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)

Return 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)

Return 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)

Return 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)

Return 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)

Return 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()

Return the denominator of self.

EXAMPLES:

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

disc()

Return the discriminant of this polynomial.

The discriminant \(R_n\) is defined as

\[R_n = a_n^{2 n-2} \prod_{1 \le i < j \le n} (r_i - r_j)^2,\]

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 \(R_n(f) := (-1)^(n (n-1)/2) R(f,f') a_n^(n-k-2)\), 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()

Return the discriminant of this polynomial.

The discriminant \(R_n\) is defined as

\[R_n = a_n^{2 n-2} \prod_{1 \le i < j \le n} (r_i - r_j)^2,\]

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 \(R_n(f) := (-1)^(n (n-1)/2) R(f,f') a_n^(n-k-2)\), 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)

Return 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 number

• prec - 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: p-adic factorization not well-defined since the discriminant is zero up to the requestion p-adic 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 non-negative
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')

Return the Galois group of this polynomial as a permutation group.

INPUT:

• self - Irreducible polynomial

• pari_group - bool (default: False); if True 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 group P, type PermutationGroup(P).

• algorithm - 'pari', 'gap', 'kash', 'magma' (default: 'pari', for degrees is at most 11; 'gap', for degrees from 12 to 15; 'kash', for degrees from 16 or more).

OUTPUT:

• Galois group

ALGORITHM:

The Galois group is computed using PARI in C library mode, or possibly GAP, 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.

GAP needs an optional transitive group library installed, from database_gap spkg.

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 4-tuple. 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 or GAP to compute Galois groups as well. The advantage is that KASH (resp. GAP) can compute Galois groups of fields up to degree 23 (resp. 15), whereas PARI only goes to degree 11. (In my not-so-thorough 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='gap')  # optional - database_gap
Transitive group number 5 of degree 4
sage: f = x^13 - 17*x^3 - 2*x + 1
sage: f.galois_group(algorithm='gap')  # optional - database_gap
Transitive group number 9 of degree 13
sage: f = x^12 - 2*x^8 - x^7 + 2*x^6 + 4*x^4 - 2*x^3 - x^2 - x + 1
sage: f.galois_group(algorithm='gap')  # optional - database_gap
Transitive group number 183 of degree 12
sage: f.galois_group(algorithm='magma')  # optional - magma
Transitive group number 5 of degree 4

galois_group_davenport_smith_test(num_trials=50, assume_irreducible=False)

Use the Davenport-Smith test to attempt to certify that \(f\) has Galois group A_n or S_n.

Return 1 if the Galois group is certified as S_n, 2 if A_n, or 0 if no conclusion is reached.

By default, we first check that \(f\) is irreducible. For extra efficiency, one can override this by specifying \(assume_irreducible=True\); this yields undefined results if \(f\) is not irreducible.

A corresponding function in Magma is \(IsEasySnAn\).

EXAMPLES:

sage: P.<x> = QQ[]
sage: u = x^7 + x + 1
sage: u.galois_group_davenport_smith_test()                                                         
1
sage: u = x^7 - x^4 - x^3 + 3*x^2 - 1                                           
sage: u.galois_group_davenport_smith_test()                                                         
2
sage: u = x^7 - 2
sage: u.galois_group_davenport_smith_test()                                                         
0

gcd(right)

Return 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.

Return an empty list if this polynomial has degree less than one.

INPUT:

• p - Prime number; coerceable to Integer

• e - Exponent; coerceable to Integer

OUTPUT:

• Hensel lifts; list of polynomials over \(\ZZ / p^e \ZZ\)

EXAMPLES:

sage: R.<x> = QQ[]
sage: R((x-1)*(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 non-zero, 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()

Return 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()

Return 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)

Return 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()

Return 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)

Return 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()

Return 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)

Return 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 an unsigned long, default: degree of self) - 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 non-negative 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)

Return 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)

Return 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)