
2
sage: R(0).hamming_weight()
0
sage: f = (x+1)^100
sage: f.hamming_weight()
101
sage: S = GF(5)['y']
sage: S(f).hamming_weight()
5
sage: cyclotomic_polynomial(105).hamming_weight()
33

integral( )

Return the integral of this polynomial.

NOTE: The integral is always chosen so the constant term is 0.

sage: R.<x> = ZZ[]
sage: R(0).integral()
0
sage: f = R(2).integral(); f
2*x

Note that since the integral is defined over the same base ring the integral is actually in the base ring.

sage: f.parent()
Univariate Polynomial Ring in x over Integer Ring

If the integral isn't defined over the same base ring, then the base ring is extended:

sage: f = x^3 + x - 2
sage: g = f.integral(); g
1/4*x^4 + 1/2*x^2 - 2*x
sage: g.parent()
Univariate Polynomial Ring in x over Rational Field

inverse_mod( )

Inverts the polynomial a with respect to m, or throw a ValueError if no such inverse exists.

EXAMMPLES:

sage: S.<t> = QQ[]
sage: f = inverse_mod(t^2 + 1, t^3 + 1); f
-1/2*t^2 - 1/2*t + 1/2
sage: f * (t^2 + 1) % (t^3 + 1)
1
sage: f = t.inverse_mod((t+1)^7); f
-t^6 - 7*t^5 - 21*t^4 - 35*t^3 - 35*t^2 - 21*t - 7
sage: (f * t) + (t+1)^7
1

It also works over in-exact rings, but note that due to rounding error the product is only guerenteed to be withing epsilon of the constant polynomial 1.

sage: R.<x> = RDF[]
sage: f = inverse_mod(x^2 + 1, x^5 + x + 1); f
0.4*x^4 - 0.2*x^3 - 0.4*x^2 + 0.2*x + 0.8
sage: f * (x^2 + 1) % (x^5 + x + 1)
5.55111512313e-17*x^3 + 1.66533453694e-16*x^2 + 5.55111512313e-17*x + 1.0
sage: f = inverse_mod(x^3 - x + 1, x - 2); f
0.142857142857
sage: f * (x^3 - x + 1) % (x - 2)
1.0

ALGORITHM: Solve the system as + mt = 1, returning s as the inverse of a mod m.

Uses the Euclidean algorithm for exact rings, and solves a linear system for the coefficients of s and t for inexact rings (as the Euclidean algorithm may not converge in that case).

Author: Robert Bradshaw (2007-05-31)

inverse_of_unit( )

sage: R.<x> = QQ[]
sage: f = x - 90283
sage: f.inverse_of_unit()
Traceback (most recent call last):
...
ValueError: self is not a unit.
sage: f = R(-90283); g = f.inverse_of_unit(); g
-1/90283
sage: parent(g)
Univariate Polynomial Ring in x over Rational Field

is_constant( )

Return True if this is a constant polynomial.

Output:

bool
- True if and only if this polynomial is constant

sage: R.<x> = ZZ[]
sage: x.is_constant()
False
sage: R(2).is_constant()
True
sage: R(0).is_constant()
True

is_gen( )

Return True if this polynomial is the distinguished generator of the parent polynomial ring.

sage: R.<x> = QQ[]
sage: R(1).is_gen()
False
sage: R(x).is_gen()
True

Important - this function doesn't return True if self equals the generator; it returs True if self is the generator.

sage: f = R([0,1]); f
x
sage: f.is_gen()
False
sage: f is x
False
sage: f == x
True

is_irreducible( )

Return True precisely if this polynomial is irreducible over its base ring. Testing irreducibility over $ \mathbf{Z}/n\mathbf{Z}$ for composite $ n$ is not implemented.

sage: R.<x> = ZZ[]
sage: (x^3 + 1).is_irreducible()
False
sage: (x^2 - 1).is_irreducible()
False
sage: (x^3 + 2).is_irreducible()
True
sage: R(0).is_irreducible()
Traceback (most recent call last):
...
ValueError: self must be nonzero

$ 4$ is irreducible as a polynomial, since as a polynomial it doesn't factor:

sage: R(4).is_irreducible()
True

TESTS:

sage: F.<t> = NumberField(x^2-5)
sage: Fx.<xF> = PolynomialRing(F)
sage: f = Fx([2*t - 5, 5*t - 10, 3*t - 6, -t, -t + 2, 1])
sage: f.is_irreducible()
False
sage: f = Fx([2*t - 3, 5*t - 10, 3*t - 6, -t, -t + 2, 1])
sage: f.is_irreducible()
True

is_monic( )

Returns True if this polynomial is monic. The zero polynomial is by definition not monic.

sage: x = QQ['x'].0
sage: f = x + 33
sage: f.is_monic()
True
sage: f = 0*x
sage: f.is_monic()
False
sage: f = 3*x^3 + x^4 + x^2
sage: f.is_monic()
True
sage: f = 2*x^2 + x^3 + 56*x^5
sage: f.is_monic()
False

Author: Naqi Jaffery (2006-01-24): examples

is_nilpotent( )

Return True if this polynomial is nilpotent.

sage: R = Integers(12)
sage: S.<x> = R[]
sage: f = 5 + 6*x
sage: f.is_nilpotent()
False
sage: f = 6 + 6*x^2
sage: f.is_nilpotent()
True
sage: f^2
0

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 nilpotent if and only if every $ a_i$ is nilpotent.

is_squarefree( )

Return True if this polynomial is square free.

sage: x = polygen(QQ)
sage: f = (x-1)*(x-2)*(x^2-5)*(x^17-3); f
x^21 - 3*x^20 - 3*x^19 + 15*x^18 - 10*x^17 - 3*x^4 + 9*x^3 + 9*x^2 - 45*x +
30
sage: f.is_squarefree()
True
sage: (f*(x^2-5)).is_squarefree()
False

is_unit( )

Return True if this polynomial is a unit.

sage: a = Integers(90384098234^3)
sage: b = a(2*191*236607587)
sage: b.is_nilpotent()
True
sage: R.<x> = a[]
sage: f = 3 + b*x + b^2*x^2
sage: f.is_unit()
True
sage: f = 3 + b*x + b^2*x^2 + 17*x^3
sage: f.is_unit()
False

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.

leading_coefficient( )

Return the leading coefficient of this polynomial.

Output: element of the base ring

sage: R.<x> = QQ[]
sage: f = (-2/5)*x^3 + 2*x - 1/3
sage: f.leading_coefficient()
-2/5

list( )

Return a new copy of the list of the underlying elements of self.

sage: R.<x> = QQ[]
sage: f = (-2/5)*x^3 + 2*x - 1/3
sage: v = f.list(); v
[-1/3, 2, 0, -2/5]

Note that v is a list, it is mutable, and each call to the list method returns a new list:

sage: type(v)
<type 'list'>
sage: v[0] = 5
sage: f.list()
[-1/3, 2, 0, -2/5]

Here is an example with a generic polynomial ring:

sage: R.<x> = QQ[]
sage: S.<y> = R[]
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: type(f)
<type 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
sage: v = f.list(); v
[-3*x, x, 0, 1]
sage: v[0] = 10
sage: f.list()
[-3*x, x, 0, 1]

monic( )

Return this polynomial divided by its leading coefficient. Does not change this polynomial.

sage: x = QQ['x'].0
sage: f = 2*x^2 + x^3 + 56*x^5
sage: f.monic()
x^5 + 1/56*x^3 + 1/28*x^2
sage: f = (1/4)*x^2 + 3*x + 1
sage: f.monic()
x^2 + 12*x + 4

The following happens because $ f = 0$ cannot be made into a monic polynomial

sage: f = 0*x
sage: f.monic()
Traceback (most recent call last):
...
ZeroDivisionError: rational division by zero

Notice that the monic version of a polynomial over the integers is defined over the rationals.

sage: x = ZZ['x'].0
sage: f = 3*x^19 + x^2 - 37
sage: g = f.monic(); g
x^19 + 1/3*x^2 - 37/3
sage: g.parent()
Univariate Polynomial Ring in x over Rational Field

Author: Naqi Jaffery (2006-01-24): examples

name( )

Return the string variable name of the indeterminate of this polynomial.

sage: R.<theta> = ZZ[]; 
sage: f = (2-theta)^3; f
-theta^3 + 6*theta^2 - 12*theta + 8
sage: f.name()
'theta'

newton_raphson( )

Return a list of n iterative approximations to a root of this polynomial, computed using the Newton-Raphson method.

The Newton-Raphson method is an iterative root-finding algorithm. For f(x) a polynomial, as is the case here, this is essentially the same as Horner's method.

Input:

n
- an integer (=the number of iterations),
x0
- an initial guess x0.

Output: A list of numbers hopefully approximating a root of f(x)=0.

** If one of the iterates is a critical point of f then a ZeroDivisionError exception is raised.

sage: x = PolynomialRing(RealField(), 'x').gen()
sage: f = x^2 - 2
sage: f.newton_raphson(4, 1)
[1.50000000000000, 1.41666666666667, 1.41421568627451, 1.41421356237469]

Author: David Joyner and William Stein (2005-11-28)

newton_slopes( )

Return the $ p$ -adic slopes of the Newton polygon of self, when this makes sense.

Output:

- list of rational numbers

sage: x = QQ['x'].0
sage: f = x^3 + 2
sage: f.newton_slopes(2)
[1/3, 1/3, 1/3]

ALGORITHM: Uses PARI.

norm( )

Return the $ p$ -norm of this polynomial.

DEFINITION: For integer $ p$ , the $ p$ -norm of a polynomial is the $ p$ th root of the sum of the $ p$ th powers of the absolute values of the coefficients of the polynomial.

Input:

p
- (positive integer or +infinity) the degree of the norm

sage: R.<x> =RR[]
sage: f = x^6 + x^2 + -x^4 - 2*x^3
sage: f.norm(2)
2.64575131106459
sage: (sqrt(1^2 + 1^2 + (-1)^2 + (-2)^2)).n()
2.64575131106459

sage: f.norm(1)
5.00000000000000
sage: f.norm(infinity)
2.00000000000000

sage: f.norm(-1)
Traceback (most recent call last):
...
ValueError: The degree of the norm must be positive

TESTS:

sage: R.<x> = RR[]
sage: f = x^6 + x^2 + -x^4 -x^3
sage: f.norm(int(2))
2.00000000000000

Author Log:

ord( )

This is the same as the valuation of self at p. See the documentation for self.valuation.

sage: P,x=PolynomialRing(ZZ,'x').objgen()
sage: (x^2+x).ord(x+1)
1

padded_list( )

Return list of coefficients of self up to (but not include $ q^n$ ).

Includes 0's in the list on the right so that the list has length $ n$ .

Input:

n
- (default: None); if given, an integer that is at least 0

sage: x = polygen(QQ)
sage: f = 1 + x^3 + 23*x^5
sage: f.padded_list()
[1, 0, 0, 1, 0, 23]
sage: f.padded_list(10)
[1, 0, 0, 1, 0, 23, 0, 0, 0, 0]
sage: len(f.padded_list(10))
10
sage: f.padded_list(3)
[1, 0, 0]
sage: f.padded_list(0)
[]
sage: f.padded_list(-1)
Traceback (most recent call last):
...
ValueError: n must be at least 0

plot( )

Return a plot of this polynomial.

Input:

xmin
- float
xmax
- float
*args, **kwds
- passed to either point or point

Output: returns a graphic object.

sage: x = polygen(GF(389))
sage: plot(x^2 + 1, rgbcolor=(0,0,1)).save()
sage: x = polygen(QQ)
sage: plot(x^2 + 1, rgbcolor=(1,0,0)).save()

polynomial( )

Return a new polynomial in the parent of self.

This function doesn't have much to do with self except that it is a convenient shortcut to avoid having to write self.parent()(...).

Input:

*args, **kwds
- are passed on exactly as is to the parent polynomial ring call method.

sage: R.<x> = ZZ[]
sage: f = 2*x^2 - 3
sage: f.polynomial([12,5,7,3])
3*x^3 + 7*x^2 + 5*x + 12

The input list must of course define a polynomial in the parent:

sage: f.polynomial([12,5,7,3/2])
Traceback (most recent call last):
...
TypeError: no coercion of this rational to integer

prec( )

Return the precision of this polynomials. This is always infinity, since polynomials are of infinite precision by definition (there is no big-oh).

sage: x = polygen(ZZ)
sage: (x^5 + x + 1).prec()
+Infinity
sage: x.prec()
+Infinity

radical( )

Returns the radical of self; over a field, this is the product of the distinct irreducible factors of self. (This is also sometimes called the "square-free part" of self, but that term is ambiguous; it is sometimes used to mean the quotient of self by its maximal square factor.)

sage: P.<x> = ZZ[]
sage: t = (x^2-x+1)^3 * (3*x-1)^2
sage: t.radical()
3*x^3 - 4*x^2 + 4*x - 1

real_roots( )

Return the real roots of this polynomial, without multiplicities.

Calls self.roots(ring=RR), unless this is a polynomial with floating-point real coefficients, in which case it calls self.roots().

sage: x = polygen(ZZ)
sage: (x^2 - x - 1).real_roots()
[-0.618033988749895, 1.61803398874989]

TESTS:

sage: x = polygen(RealField(100))
sage: (x^2 - x - 1).real_roots()[0].parent()
    Real Field with 100 bits of precision
sage: x = polygen(RDF)
sage: (x^2 - x - 1).real_roots()[0].parent()
Real Double Field

resultant( )

Returns the resultant of self and other.

Input:

other
- a polynomial

Output: an element of the base ring of the polynomial ring

NOTES: Implemented using PARI's polresultant function.

sage: R.<x> = QQ[]
sage: f = x^3 + x + 1;  g = x^3 - x - 1
sage: r = f.resultant(g); r
-8
sage: r.parent() is QQ
True

We can also compute resultants over univariate and multivariate polynomial rings, provided that PARI's variable ordering requirements are respected. Usually, your resultants will work if you always ask for them in the variable x:

sage: R.<a> = QQ[]
sage: S.<x> = R[]
sage: f = x^2 + a; g = x^3 + a
sage: r = f.resultant(g); r
a^3 + a^2
sage: r.parent() is R
True

sage: R.<a, b> = QQ[]
sage: S.<x> = R[]
sage: f = x^2 + a; g = x^3 + b
sage: r = f.resultant(g); r
a^3 + b^2
sage: r.parent() is R
True

Unfortunately SAGE does not handle PARI's variable ordering requirements gracefully, so the following fails:

sage: R.<x, y> = QQ[]
sage: S.<a> = R[]
sage: f = x^2 + a; g = y^3 + a
sage: f.resultant(g)
Traceback (most recent call last):
...
PariError: (8)

reverse( )

Return polynomial but with the coefficients reversed.

sage: R.<x> = ZZ[]; S.<y> = R[]
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: f.reverse()
(-3*x)*y^3 + x*y^2 + 1

root_field( )

Return the field generated by the roots of the irreducible polynomial self. The output is either a number field, relative number field, a quotient of a polynomial ring over a field, or the fraction field of the base ring.

sage: R.<x> = QQ['x']
sage: f = x^3 + x + 17
sage: f.root_field('a')
Number Field in a with defining polynomial x^3 + x + 17

sage: R.<x> = QQ['x']
sage: f = x - 3
sage: f.root_field('b')
Rational Field

sage: R.<x> = ZZ['x']
sage: f = x^3 + x + 17
sage: f.root_field('b')
Number Field in b with defining polynomial x^3 + x + 17

sage: y = QQ['x'].0
sage: L.<a> = NumberField(y^3-2)
sage: R.<x> = L['x']
sage: f = x^3 + x + 17
sage: f.root_field('c')
Number Field in c with defining polynomial x^3 + x + 17 over its base field

sage: R.<x> = PolynomialRing(GF(9,'a'))
sage: f = x^3 + x^2 + 8
sage: K.<alpha> = f.root_field(); K
Univariate Quotient Polynomial Ring in alpha over Finite Field in a of size
3^2 with modulus x^3 + x^2 + 2
sage: alpha^2 + 1
alpha^2 + 1
sage: alpha^3 + alpha^2
1

sage: R.<x> = QQ[]
sage: f = x^2
sage: K.<alpha> = f.root_field()
Traceback (most recent call last):
...
ValueError: polynomial must be irreducible

TESTS:

sage: (PolynomialRing(Integers(31),name='x').0+5).root_field('a')
Ring of integers modulo 31

roots( )

Return the roots of this polynomial (by default, in the base ring of this polynomial).

Input:

ring
- the ring to find roots in
multiplicities
- bool (default: True) if True return list of pairs (r, n), where r is the root and n is the multiplicity. If False, just return the unique roots, with no information about multiplicities.
algorithm
- the root-finding algorithm to use. We attempt to select a reasonable algorithm by default, but this lets the caller override our choice.

By default, this finds all the roots that lie in the base ring of the polynomial. However, the ring parameter can be used to specify a ring to look for roots in.

If the polynomial and the output ring are both exact (integers, rationals, finite fields, etc.), then the output should always be correct (or raise an exception, if that case is not yet handled).

If the output ring is approximate (floating-point real or complex numbers), then the answer will be estimated numerically, using floating-point arithmetic of at least the precision of the output ring. If the polynomial is ill-conditioned, meaning that a small change in the coefficients of the polynomial will lead to a relatively large change in the location of the roots, this may give poor results. Distinct roots may be returned as multiple roots, multiple roots may be returned as distinct roots, real roots may be lost entirely (because the numerical estimate thinks they are complex roots). Note that polynomials with multiple roots are always ill-conditioned; there's a footnote at the end of the docstring about this.

If the output ring is a RealIntervalField or ComplexIntervalField of a given precision, then the answer will always be correct (or an exception will be raised, if a case is not implemented). Each root will be contained in one of the returned intervals, and the intervals will be disjoint. (The returned intervals may be of higher precision than the specified output ring.)

At the end of this docstring (after the examples) is a description of all the cases implemented in this function, and the algorithms used. That section also describes the possibilities for "algorithm=", for the cases where multiple algorithms exist.

sage: x = QQ['x'].0
sage: f = x^3 - 1
sage: f.roots()
[(1, 1)]
sage: f.roots(ring=CC)   # note -- low order bits slightly different on ppc.
[(1.00000000000000, 1), (-0.500000000000000 + 0.86602540378443...*I, 1),
(-0.500000000000000 - 0.86602540378443...*I, 1)]
sage: f = (x^3 - 1)^2
sage: f.roots()
[(1, 2)]

sage: f = -19*x + 884736
sage: f.roots()
[(884736/19, 1)]
sage: (f^20).roots()
[(884736/19, 20)]

sage: K.<z> = CyclotomicField(3)
sage: f = K.defining_polynomial()
sage: f.roots(ring=GF(7))
[(4, 1), (2, 1)]
sage: g = f.change_ring(GF(7))
sage: g.roots()
[(4, 1), (2, 1)]
sage: g.roots(multiplicities=False)
[4, 2]

An example over RR, which illustrates that only the roots in RR are returned:

sage: x = RR['x'].0
sage: f = x^3 -2
sage: f.roots()
[(1.25992104989487, 1)]
sage: f.factor()
(1.00000000000000*x - 1.25992104989487) * (1.00000000000000*x^2 +
1.25992104989487*x + 1.58740105196820)
sage: x = RealField(100)['x'].0
sage: f = x^3 -2
sage: f.roots()
[(1.2599210498948731647672106073, 1)]

sage: x = CC['x'].0
sage: f = x^3 -2
sage: f.roots()
[(1.25992104989487, 1), (-0.62996052494743... + 1.09112363597172*I, 1),
(-0.62996052494743... - 1.09112363597172*I, 1)]
sage: f.roots(algorithm='pari')
[(1.25992104989487, 1), (-0.629960524947437 + 1.09112363597172*I, 1),
(-0.629960524947437 - 1.09112363597172*I, 1)]

Another example showing that only roots in the base ring are returned:

sage: x = polygen(ZZ)
sage: f = (2*x-3) * (x-1) * (x+1)
sage: f.roots()
[(1, 1), (-1, 1)]
sage: f.roots(ring=QQ)
[(3/2, 1), (1, 1), (-1, 1)]

An example involving large numbers:

sage: x = RR['x'].0
sage: f = x^2 - 1e100
sage: f.roots()
[(-1.00000000000000e50, 1), (1.00000000000000e50, 1)]
sage: f = x^10 - 2*(5*x-1)^2
sage: f.roots(multiplicities=False)
[-1.6772670339941..., 0.19995479628..., 0.20004530611...,
1.5763035161844...]

sage: x = CC['x'].0
sage: i = CC.0
sage: f = (x - 1)*(x - i)
sage: f.roots(multiplicities=False) #random - this example is numerically rather unstable
[2.22044604925031e-16 + 1.00000000000000*I, 1.00000000000000 +
8.32667268468867e-17*I]
sage: g=(x-1.33+1.33*i)*(x-2.66-2.66*i)
sage: g.roots(multiplicities=False)
[2.66000000000000 + 2.66000000000000*I, 1.33000000000000 -
1.33000000000000*I]

A purely symbolic roots example:

sage: X = var('X')
sage: f = expand((X-1)*(X-I)^3*(X^2 - sqrt(2))); f
X^6 - 3*I*X^5 - X^5 + 3*I*X^4 - sqrt(2)*X^4 - 3*X^4 + 3*sqrt(2)*I*X^3 +
I*X^3 + sqrt(2)*X^3 + 3*X^3 - 3*sqrt(2)*I*X^2 - I*X^2 + 3*sqrt(2)*X^2 -
sqrt(2)*I*X - 3*sqrt(2)*X + sqrt(2)*I
sage: print f.roots()
[(I, 3), (-2^(1/4), 1), (2^(1/4), 1), (1, 1)]

A couple of examples where the base ring doesn't have a factorization algorithm (yet). Note that this is currently done via naive enumeration, so could be very slow:

sage: R = Integers(6)
sage: S.<x> = R['x']
sage: p = x^2-1
sage: p.roots()
Traceback (most recent call last):
...
NotImplementedError: root finding with multiplicities for this polynomial
not implemented (try the multiplicities=False option)
sage: p.roots(multiplicities=False)
[1, 5]
sage: R = Integers(9)
sage: A = PolynomialRing(R, 'y')
sage: y = A.gen()
sage: f = 10*y^2 - y^3 - 9
sage: f.roots(multiplicities=False)
[0, 1, 3, 6]

An example over the complex double field (where root finding is fast, thanks to numpy):

sage: R.<x> = CDF[]
sage: f = R.cyclotomic_polynomial(5); f
1.0*x^4 + 1.0*x^3 + 1.0*x^2 + 1.0*x + 1.0
sage: f.roots(multiplicities=False)   # slightly random
[0.309016994375 + 0.951056516295*I, 0.309016994375 - 0.951056516295*I,
-0.809016994375 + 0.587785252292*I, -0.809016994375 - 0.587785252292*I]
sage: [z^5 for z in f.roots(multiplicities=False)]     # slightly random
[1.0 - 2.44929359829e-16*I, 1.0 + 2.44929359829e-16*I, 1.0 -
4.89858719659e-16*I, 1.0 + 4.89858719659e-16*I]
sage: f = CDF['x']([1,2,3,4]); f
4.0*x^3 + 3.0*x^2 + 2.0*x + 1.0
sage: r = f.roots(multiplicities=False)
sage: [f(a) for a in r]    # slightly random
[2.55351295664e-15, -4.4408920985e-16 - 2.08166817117e-16*I,
-4.4408920985e-16 + 2.08166817117e-16*I]

Another example over RDF:

sage: x = RDF['x'].0
sage: ((x^3 -1)).roots()
[(1.0, 1)]
sage: ((x^3 -1)).roots(multiplicities=False)
[1.0]

Another examples involving the complex double field:

sage: x = CDF['x'].0
sage: i = CDF.0
sage: f = x^3 + 2*i; f
1.0*x^3 + 2.0*I
sage: f.roots()  # random low-order bits
[(-1.09112363597 - 0.629960524947*I, 1), (6.66133814775e-16 +
1.25992104989*I, 1), (1.09112363597 - 0.629960524947*I, 1)]
sage: f.roots(multiplicities=False)   # random low-order bits
[-1.09112363597 - 0.629960524947*I, 6.66133814775e-16 + 1.25992104989*I,
1.09112363597 - 0.629960524947*I]
sage: [f(z) for z in f.roots(multiplicities=False)]  # random low-order bits
[-3.10862446895e-15 - 4.4408920985e-16*I, -3.17226455498e-15 +
3.99680288865e-15*I, -5.55111512313e-16 - 8.881784197e-16*I]
sage: f = i*x^3 + 2; f
1.0*I*x^3 + 2.0
sage: f.roots()     # random low-order bits
[(-1.09112363597 + 0.629960524947*I, 1), (6.66133814775e-16 -
1.25992104989*I, 1), (1.09112363597 + 0.629960524947*I, 1)]
sage: f(f.roots()[0][0])         # random low-order bits
-4.4408920985e-16 - 3.10862446895e-15*I

Examples using real root isolation:

sage: x = polygen(ZZ)
sage: f = x^2 - x - 1
sage: f.roots()
[]
sage: f.roots(ring=RIF)
[([-0.618033988749894848204586834365642 ..
-0.618033988749894848204586834365629], 1),
([1.61803398874989484820458683436561 ..
1.61803398874989484820458683436565], 1)]
sage: f.roots(ring=RIF, multiplicities=False)
[[-0.618033988749894848204586834365642 ..
-0.618033988749894848204586834365629], [1.61803398874989484820458683436561
.. 1.61803398874989484820458683436565]]
sage: f.roots(ring=RealIntervalField(150))
[([-0.61803398874989484820458683436563811772030917980576286213544862277 ..
-0.61803398874989484820458683436563811772030917980576286213544862260], 1),
([1.6180339887498948482045868343656381177203091798057628621354486226 ..
1.6180339887498948482045868343656381177203091798057628621354486230], 1)]
sage: f.roots(ring=AA)
[([-0.61803398874989491 .. -0.61803398874989479], 1), ([1.6180339887498946
.. 1.6180339887498950], 1)]
sage: f = f^2 * (x - 1)
sage: f.roots(ring=RIF)
[([-0.618033988749894848204586834365642 ..
-0.618033988749894848204586834365629], 2),
([0.999999999999999999999999999999987 ..
1.00000000000000000000000000000003], 1),
([1.61803398874989484820458683436561 ..
1.61803398874989484820458683436565], 2)]
sage: f.roots(ring=RIF, multiplicities=False)
[[-0.618033988749894848204586834365642 ..
-0.618033988749894848204586834365629], [0.999999999999999999999999999999987
.. 1.00000000000000000000000000000003], [1.61803398874989484820458683436561
.. 1.61803398874989484820458683436565]]

Examples using complex root isolation:

sage: x = polygen(ZZ)
sage: p = x^5 - x - 1
sage: p.roots()
[]
sage: p.roots(ring=CIF)
[([1.1673039782614185 .. 1.16730397826141...], 1), ([0.18123244446987518 ..
0.18123244446987558] + [1.0839541013177103 .. 1.0839541013177110]*I, 1),
([0.181232444469875... .. 0.1812324444698755...] - [1.083954101317710... ..
1.0839541013177110]*I, 1), ([-0.76488443360058489 .. -0.76488443360058455]
+ [0.35247154603172609 .. 0.3524715460317264...]*I, 1),
([-0.76488443360058489 .. -0.76488443360058455] - [0.35247154603172609 ..
0.35247154603172643]*I, 1)]
sage: p.roots(ring=ComplexIntervalField(200))
[([1.1673039782614186842560458998548421807205603715254890391400816 ..
1.1673039782614186842560458998548421807205603715254890391400829], 1),
([0.18123244446987538390180023778112063996871646618462304743773153 ..
0.18123244446987538390180023778112063996871646618462304743773341] +
[1.0839541013177106684303444929807665742736402431551156543011306 ..
1.0839541013177106684303444929807665742736402431551156543011344]*I, 1),
([0.18123244446987538390180023778112063996871646618462304743773153 ..
0.18123244446987538390180023778112063996871646618462304743773341] -
[1.0839541013177106684303444929807665742736402431551156543011306 ..
1.0839541013177106684303444929807665742736402431551156543011344]*I, 1),
([-0.76488443360058472602982318770854173032899665194736756700777... ..
-0.76488443360058472602982318770854173032899665194736756700777...] +
[0.35247154603172624931794709140258105439420648082424733283769... ..
0.35247154603172624931794709140258105439420648082424733283769...]*I, 1),
([-0.76488443360058472602982318770854173032899665194736756700777454 ..
-0.764884433600584726029823187708541730328996651947367567007772...] -
[0.35247154603172624931794709140258105439420648082424733283769... ..
0.352471546031726249317947091402581054394206480824247332837693...]*I, 1)]
sage: rts = p.roots(ring=QQbar); rts
[([1.1673039782614185 .. 1.1673039782614188], 1), ([0.18123244446987538 ..
0.18123244446987541] + [1.0839541013177105 .. 1.0839541013177108]*I, 1),
([0.18123244446987538 .. 0.18123244446987541] - [1.0839541013177105 ..
1.0839541013177108]*I, 1), ([-0.76488443360058478 .. -0.76488443360058466]
+ [0.35247154603172620 .. 0.35247154603172626]*I, 1),
([-0.76488443360058478 .. -0.76488443360058466] - [0.35247154603172620 ..
0.35247154603172626]*I, 1)]
sage: p.roots(ring=AA)
[([1.1673039782614185 .. 1.1673039782614188], 1)]
sage: p = (x - rts[1][0])^2 * (3*x^2 + x + 1)
sage: p.roots(ring=QQbar)
[([-0.16666666666666669 .. -0.16666666666666665] + [0.55277079839256659 ..
0.55277079839256671]*I, 1), ([-0.16666666666666669 .. -0.16666666666666665]
- [0.55277079839256659 .. 0.55277079839256671]*I, 1), ([0.18123244446987538
.. 0.18123244446987541] + [1.0839541013177105 .. 1.0839541013177108]*I, 2)]
sage: p.roots(ring=CIF)
[([-0.16666666666666672 .. -0.16666666666666662] + [0.55277079839256648 ..
0.55277079839256671]*I, 1), ([-0.16666666666666672 .. -0.16666666666666662]
- [0.55277079839256648 .. 0.55277079839256671]*I, 1), ([0.18123244446987538
.. 0.18123244446987541] + [1.0839541013177105 .. 1.0839541013177108]*I, 2)]

Note that coefficients in a number field with defining polynomial $ x^2 + 1$ are considered to be Gaussian rationals (with the generator mapping to +I), if you ask for complex roots.

sage: K.<im> = NumberField(x^2 + 1)
sage: y = polygen(K)
sage: p = y^4 - 2 - im
sage: p.roots(ring=CC)
[(-1.2146389322441... - 0.14142505258239...*I, 1), (-0.14142505258239... +
1.2146389322441...*I, 1), (0.14142505258239... - 1.2146389322441...*I, 1),
(1.2146389322441... + 0.14142505258239...*I, 1)]
sage: p = p^2 * (y^2 - 2)
sage: p.roots(ring=CIF)
[([-1.41421356237309... .. -1.41421356237309...], 1), ([1.41421356237309...
.. 1.41421356237309...], 1), ([-1.214638932244182... ..
-1.21463893224418...] - [0.1414250525823937... .. 0.1414250525823939...]*I,
2), ([-0.141425052582393... .. -0.1414250525823937...] +
[1.21463893224418... .. 1.214638932244182...]*I, 2), ([0.141425052582393...
.. 0.141425052582393...] - [1.21463893224418... .. 1.21463893224418...]*I,
2), ([1.21463893224418... .. 1.21463893224418...] + [0.141425052582393...
.. 0.141425052582393...]*I, 2)]

There are many combinations of floating-point input and output types that work. (Note that some of them are quite pointless... there's no reason to use high-precision input and output, and still use numpy to find the roots.)

sage: rflds = (RR, RDF, RealField(100))
sage: cflds = (CC, CDF, ComplexField(100))
sage: def cross(a, b):
...       return list(cartesian_product_iterator([a, b]))
sage: flds = cross(rflds, rflds) + cross(rflds, cflds) + cross(cflds, cflds)
sage: for (fld_in, fld_out) in flds:
...       x = polygen(fld_in)
...       f = x^3 - fld_in(2)
...       x2 = polygen(fld_out)
...       f2 = x2^3 - fld_out(2)
...       for algo in (None, 'pari', 'numpy'):
...           rts = f.roots(ring=fld_out, multiplicities=False)
...           if fld_in == fld_out and algo is None:
...               print fld_in, rts
...           for rt in rts:
...               assert(abs(f2(rt)) <= 1e-10)
...               assert(rt.parent() == fld_out)
Real Field with 53 bits of precision [1.25992104989487]
Real Double Field [1.25992104989]
Real Field with 100 bits of precision [1.2599210498948731647672106073]
Complex Field with 53 bits of precision [1.25992104989487,
-0.62996052494743... + 1.09112363597172*I, -0.62996052494743... -
1.09112363597172*I]
Complex Double Field [1.25992104989, -0.62996052494... + 1.09112363597*I,
-0.62996052494... - 1.09112363597*I]
Complex Field with 100 bits of precision [1.2599210498948731647672106073,
-0.62996052494743658238360530364 + 1.0911236359717214035600726142*I,
-0.62996052494743658238360530364 - 1.0911236359717214035600726142*I]

Note that we can find the roots of a polynomial with algebraic coefficients:

            sage: rt2 = sqrt(AA(2))
            sage: rt3 = sqrt(AA(3))
            sage: x = polygen(AA)
            sage: f = (x - rt2) * (x - rt3); f
            x^2 + [-3.1462643699419726 .. -3.1462643699419721]*x +
[2.4494897427831778 .. 2.4494897427831784]
            sage: rts = f.roots(); rts
            [([1.4142135623730949 .. 1.4142135623730952], 1),
([1.7320508075688771 .. 1.7320508075688775], 1)]
sage: rts[0][0] == rt2
            True
            sage: f.roots(ring=RealIntervalField(150))
            [([1.4142135623730950488016887242096980785696718753769480731766
797377 .. 1.414213562373095048801688724209698078569671875376948073176679738
1], 1), ([1.732050807568877293527446341505872366942805253810380628055806979
3 .. 1.7320508075688772935274463415058723669428052538103806280558069797],
1)]

Algorithms used:

For brevity, we will use RR to mean any RealField of any precision; similarly for RIF, CC, and CIF. Since Sage has no specific implementation of Gaussian rationals (or of number fields with embedding, at all), when we refer to Gaussian rationals below we will accept any number field with defining polynomial $ x^2 + 1$ , mapping the field generator to +I.

We call the base ring of the polynomial K, and the ring given by the ring= argument L. (If ring= is not specified, then L is the same as K.)

If K and L are floating-point (RDF, CDF, RR, or CC), then a floating-point root-finder is used. If L has precision 53 bits or less (RDF and CDF both have precision exactly 53 bits, as do the default RR=RealField() and CC=ComplexField()) then we default to using numpy's roots(); otherwise, we use Pari's polroots(). This choice can be overridden with algorithm='pari' or algorithm='numpy'.

If L is AA or RIF, and K is ZZ, QQ, or AA, then the root isolation algorithm sage.rings.polynomial.real_roots.real_roots() is used. (You can call real_roots() directly to get more control than this method gives.)

If L is QQbar or CIF, and K is ZZ, QQ, AA, QQbar, or the Gaussian rationals, then the root isolation algorithm sage.rings.polynomial.complex_roots.complex_roots() is used. (You can call complex_roots() directly to get more control than this method gives.)

If L is AA and K is QQbar or the Gaussian rationals, then complex_roots() is used (as above) to find roots in QQbar, then these roots are filtered to select only the real roots.

If L is floating-point and K is not, then we attempt to change the polynomial ring to L (using .change_ring()) (or, if L is complex and K is not, to the corresponding real field). Then we use either Pari or numpy as specified above.

For all other cases where K is different than L, we just use .change_ring(L) and proceed as below.

The next method, which is used if K is an integral domain, is to attempt to factor the polynomial. If this succeeds, then for every degree-one factor a*x+b, we add -b/a as a root (as long as this quotient is actually in the desired ring).

If factoring over K is not implemented (or K is not an integral domain), and K is finite, then we find the roots by enumerating all elements of K and checking whether the polynomial evaluates to zero at that value.

NOTE: We mentioned above that polynomials with multiple roots are always ill-conditioned; if your input is given to n bits of precision, you should not expect more than n/k good bits for a k-fold root. (You can get solutions that make the polynomial evaluate to a number very close to zero; basically the problem is that with a multiple root, there are many such numbers, and it's difficult to choose between them.)

To see why this is true, consider the naive floating-point error analysis model where you just pretend that all floating-point numbers are somewhat imprecise - a little "fuzzy", if you will. Then the graph of a floating-point polynomial will be a fuzzy line. Consider the graph of $ (x-1)^3$ ; this will be a fuzzy line with a horizontal tangent at $ x=1$ , $ y=0$ . If the fuzziness extends up and down by about j, then it will extend left and right by about cube_root(j).

shift( )

Returns this polynomial multiplied by the power $ x^n$ . If $ n$ is negative, terms below $ x^n$ will be discarded. Does not change this polynomial (since polynomials are immutable).

sage: R.<x> = PolynomialRing(PolynomialRing(QQ,'w'),'x') 
sage: p = x^2 + 2*x + 4
sage: p.shift(0)
 x^2 + 2*x + 4
sage: p.shift(-1)
 x + 2
sage: p.shift(-5)
 0
sage: p.shift(2)
 x^4 + 2*x^3 + 4*x^2

One can also use the infix shift operator:

sage: f = x^3 + x
sage: f >> 2
x
sage: f << 2
x^5 + x^3

Author Log:

square( )

Returns the square of this polynomial.

TODO: - This is just a placeholder; for now it just uses ordinary multiplication. But generally speaking, squaring is faster than ordinary multiplication, and it's frequently used, so subclasses may choose to provide a specialised squaring routine.

- Perhaps this even belongs at a lower level? ring_element or something?

Author: David Harvey (2006-09-09)

sage: R.<x> = QQ[]
sage: f = x^3 + 1
sage: f.square()
x^6 + 2*x^3 + 1
sage: f*f
x^6 + 2*x^3 + 1

squarefree_decomposition( )

Return the square-free decomposition of self. This is a partial factorization of self into square-free, relatively prime polynomials.

This is the straightforward algorithm, using only polynomial GCD and polynomial division. Faster algorithms exist. The algorithm comes from the Wikipedia article, "Square-free polynomial".

sage: x = polygen(QQ)
sage: p = 37 * (x-1)^3 * (x-2)^3 * (x-1/3)^7 * (x-3/7)
sage: p.squarefree_decomposition()
(37*x - 111/7) * (x^2 - 3*x + 2)^3 * (x - 1/3)^7
sage: p = 37 * (x-2/3)^2
sage: p.squarefree_decomposition()
(37) * (x - 2/3)^2

subs( )

Identical to self(*x).

See the docstring for self.__call__.

sage: R.<x> = QQ[]
sage: f = x^3 + x - 3
sage: f.subs(x=5)
127
sage: f.subs(5)
127
sage: f.subs({x:2})
7
sage: f.subs({})
x^3 + x - 3            
sage: f.subs({'x':2})
Traceback (most recent call last):
...
TypeError: keys do not match self's parent

substitute( )

Identical to self(*x).

See the docstring for self.__call__.

sage: R.<x> = QQ[]
sage: f = x^3 + x - 3
sage: f.subs(x=5)
127
sage: f.subs(5)
127
sage: f.subs({x:2})
7
sage: f.subs({})
x^3 + x - 3            
sage: f.subs({'x':2})
Traceback (most recent call last):
...
TypeError: keys do not match self's parent

truncate( )

Returns the polynomial of degree $ < n$ which is equivalent to self modulo $ x^n$ .

sage: R.<x> = ZZ[]; S.<y> = R[]
sage: f = y^3 + x*y -3*x; f
y^3 + x*y - 3*x
sage: f.truncate(2)
x*y - 3*x
sage: f.truncate(1)
-3*x
sage: f.truncate(0)
0

valuation( )

If $ f = a_r x^r + a_{r+1}x^{r+1} + \cdots$ , with $ a_r$ nonzero, then the valuation of $ f$ is $ r$ . The valuation of the zero polynomial is $ \infty$ .

If a prime (or non-prime) $ p$ is given, then the valuation is the largest power of $ p$ which divides self.

The valuation at $ \infty$ is -self.degree().

sage: P,x=PolynomialRing(ZZ,'x').objgen()
sage: (x^2+x).valuation()
1
sage: (x^2+x).valuation(x+1)
1
sage: (x^2+1).valuation()
0
sage: (x^3+1).valuation(infinity)
-3
sage: P(0).valuation()
+Infinity

variable_name( )

Return name of variable used in this polynomial as a string.

Output: string

sage: R.<t> = QQ[]
sage: f = t^3 + 3/2*t + 5
sage: f.variable_name()
't'

variables( )

Returns the list of variables occuring in this polynomial.

sage: R.<x> = QQ[]
sage: x.variables()
(x,)

A constant polynomial has no variables.

sage: R(2).variables()
()

Special Functions: __call__,$ \,$ __copy__,$ \,$ __delitem__,$ \,$ __div__,$ \,$ __eq__,$ \,$ __float__,$ \,$ __floordiv__,$ \,$ __ge__,$ \,$ __getitem__,$ \,$ __gt__,$ \,$ __init__,$ \,$ __int__,$ \,$ __invert__,$ \,$ __iter__,$ \,$ __le__,$ \,$ __long__,$ \,$ __lshift__,$ \,$ __lt__,$ \,$ __mod__,$ \,$ __ne__,$ \,$ __pow__,$ \,$ __rdiv__,$ \,$ __rfloordiv__,$ \,$ __rlshift__,$ \,$ __rmod__,$ \,$ __rpow__,$ \,$ __rrshift__,$ \,$ __rshift__,$ \,$ __setitem__,$ \,$ _compile,$ \,$ _derivative,$ \,$ _dict_to_list,$ \,$ _factor_pari_helper,$ \,$ _fast_float_,$ \,$ _gap_,$ \,$ _gap_init_,$ \,$ _im_gens_,$ \,$ _integer_,$ \,$ _is_atomic,$ \,$ _latex_,$ \,$ _lcm,$ \,$ _magma_,$ \,$ _magma_init_,$ \,$ _mpoly_dict_recursive,$ \,$ _mul_fateman,$ \,$ _mul_generic,$ \,$ _mul_karatsuba,$ \,$ _pari_,$ \,$ _pari_init_,$ \,$ _pari_with_name,$ \,$ _pow,$ \,$ _rational_,$ \,$ _repr,$ \,$ _repr_,$ \,$ _square_generic,$ \,$ _xgcd

__call__( )

Evaluate polynomial at x=a.

Input:

a
- ring element a; need not be in the coefficient ring of the polynomial.
- or - a dictionary for kwds:value pairs. If the variable name of the polynomial is a kwds it is substituted in; otherwise this polynomial is returned unchanged.

Output: the value of f at a.

sage: R.<x> = QQ[]
sage: f = x/2 - 5
sage: f(3)
-7/2
sage: R.<x> = ZZ[]
sage: f = (x-1)^5
sage: f(2/3)
-1/243

We evaluate a polynomial over a quaternion algebra:

sage: A.<i,j,k> = QuaternionAlgebra(QQ, -1,-1)
sage: R.<w> = PolynomialRing(A,sparse=True)
sage: f = i*j*w^5 - 13*i*w^2 + (i+j)*w + i
sage: f(i+j+1)
24 + 26*i - 10*j - 25*k
sage: w = i+j+1; i*j*w^5 - 13*i*w^2 + (i+j)*w + i
24 + 26*i - 10*j - 25*k

The parent ring of the answer always "starts" with the parent of the object at which we are evaluating. Thus, e.g., if we input a matrix, we are guaranteed to get a matrix out, though the base ring of that matrix may change depending on the base of the polynomial ring.

sage: R.<x> = QQ[]
sage: f = R(2/3)
sage: a = matrix(ZZ,2)
sage: b = f(a); b
[2/3   0]
[  0 2/3]
sage: b.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: f = R(1)
sage: b = f(a); b
[1 0]
[0 1]
sage: b.parent()
Full MatrixSpace of 2 by 2 dense matrices over Rational Field

sage: R.<w> = GF(17)[]
sage: f = w^3 + 3*w +2  
sage: f(5)
6
sage: f(w=5)
6
sage: f(x=10)   # x isn't mention
w^3 + 3*w + 2

Nested polynomial ring elements can be called like multi-variate polynomials.

sage: R.<x> = QQ[]; S.<y> = R[]
sage: f = x+y*x+y^2
sage: f.parent()
Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over
Rational Field
sage: f(2)
3*x + 4
sage: f(2,4)
16
sage: f(y=2,x=4)
16
sage: f(2,x=4)
16
sage: f(2,x=4,z=5)
16
sage: f(2,4, z=10)
16            
sage: f(y=x)
2*x^2 + x
sage: f(x=y)
2*y^2 + y

The following results in an element of the symbolic ring.

sage: f(x=sqrt(2))
y*(y + sqrt(2)) + sqrt(2)

sage: R.<t> = PowerSeriesRing(QQ, 't'); S.<x> = R[]
sage: f = 1 + x*t^2 + 3*x*t^4
sage: f(2)
1 + 2*t^2 + 6*t^4
sage: f(2, 1/2)
15/8

Author Log:

__copy__( )

Return a "copy" of self. This is just self, since in SAGE polynomials are immutable this just returns self again.

We create the polynomial $ f=x+3$ , then note that the copy is just the same polynomial again, which is fine since polynomials are immutable.

sage: x = ZZ['x'].0 
sage: f = x + 3
sage: g = copy(f)
sage: g is f
True

__div__( )

sage: x = QQ['x'].0
sage: f = (x^3 + 5)/3; f
1/3*x^3 + 5/3
sage: f.parent()
Univariate Polynomial Ring in x over Rational Field

If we do the same over $ \mathbf{Z}$ the result is in the polynomial ring over $ \mathbf{Q}$ .

sage: x  = ZZ['x'].0
sage: f = (x^3 + 5)/3; f
1/3*x^3 + 5/3
sage: f.parent()
Univariate Polynomial Ring in x over Rational Field

Divides can make elements of the fraction field:

sage: R.<x> = QQ['x']
sage: f = x^3 + 5
sage: g = R(3)
sage: h = f/g; h
1/3*x^3 + 5/3
sage: h.parent()
Fraction Field of Univariate Polynomial Ring in x over Rational Field

This is another example over a non-prime finite field (submited by a student of Jon Hanke). It illustrates cancellation between the numerator and denominator over a non-prime finite field.

sage: R.<x> = PolynomialRing(GF(5^2, 'a'), 'x')
sage: f = x^3 + 4*x
sage: f / (x - 1)
x^2 + x

Be careful about coercions (this used to be broken):

sage: R.<x> = ZZ['x']
sage: f = x / Mod(2,5); f
3*x
sage: f.parent()
Univariate Polynomial Ring in x over Ring of integers modulo 5

__float__( )

__floordiv__( )

Quotient of division of self by other. This is denoted //.

If self = quotient * right + remainder, this function returns quotient.

sage: R.<x> = ZZ[]
sage: f = x^3 + x + 1
sage: g = f*(x^2-2) + x
sage: g.__floordiv__(f)
x^2 - 2
sage: g//f
x^2 - 2

__getitem__( )

__int__( )

__invert__( )

sage: R.<x> = QQ[]
sage: f = x - 90283
sage: f.__invert__()
1/(x - 90283)
sage: ~f
1/(x - 90283)

__iter__( )

__long__( )

sage: R.<x> = ZZ[]
sage: f = x - 902384
sage: long(f)
Traceback (most recent call last):
...
TypeError: cannot coerce nonconstant polynomial to long
sage: long(R(939392920202))
939392920202L

__lshift__( )

__mod__( )

Remainder of division of self by other.

sage: R.<x> = ZZ[]
sage: x % (x+1)
-1
sage: (x^3 + x - 1) % (x^2 - 1)
2*x - 1

__rshift__( )

_compile( )

_derivative( )

Return the formal derivative of this polynomial with respect to the variable var.

If var is the generator of this polynomial ring (or the default value None), this is the usual formal derivative.

Otherwise, _derivative(var) is called recursively for each of the coefficients of this polynomial.

SEE ALSO: self.derivative()

sage: R.<x> = ZZ[]
sage: R(0)._derivative()
0
sage: parent(R(0)._derivative())
Univariate Polynomial Ring in x over Integer Ring

sage: f = 7*x^5 + x^2 - 2*x - 3
sage: f._derivative()
35*x^4 + 2*x - 2
sage: f._derivative(None)
35*x^4 + 2*x - 2
sage: f._derivative(x)
35*x^4 + 2*x - 2

In the following example, it doesn't recognise 2*x as the generator, so it tries to differentiate each of the coefficients with respect to 2*x, which doesn't work because the integer coefficients don't have a _derivative() method:

sage: f._derivative(2*x)
Traceback (most recent call last):
...
AttributeError: 'sage.rings.integer.Integer' object has no attribute
'_derivative'

Examples illustrating recursive behaviour:

sage: R.<x> = ZZ[]
sage: S.<y> = PolynomialRing(R)
sage: f = x^3 + y^3
sage: f._derivative()
3*y^2
sage: f._derivative(y)
3*y^2
sage: f._derivative(x)
3*x^2

sage: R = ZZ['x']
sage: S = R.fraction_field(); x = S.gen()
sage: R(1).derivative(R(x))
0

_dict_to_list( )

_factor_pari_helper( )

_fast_float_( )

Returns a quickly-evaluating function on floats.

sage: R.<t> = QQ[]
sage: f = t^3-t
sage: ff = f._fast_float_()
sage: ff(10)
990.0

Horner's method is used:

sage: f = (t+10)^3; f
t^3 + 30*t^2 + 300*t + 1000
sage: list(f._fast_float_())
['load 0', 'push 30.0', 'add', 'load 0', 'mul', 'push 300.0', 'add', 'load
0', 'mul', 'push 1000.0', 'add']

_gap_( )

sage: R.<y> = ZZ[]
sage: f = y^3 - 17*y + 5
sage: g = gap(f); g
y^3-17*y+5
sage: f._gap_init_()
'y^3 - 17*y + 5'
sage: R.<z> = ZZ[]
sage: gap(R)
PolynomialRing( Integers, ["z"] )
sage: g
y^3-17*y+5
sage: gap(z^2 + z)
z^2+z

We coerce a polynomial with coefficients in a finite field:

sage: R.<y> = GF(7)[]
sage: f = y^3 - 17*y + 5
sage: g = gap(f); g
y^3+Z(7)^4*y+Z(7)^5
sage: g.Factors()
[ y+Z(7)^0, y+Z(7)^0, y+Z(7)^5 ]
sage: f.factor()
(y + 5) * (y + 1)^2

_gap_init_( )

_im_gens_( )

sage: R.<x> = ZZ[]
sage: H = Hom(R, QQ); H
Set of Homomorphisms from Univariate Polynomial Ring in x over Integer Ring
to Rational Field
sage: f = H([5]); f
Ring morphism:
  From: Univariate Polynomial Ring in x over Integer Ring
  To:   Rational Field
  Defn: x |--> 5
sage: f(x)
5
sage: f(x^2 + 3)
28

_integer_( )

sage: k = GF(47)
sage: R.<x> = PolynomialRing(k)
sage: ZZ(R(45))
45
sage: ZZ(3*x + 45)
Traceback (most recent call last):
...
TypeError: cannot coerce nonconstant polynomial

_is_atomic( )

_latex_( )

Return the latex representation of this polynomial.

A fairly simple example over $ \mathbf{Q}$ .

sage: x = polygen(QQ)
sage: latex(x^3+2/3*x^2 - 5/3)
x^{3} + \frac{2}{3} x^{2} - \frac{5}{3}

A $ p$ -adic example where the coefficients are 0 to some precision.

sage: K = Qp(3,20)
sage: R.<x> = K[]
sage: f = K(0,-2)*x + K(0,-1)
sage: f
(O(3^-2))*x + (O(3^-1))
sage: latex(f)
\left(O(3^{-2})\right) x + O(3^{-1})

The following illustrates the fix of trac #2586:

sage: latex(ZZ['alpha']['b']([0, ZZ['alpha'].0]))
\alpha b

_lcm( )

Let f and g be two polynomials. Then this function returns the monic least common multiple of f and g.

_magma_( )

Return the Magma version of this polynomial.

sage: R.<y> = ZZ[]
sage: f = y^3 - 17*y + 5
sage: g = magma(f); g              # optional -- requires Magma
y^3 - 17*y + 5

Note that in Magma there is only one polynomial ring over each base, so if we make the polynomial ring over ZZ with variable $ z$ , then this changes the variable name of the polynomial we already defined:

sage: R.<z> = ZZ[]
sage: magma(R)                     # optional -- requires Magma
Univariate Polynomial Ring in z over Integer Ring
sage: g                            # optional -- requires Magma
z^3 - 17*z + 5

In SAGE the variable name does not change:

sage: f
y^3 - 17*y + 5

_magma_init_( )

Return a string that evaluates in Magma to this polynomial.

sage: R.<y> = ZZ[]
sage: f = y^3 - 17*y + 5
sage: f._magma_init_()
'Polynomial(IntegerRing(), [5,-17,0,1])'

_mpoly_dict_recursive( )

Return a dict of coefficent entries suitable for construction of a MPolynomial_polydict with the given variables.

sage: R.<x> = ZZ[]
sage: R(0)._mpoly_dict_recursive()
{}
sage: f = 7*x^5 + x^2 - 2*x - 3
sage: f._mpoly_dict_recursive()
{(0,): -3, (1,): -2, (5,): 7, (2,): 1}

_mul_fateman( )

Returns the product of two polynomials using Kronecker's trick to do the multiplication. This could be used used over a generic base ring.

NOTES:

Input:

self
- Polynomial
right
- Polynomial (over same base ring as self)

Output: polynomial The product self*right.

ALGORITHM: Based on a paper by R. Fateman

http://www.cs.berkeley.edu/ fateman/papers/polysbyGMP.pdf

The idea is to encode dense univariate polynomials as big integers, instead of sequences of coefficients. The paper argues that because integer multiplication is so cheap, that encoding 2 polynomials to big numbers and then decoding the result might be faster than popular multiplication algorithms. This seems true when the degree is larger than 200.

sage: S.<y> = PolynomialRing(RR)
sage: f = y^10 - 1.393493*y + 0.3
sage: f._mul_karatsuba(f)
1.00000000000000*y^20 - 2.78698600000000*y^11 + 0.600000000000000*y^10 +
1.11022302462516e-16*y^8 - 1.11022302462516e-16*y^6 -
1.11022302462516e-16*y^3 + 1.94182274104900*y^2 - 0.836095800000000*y +
0.0900000000000000
sage: f._mul_fateman(f)
1.00000000000000*y^20 - 2.78698600000000*y^11 + 0.600000000000000*y^10 +
1.94182274104900*y^2 - 0.836095800000000*y + 0.0900000000000000

Advantages:

Drawbacks:

Author: Didier Deshommes (2006-05-25)

_mul_generic( )

_mul_karatsuba( )

Returns the product of two polynomials using the Karatsuba divide and conquer multiplication algorithm. This is only used over a generic base ring. (Special libraries like NTL are used, e.g., for the integers and rationals, which are much faster.)

Input: self: Polynomial right: Polynomial (over same base ring as self)

Output: polynomial The product self*right.

ALGORITHM: The basic idea is to use that

$\displaystyle (aX + b) (cX + d) = acX^2 + ((a+b)(c+d)-ac-bd)X + bd
$

where ac=a*c and bd=b*d, which requires three multiplications instead of the naive four. (In my examples, strangely just doing the above with four multiplications does tend to speed things up noticeably.) Given f and g of arbitrary degree bigger than one, let e be min(deg(f),deg(g))/2. Write

$\displaystyle f = a X^e + b$    and $\displaystyle g = c X^e + d
$

and use the identity

$\displaystyle (aX^e + b) (cX^e + d) = ac X^{2e} +((a+b)(c+d) - ac - bd)X^e + bd
$

to recursively compute $ fg$ .

TIMINGS: On a Pentium M 1.8Ghz laptop: f=R.random(1000,bound=100) g=R.random(1000,bound=100) time h=f._mul_karatsuba(g) Time: 0.42 seconds The naive multiplication algorithm takes 14.58 seconds. In contrast, MAGMA does this sort of product almost instantly, and can easily deal with degree 5000. Basically MAGMA is 100 times faster at polynomial multiplication.

Over Z using NTL, multiplying two polynomials constructed using R.random(10000,bound=100) takes 0.10 seconds. Using MAGMA V2.11-10 the same takes 0.14 seconds. So in this case NTL is somewhat faster than MAGMA.

Over Q using PARI, multiplying two polynomials constructed using R.random(10000,bound=100) takes 1.23 seconds. Not good! TODO: use NTL polynomials over Z with a denominator instead of PARI.

NOTES: * Karatsuba multiplication of polynomials is also implemented in PARI in src/basemath/polarit3.c * The MAGMA documentation appears to give no information about how polynomial multiplication is implemented.

_pari_( )

Return polynomial as a PARI object.

SAGE does not handle PARI's variable ordering requirements gracefully at this time. In practice, this means that the variable x needs to be the topmost variable, as in the example.

sage: f = QQ['x']([0,1,2/3,3])
sage: pari(f)
3*x^3 + 2/3*x^2 + x

sage: S.<a> = QQ['a']
sage: R.<x> = S['x']
sage: f = R([0, a]) + R([0, 0, 2/3])
sage: pari(f)
2/3*x^2 + a*x

TESTS: Unfortunately, variable names matter:

sage: R.<x, y> = QQ[]
sage: S.<a> = R[]
sage: f = x^2 + a; g = y^3 + a
sage: pari(f)
Traceback (most recent call last):
...
PariError: (8)

Stacked polynomial rings, first with a univariate ring on the bottom:

sage: S.<a> = QQ['a']
sage: R.<x> = S['x']
sage: pari(x^2 + 2*x)
x^2 + 2*x
sage: pari(a*x + 2*x^3)
2*x^3 + a*x

Stacked polynomial rings, second with a multivariate ring on the bottom:

sage: S.<a, b> = ZZ['a', 'b']
sage: R.<x> = S['x']
sage: pari(x^2 + 2*x)
x^2 + 2*x
sage: pari(a*x + 2*b*x^3)
2*b*x^3 + a*x

Stacked polynomial rings with exotic base rings:

sage: S.<a, b> = GF(7)['a', 'b']
sage: R.<x> = S['x']
sage: pari(x^2 + 9*x)
x^2 + 2*x
sage: pari(a*x + 9*b*x^3)
2*b*x^3 + a*x

sage: S.<a> = Integers(8)['a']
sage: R.<x> = S['x']
sage: pari(x^2 + 2*x)
Mod(1, 8)*x^2 + Mod(2, 8)*x
sage: pari(a*x + 10*x^3)
Mod(2, 8)*x^3 + (Mod(1, 8)*a)*x

_pari_init_( )

_pari_with_name( )

Return polynomial as a PARI object with topmost variable name.

For internal use only.

_pow( )

_rational_( )

sage: R.<x> = PolynomialRing(QQ)
sage: QQ(R(45/4))
45/4
sage: QQ(3*x + 45)
Traceback (most recent call last):
...
TypeError: cannot coerce nonconstant polynomial

_repr( )

Return the string representation of this polynomial.

Input:

name
- None or a string; used for printing the variable.

sage: R.<x> = QQ[]
sage: f = x^3 + x + 1
sage: f._repr()
'x^3 + x + 1'
sage: f._repr('theta')
'theta^3 + theta + 1'
sage: f._repr('sage math')
'sage math^3 + sage math + 1'

_repr_( )

Return string representatin of this polynomial.

sage: x = polygen(QQ)
sage: f = x^3+2/3*x^2 - 5/3
sage: f._repr_()
'x^3 + 2/3*x^2 - 5/3'
sage: f.rename('vaughn')
sage: f
vaughn

_square_generic( )

_xgcd( )

Extended gcd of self and polynomial other.

Returns g, u, and v such that g = u*self + v*other.

sage: P.<x> = QQ[]
sage: F = (x^2 + 2)*x^3; G = (x^2+2)*(x-3)
sage: g, u, v = F.xgcd(G)
sage: g, u, v
(27*x^2 + 54, 1, -x^2 - 3*x - 9)
sage: u*F + v*G
27*x^2 + 54

sage: g, u, v = x.xgcd(P(0)); g, u, v
(x, 1, 0)
sage: g == u*x + v*P(0)
True
sage: g, u, v = P(0).xgcd(x); g, u, v
(x, 0, 1)            
sage: g == u*P(0) + v*x
True

Class: Polynomial_generic_dense

class Polynomial_generic_dense
A generic dense polynomial.

sage: R.<x> = PolynomialRing(PolynomialRing(QQ,'y'))
sage: f = x^3 - x + 17
sage: type(f)
<type 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
sage: loads(f.dumps()) == f
True

Functions: degree,$ \,$ list,$ \,$ shift,$ \,$ truncate

degree( )

sage: R.<x> = RDF[]
sage: f = (1+2*x^7)^5
sage: f.degree()
35

list( )

Return a new copy of the list of the underlying elements of self.

sage: R.<x> = GF(17)[]
sage: f = (1+2*x)^3 + 3*x; f
8*x^3 + 12*x^2 + 9*x + 1
sage: f.list()
[1, 9, 12, 8]

shift( )

Returns this polynomial multiplied by the power $ x^n$ . If $ n$ is negative, terms below $ x^n$ will be discarded. Does not change this polynomial.

sage: R.<x> = PolynomialRing(PolynomialRing(QQ,'y'), 'x')
sage: p = x^2 + 2*x + 4
sage: type(p)
<type 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'>
sage: p.shift(0)
 x^2 + 2*x + 4
sage: p.shift(-1)
 x + 2
sage: p.shift(2)
 x^4 + 2*x^3 + 4*x^2

Author: David Harvey (2006-08-06)

truncate( )

Returns the polynomial of degree $ < n$ which is equivalent to self modulo $ x^n$ .

Special Functions: __eq__,$ \,$ __floordiv__,$ \,$ __ge__,$ \,$ __getitem__,$ \,$ __getslice__,$ \,$ __gt__,$ \,$ __init__,$ \,$ __le__,$ \,$ __lt__,$ \,$ __ne__,$ \,$ __reduce__,$ \,$ __rfloordiv__,$ \,$ _unsafe_mutate

__floordiv__( )

Return the quotient upon division (no remainder).

sage: R.<x> = QQ[]
sage: f = (1+2*x)^3 + 3*x; f
8*x^3 + 12*x^2 + 9*x + 1
sage: g = f // (1+2*x); g
4*x^2 + 4*x + 5/2
sage: f - g * (1+2*x)
-3/2
sage: f.quo_rem(1+2*x)
(4*x^2 + 4*x + 5/2, -3/2)

__getitem__( )

sage: R.<x> = ZZ[]
sage: f = (1+2*x)^5; f
32*x^5 + 80*x^4 + 80*x^3 + 40*x^2 + 10*x + 1
sage: f[-1]
0
sage: f[2]
40
sage: f[6]
0

__reduce__( )

_unsafe_mutate( )

Never use this unless you really know what you are doing.

WARNING: This could easily introduce subtle bugs, since SAGE assumes everywhere that polynomials are immutable. It's OK to use this if you really know what you're doing.

sage: R.<x> = ZZ[]
sage: f = (1+2*x)^2; f
4*x^2 + 4*x + 1
sage: f._unsafe_mutate(1, -5)
sage: f
4*x^2 - 5*x + 1

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