Univariate polynomials over \(\QQ\) implemented via FLINT#
AUTHOR:
Sebastian Pancratz
- class sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint#
Bases:
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
, whereright
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
, whereleft
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
andright
.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 github issue #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 polynomialEXAMPLES:
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 \(\QQ_p\) and then factoring (see also github issue #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 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'
,'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 uses the “Transitive Groups Libraries” from the “TransGrp” GAP package which comes installed with the “gap” Sage package.
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') Transitive group number 5 of degree 4 sage: f = x^13 - 17*x^3 - 2*x + 1 sage: f.galois_group(algorithm='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') 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
andright
.Corner cases: if
self
andright
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:
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
andright
.Corner cases: if either of
self
andright
are zero, returns zero. This behaviour is ensures that the relation \(\lcm(a,b)\cdot \gcd(a,b) = a\cdot b\) 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
andright
.Raises a
ZeroDivisionError
ifright
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
andright
.Enumerating the roots over \(\QQ\) as \(r_1, \dots, r_m\) and \(s_1, \dots, s_n\) 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 where 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 ofself
, 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 ofself
, 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 whichself
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 ofself
andright
. The choice of \(s\) and \(t\) is not specified any further.Corner cases: if
self
andright
are zero, returns zero polynomials. Otherwise, if onlyself
is zero, returns(d, s, t) = (right, 0, 1)
up to normalisation, and similarly if onlyright
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)