Univariate polynomials over \(\QQ\) implemented via FLINT¶
AUTHOR:
 Sebastian Pancratz

class
sage.rings.polynomial.polynomial_rational_flint.
Polynomial_rational_flint
¶ Bases:
sage.rings.polynomial.polynomial_element.Polynomial
Univariate polynomials over the rationals, implemented via FLINT.
Internally, we represent rational polynomial as the quotient of an integer polynomial and a positive denominator which is coprime to the content of the numerator.

_add_
(right)¶ Returns the sum of two rational polynomials.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = 2/3 + t + 2*t^3 sage: g = 1 + t/3  10/11*t^4 sage: f + g 10/11*t^4 + 2*t^3 + 4/3*t  1/3

_sub_
(right)¶ Returns the difference of two rational polynomials.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = 10/11*t^4 + 2*t^3 + 4/3*t  1/3 sage: g = 2*t^3 sage: f  g # indirect doctest 10/11*t^4 + 4/3*t  1/3

_lmul_
(right)¶ Returns self * right, where right is a rational number.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = 3/2*t^3  t + 1/3 sage: f * 6 # indirect doctest 9*t^3  6*t + 2

_rmul_
(left)¶ Returns left * self, where left is a rational number.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = 3/2*t^3  t + 1/3 sage: 6 * f # indirect doctest 9*t^3  6*t + 2

_mul_
(right)¶ Returns the product of self and right.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = 1 + 3*t/2  t^3 sage: g = 2/3 + 7/3*t + 3*t^2 sage: f * g # indirect doctest 3*t^5  7/3*t^4 + 23/6*t^3 + 1/2*t^2  4/3*t  2/3

_mul_trunc_
(right, n)¶ Truncated multiplication.
EXAMPLES:
sage: x = polygen(QQ) sage: p1 = 1/2  3*x + 2/7*x**3 sage: p2 = x + 2/5*x**5 + x**7 sage: p1._mul_trunc_(p2, 5) 2/7*x^4  3*x^2 + 1/2*x sage: (p1*p2).truncate(5) 2/7*x^4  3*x^2 + 1/2*x sage: p1._mul_trunc_(p2, 1) 0 sage: p1._mul_trunc_(p2, 0) Traceback (most recent call last): ... ValueError: n must be > 0
ALGORITHM:
Call the FLINT method
fmpq_poly_mullow
.

degree
()¶ Return the degree of
self
.By convention, the degree of the zero polynomial is 1.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = 1 + t + t^2/2 + t^3/3 + t^4/4 sage: f.degree() 4 sage: g = R(0) sage: g.degree() 1

denominator
()¶ Returns the denominator of self.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = (3 * t^3 + 1) / 3 sage: f.denominator() 3

disc
()¶ Returns the discriminant of this polynomial.
The discriminant \(R_n\) is defined as
\[R_n = a_n^{2 n2} \prod_{1 \le i < j \le n} (r_i  r_j)^2,\]where \(n\) is the degree of this polynomial, \(a_n\) is the leading coefficient and the roots over \(\QQbar\) are \(r_1, \ldots, r_n\).
The discriminant of constant polynomials is defined to be 0.
OUTPUT:
 Discriminant, an element of the base ring of the polynomial ring
Note
Note the identity \(R_n(f) := (1)^(n (n1)/2) R(f,f') a_n^(nk2)\), where \(n\) is the degree of this polynomial, \(a_n\) is the leading coefficient, \(f'\) is the derivative of \(f\), and \(k\) is the degree of \(f'\). Calls
resultant()
.ALGORITHM:
Use PARI.
EXAMPLES:
In the case of elliptic curves in special form, the discriminant is easy to calculate:
sage: R.<t> = QQ[] sage: f = t^3 + t + 1 sage: d = f.discriminant(); d 31 sage: d.parent() is QQ True sage: EllipticCurve([1, 1]).discriminant() / 16 31
sage: R.<t> = QQ[] sage: f = 2*t^3 + t + 1 sage: d = f.discriminant(); d 116
sage: R.<t> = QQ[] sage: f = t^3 + 3*t  17 sage: f.discriminant() 7911

discriminant
()¶ Returns the discriminant of this polynomial.
The discriminant \(R_n\) is defined as
\[R_n = a_n^{2 n2} \prod_{1 \le i < j \le n} (r_i  r_j)^2,\]where \(n\) is the degree of this polynomial, \(a_n\) is the leading coefficient and the roots over \(\QQbar\) are \(r_1, \ldots, r_n\).
The discriminant of constant polynomials is defined to be 0.
OUTPUT:
 Discriminant, an element of the base ring of the polynomial ring
Note
Note the identity \(R_n(f) := (1)^(n (n1)/2) R(f,f') a_n^(nk2)\), where \(n\) is the degree of this polynomial, \(a_n\) is the leading coefficient, \(f'\) is the derivative of \(f\), and \(k\) is the degree of \(f'\). Calls
resultant()
.ALGORITHM:
Use PARI.
EXAMPLES:
In the case of elliptic curves in special form, the discriminant is easy to calculate:
sage: R.<t> = QQ[] sage: f = t^3 + t + 1 sage: d = f.discriminant(); d 31 sage: d.parent() is QQ True sage: EllipticCurve([1, 1]).discriminant() / 16 31
sage: R.<t> = QQ[] sage: f = 2*t^3 + t + 1 sage: d = f.discriminant(); d 116
sage: R.<t> = QQ[] sage: f = t^3 + 3*t  17 sage: f.discriminant() 7911

factor_mod
(p)¶ Returns the factorization of self modulo the prime
p
.Assumes that the degree of this polynomial is at least one, and raises a
ValueError
otherwise.INPUT:
p
 Prime number
OUTPUT:
 Factorization of this polynomial modulo
p
EXAMPLES:
sage: R.<x> = QQ[] sage: (x^5 + 17*x^3 + x+ 3).factor_mod(3) x * (x^2 + 1)^2 sage: (x^5 + 2).factor_mod(5) (x + 2)^5
Variable names that are reserved in PARI, such as
zeta
, are supported (see trac ticket #20631):sage: R.<zeta> = QQ[] sage: (zeta^2 + zeta + 1).factor_mod(7) (zeta + 3) * (zeta + 5)

factor_padic
(p, prec=10)¶ Return the \(p\)adic factorization of this polynomial to the given precision.
INPUT:
p
 Prime numberprec
 Integer; the precision
OUTPUT:
 factorization of
self
viewed as a \(p\)adic polynomial
EXAMPLES:
sage: R.<x> = QQ[] sage: f = x^3  2 sage: f.factor_padic(2) (1 + O(2^10))*x^3 + O(2^10)*x^2 + O(2^10)*x + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + O(2^10) sage: f.factor_padic(3) (1 + O(3^10))*x^3 + O(3^10)*x^2 + O(3^10)*x + 1 + 2*3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + O(3^10) sage: f.factor_padic(5) ((1 + O(5^10))*x + 2 + 4*5 + 2*5^2 + 2*5^3 + 5^4 + 3*5^5 + 4*5^7 + 2*5^8 + 5^9 + O(5^10)) * ((1 + O(5^10))*x^2 + (3 + 2*5^2 + 2*5^3 + 3*5^4 + 5^5 + 4*5^6 + 2*5^8 + 3*5^9 + O(5^10))*x + 4 + 5 + 2*5^2 + 4*5^3 + 4*5^4 + 3*5^5 + 3*5^6 + 4*5^7 + 4*5^9 + O(5^10))
The input polynomial is considered to have “infinite” precision, therefore the \(p\)adic factorization of the polynomial is not the same as first coercing to \(Q_p\) and then factoring (see also trac ticket #15422):
sage: f = x^2  3^6 sage: f.factor_padic(3,5) ((1 + O(3^5))*x + 3^3 + O(3^5)) * ((1 + O(3^5))*x + 2*3^3 + 2*3^4 + O(3^5)) sage: f.change_ring(Qp(3,5)).factor() Traceback (most recent call last): ... PrecisionError: padic factorization not welldefined since the discriminant is zero up to the requestion padic precision
A more difficult example:
sage: f = 100 * (5*x + 1)^2 * (x + 5)^2 sage: f.factor_padic(5, 10) (4*5^4 + O(5^14)) * ((1 + O(5^9))*x + 5^1 + O(5^9))^2 * ((1 + O(5^10))*x + 5 + O(5^10))^2
Try some bogus inputs:
sage: f.factor_padic(3,1) Traceback (most recent call last): ... ValueError: prec_cap must be nonnegative. sage: f.factor_padic(6,10) Traceback (most recent call last): ... ValueError: p must be prime sage: f.factor_padic('hello', 'world') Traceback (most recent call last): ... TypeError: unable to convert 'hello' to an integer

galois_group
(pari_group=False, algorithm='pari')¶ Returns the Galois group of self as a permutation group.
INPUT:
self
 Irreducible polynomialpari_group
 bool (default:False
); ifTrue
instead return the Galois group as a PARI group. This has a useful label in it, and may be slightly faster since it doesn’t require looking up a group in Gap. To get a permutation group from a PARI groupP
, typePermutationGroup(P)
.algorithm
'pari'
,'kash'
,'magma'
(default:'pari'
, except when the degree is at least 12 in which case'kash'
is tried).
OUTPUT:
 Galois group
ALGORITHM:
The Galois group is computed using PARI in C library mode, or possibly KASH or MAGMA.
Note
The PARI documentation contains the following warning: The method used is that of resolvent polynomials and is sensitive to the current precision. The precision is updated internally but, in very rare cases, a wrong result may be returned if the initial precision was not sufficient.
MAGMA does not return a provably correct result. Please see the MAGMA documentation for how to obtain a provably correct result.
EXAMPLES:
sage: R.<x> = QQ[] sage: f = x^4  17*x^3  2*x + 1 sage: G = f.galois_group(); G Transitive group number 5 of degree 4 sage: G.gens() [(1,2), (1,2,3,4)] sage: G.order() 24
It is potentially useful to instead obtain the corresponding PARI group, which is little more than a 4tuple. See the PARI manual for the exact details. (Note that the third entry in the tuple is in the new standard ordering.)
sage: f = x^4  17*x^3  2*x + 1 sage: G = f.galois_group(pari_group=True); G PARI group [24, 1, 5, "S4"] of degree 4 sage: PermutationGroup(G) Transitive group number 5 of degree 4
You can use KASH to compute Galois groups as well. The advantage is that KASH can compute Galois groups of fields up to degree 21, whereas PARI only goes to degree 11. (In my notsothorough experiments PARI is faster than KASH.)
sage: f = x^4  17*x^3  2*x + 1 sage: f.galois_group(algorithm='kash') # optional  kash Transitive group number 5 of degree 4 sage: f = x^4  17*x^3  2*x + 1 sage: f.galois_group(algorithm='magma') # optional  magma Transitive group number 5 of degree 4

gcd
(right)¶ Returns the (monic) greatest common divisor of self and right.
Corner cases: if self and right are both zero, returns zero. If only one of them is zero, returns the other polynomial, up to normalisation.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = 2 + 3*t/2 + 4*t^2/7  t^3 sage: g = 1/2 + 4*t + 2*t^4/3 sage: f.gcd(g) 1 sage: f = (3*t + 1/2) * f sage: g = (3*t + 1/2) * (4*t^2/3  1) * g sage: f.gcd(g) t  1/6

hensel_lift
(p, e)¶ Assuming that this polynomial factors modulo \(p\) into distinct monic factors, computes the Hensel lifts of these factors modulo \(p^e\). We assume that
self
has integer coefficients.Returns an empty list if this polynomial has degree less than one.
INPUT:
p
 Prime number; coerceable toInteger
e
 Exponent; coerceable toInteger
OUTPUT:
 Hensel lifts; list of polynomials over \(\ZZ / p^e \ZZ\)
EXAMPLES:
sage: R.<x> = QQ[] sage: R((x1)*(x+1)).hensel_lift(7, 2) [x + 1, x + 48]
If the input polynomial \(f\) is not monic, we get a factorization of \(f / lc(f)\):
sage: R(2*x^2  2).hensel_lift(7, 2) [x + 1, x + 48]

inverse_series_trunc
(prec)¶ Return a polynomial approximation of precision
prec
of the inverse series of this polynomial.EXAMPLES:
sage: x = polygen(QQ) sage: p = 2 + x  3/5*x**2 sage: q5 = p.inverse_series_trunc(5) sage: q5 151/800*x^4  17/80*x^3 + 11/40*x^2  1/4*x + 1/2 sage: q5 * p 453/4000*x^6 + 253/800*x^5 + 1 sage: q100 = p.inverse_series_trunc(100) sage: (q100 * p).truncate(100) 1

is_irreducible
()¶ Return whether this polynomial is irreducible.
This method computes the primitive part as an element of \(\ZZ[t]\) and calls the method
is_irreducible
for elements of that polynomial ring.By definition, over any integral domain, an element \(r\) is irreducible if and only if it is nonzero, not a unit and whenever \(r = ab\) then \(a\) or \(b\) is a unit.
EXAMPLES:
sage: R.<t> = QQ[] sage: (t^2 + 2).is_irreducible() True sage: (t^2  1).is_irreducible() False

is_one
()¶ Returns whether or not this polynomial is one.
EXAMPLES:
sage: R.<x> = QQ[] sage: R([0,1]).is_one() False sage: R([1]).is_one() True sage: R([0]).is_one() False sage: R([1]).is_one() False sage: R([1,1]).is_one() False

is_zero
()¶ Returns whether or not self is the zero polynomial.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = 1  t + 1/2*t^2  1/3*t^3 sage: f.is_zero() False sage: R(0).is_zero() True

lcm
(right)¶ Returns the monic (or zero) least common multiple of self and right.
Corner cases: if either of self and right are zero, returns zero. This behaviour is ensures that the relation lcm(a,b) gcd(a,b) == a b holds up to multiplication by rationals.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = 2 + 3*t/2 + 4*t^2/7  t^3 sage: g = 1/2 + 4*t + 2*t^4/3 sage: f.lcm(g) t^7  4/7*t^6  3/2*t^5 + 8*t^4  75/28*t^3  66/7*t^2 + 87/8*t + 3/2 sage: f.lcm(g) * f.gcd(g) // (f * g) 3/2

list
(copy=True)¶ Return a list with the coefficients of
self
.EXAMPLES:
sage: R.<t> = QQ[] sage: f = 1 + t + t^2/2 + t^3/3 + t^4/4 sage: f.list() [1, 1, 1/2, 1/3, 1/4] sage: g = R(0) sage: g.list() []

numerator
()¶ Returns the numerator of self.
Representing self as the quotient of an integer polynomial and a positive integer denominator (coprime to the content of the polynomial), returns the integer polynomial.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = (3 * t^3 + 1) / 3 sage: f.numerator() 3*t^3  1

quo_rem
(right)¶ Returns the quotient and remainder of the Euclidean division of self and right.
Raises a ZerodivisionError if right is zero.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = R.random_element(2000) sage: g = R.random_element(1000) sage: q, r = f.quo_rem(g) sage: f == q*g + r True

real_root_intervals
()¶ Returns isolating intervals for the real roots of self.
EXAMPLES:
We compute the roots of the characteristic polynomial of some Salem numbers:
sage: R.<t> = QQ[] sage: f = 1  t^2  t^3  t^4 + t^6 sage: f.real_root_intervals() [((1/2, 3/4), 1), ((1, 3/2), 1)]

resultant
(right)¶ Returns the resultant of self and right.
Enumerating the roots over \(\QQ\) as \(r_1, \cdots, r_m\) and \(s_1, \cdots, s_n\) and letting \(x\) and \(y\) denote the leading coefficients of \(f\) and \(g\), the resultant of the two polynomials is defined by
\[x^{\deg g} y^{\deg f} \prod_{i,j} (r_i  s_j).\]Corner cases: if one of the polynomials is zero, the resultant is zero. Note that otherwise if one of the polynomials is constant, the last term in the above is the empty product.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = (t  2/3) * (t + 4/5) * (t  1) sage: g = (t  1/3) * (t + 1/2) * (t + 1) sage: f.resultant(g) 119/1350 sage: h = (t  1/3) * (t + 1/2) * (t  1) sage: f.resultant(h) 0

reverse
(degree=None)¶ Reverse the coefficients of this polynomial (thought of as a polynomial of degree
degree
).INPUT:
degree
(None
or integral value that fits in anunsigned long
, default: degree ofself
)  if specified, truncate or zero pad the list of coefficients to this degree before reversing it.
EXAMPLES:
We first consider the simplest case, where we reverse all coefficients of a polynomial and obtain a polynomial of the same degree:
sage: R.<t> = QQ[] sage: f = 1 + t + t^2 / 2 + t^3 / 3 + t^4 / 4 sage: f.reverse() t^4 + t^3 + 1/2*t^2 + 1/3*t + 1/4
Next, an example we the returned polynomial has lower degree because the original polynomial has low coefficients equal to zero:
sage: R.<t> = QQ[] sage: f = 3/4*t^2 + 6*t^7 sage: f.reverse() 3/4*t^5 + 6
The next example illustrates the passing of a value for
degree
less than the length of self, notationally resulting in truncation prior to reversing:sage: R.<t> = QQ[] sage: f = 1 + t + t^2 / 2 + t^3 / 3 + t^4 / 4 sage: f.reverse(2) t^2 + t + 1/2
Now we illustrate the passing of a value for
degree
greater than the length of self, notationally resulting in zero padding at the top end prior to reversing:sage: R.<t> = QQ[] sage: f = 1 + t + t^2 / 2 + t^3 / 3 sage: f.reverse(4) t^4 + t^3 + 1/2*t^2 + 1/3*t

revert_series
(n)¶ Return a polynomial \(f\) such that \(f(self(x)) = self(f(x)) = x mod x^n\).
EXAMPLES:
sage: R.<t> = QQ[] sage: f = t  t^3/6 + t^5/120 sage: f.revert_series(6) 3/40*t^5 + 1/6*t^3 + t sage: f.revert_series(1) Traceback (most recent call last): ValueError: argument n must be a nonnegative integer, got 1 sage: g =  t^3/3 + t^5/5 sage: g.revert_series(6) Traceback (most recent call last): ... ValueError: self must have constant coefficient 0 and a unit for coefficient t^1

truncate
(n)¶ Returns self truncated modulo \(t^n\).
INPUT:
n
 The power of \(t\) modulo which self is truncated
EXAMPLES:
sage: R.<t> = QQ[] sage: f = 1  t + 1/2*t^2  1/3*t^3 sage: f.truncate(0) 0 sage: f.truncate(2) t + 1

xgcd
(right)¶ Returns polynomials d, s, and t such that d == s * self + t * right, where d is the (monic) greatest common divisor of self and right. The choice of s and t is not specified any further.
Corner cases: if self and right are zero, returns zero polynomials. Otherwise, if only self is zero, returns (d, s, t) = (right, 0, 1) up to normalisation, and similarly if only right is zero.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = 2/3 + 3/4 * t  t^2 sage: g = 3 + 1/7 * t sage: f.xgcd(g) (1, 12/5095, 84/5095*t  1701/5095)
