Polynomials - Tutorial (original) (raw)
In this section we illustrate how to create and use polynomials in Sage.
Univariate Polynomials¶
There are three ways to create polynomial rings.
Sage
sage: R = PolynomialRing(QQ, 't') sage: R Univariate Polynomial Ring in t over Rational Field
Python
from sage.all import * R = PolynomialRing(QQ, 't') R Univariate Polynomial Ring in t over Rational Field
This creates a polynomial ring and tells Sage to use (the string) ‘t’ as the indeterminate when printing to the screen. However, this does not define the symbol t for use in Sage, so you cannot use it to enter a polynomial (such as \(t^2+1\)) belonging to R.
An alternate way is
Sage
sage: S = QQ['t'] sage: S == R True
Python
from sage.all import * S = QQ['t'] S == R True
This has the same issue regarding t.
A third very convenient way is
Sage
sage: R. = PolynomialRing(QQ)
Python
from sage.all import * R = PolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
or
Sage
Python
from sage.all import * R = QQ['t']; (t,) = R._first_ngens(1)
or even
Sage
Python
from sage.all import * R = QQ['t']; (t,) = R._first_ngens(1)
This has the additional side effect that it defines the variablet to be the indeterminate of the polynomial ring, so you can easily construct elements of R, as follows. (Note that the third way is very similar to the constructor notation in Magma, and just as in Magma it can be used for a wide range of objects.)
Sage
sage: poly = (t+1) * (t+2); poly t^2 + 3*t + 2 sage: poly in R True
Python
from sage.all import * poly = (t+Integer(1)) * (t+Integer(2)); poly t^2 + 3*t + 2 poly in R True
Whatever method you use to define a polynomial ring, you can recover the indeterminate as the \(0^{th}\) generator:
Sage
sage: R = PolynomialRing(QQ, 't') sage: t = R.0 sage: t in R True
Python
from sage.all import * R = PolynomialRing(QQ, 't') t = R.gen(0) t in R True
Note that a similar construction works with the complex numbers: the complex numbers can be viewed as being generated over the real numbers by the symbol i; thus we have the following:
Sage
sage: CC Complex Field with 53 bits of precision sage: CC.0 # 0th generator of CC 1.00000000000000*I
Python
from sage.all import * CC Complex Field with 53 bits of precision CC.gen(0) # 0th generator of CC 1.00000000000000*I
For polynomial rings, you can obtain both the ring and its generator, or just the generator, during ring creation as follows:
Sage
sage: R, t = QQ['t'].objgen() sage: t = QQ['t'].gen() sage: R, t = objgen(QQ['t']) sage: t = gen(QQ['t'])
Python
from sage.all import * R, t = QQ['t'].objgen() t = QQ['t'].gen() R, t = objgen(QQ['t']) t = gen(QQ['t'])
Finally we do some arithmetic in \(\QQ[t]\).
Sage
sage: R, t = QQ['t'].objgen() sage: f = 2t^7 + 3t^2 - 15/19 sage: f^2 4t^14 + 12t^9 - 60/19t^7 + 9t^4 - 90/19t^2 + 225/361 sage: cyclo = R.cyclotomic_polynomial(7); cyclo t^6 + t^5 + t^4 + t^3 + t^2 + t + 1 sage: g = 7 * cyclo * t^5 * (t^5 + 10t + 2) sage: g 7t^16 + 7t^15 + 7t^14 + 7t^13 + 77t^12 + 91t^11 + 91t^10 + 84t^9 + 84t^8 + 84t^7 + 84t^6 + 14t^5 sage: F = factor(g); F (7) * t^5 * (t^5 + 10t + 2) * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1) sage: F.unit() 7 sage: list(F) [(t, 5), (t^5 + 10t + 2, 1), (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1, 1)]
Python
from sage.all import * R, t = QQ['t'].objgen() f = Integer(2)t*Integer(7) + Integer(3)*t**Integer(2) - Integer(15)/Integer(19) fInteger(2) 4t^14 + 12t^9 - 60/19t^7 + 9t^4 - 90/19t^2 + 225/361 cyclo = R.cyclotomic_polynomial(Integer(7)); cyclo t^6 + t^5 + t^4 + t^3 + t^2 + t + 1 g = Integer(7) * cyclo * tInteger(5) * (tInteger(5) + Integer(10)t + Integer(2)) g 7t^16 + 7t^15 + 7t^14 + 7t^13 + 77t^12 + 91t^11 + 91t^10 + 84t^9 + 84t^8 + 84t^7 + 84t^6 + 14t^5 F = factor(g); F (7) * t^5 * (t^5 + 10t + 2) * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1) F.unit() 7 list(F) [(t, 5), (t^5 + 10t + 2, 1), (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1, 1)]
Notice that the factorization correctly takes into account and records the unit part.
If you were to use, e.g., the R.cyclotomic_polynomial function a lot for some research project, in addition to citing Sage you should make an attempt to find out what component of Sage is being used to actually compute the cyclotomic polynomial and cite that as well. In this case, if you type R.cyclotomic_polynomial?? to see the source code, you’ll quickly see a line f = pari.polcyclo(n) which means that PARI is being used for computation of the cyclotomic polynomial. Cite PARI in your work as well.
Dividing two polynomials constructs an element of the fraction field (which Sage creates automatically).
Sage
sage: x = QQ['x'].0 sage: f = x^3 + 1; g = x^2 - 17 sage: h = f/g; h (x^3 + 1)/(x^2 - 17) sage: h.parent() Fraction Field of Univariate Polynomial Ring in x over Rational Field
Python
from sage.all import * x = QQ['x'].gen(0) f = xInteger(3) + Integer(1); g = xInteger(2) - Integer(17) h = f/g; h (x^3 + 1)/(x^2 - 17) h.parent() Fraction Field of Univariate Polynomial Ring in x over Rational Field
Using Laurent series, one can compute series expansions in the fraction field of QQ[x]:
Sage
sage: R. = LaurentSeriesRing(QQ); R Laurent Series Ring in x over Rational Field sage: 1/(1-x) + O(x^10) 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + O(x^10)
Python
from sage.all import * R = LaurentSeriesRing(QQ, names=('x',)); (x,) = R._first_ngens(1); R Laurent Series Ring in x over Rational Field Integer(1)/(Integer(1)-x) + O(x**Integer(10)) 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + O(x^10)
If we name the variable differently, we obtain a different univariate polynomial ring.
Sage
sage: R. = PolynomialRing(QQ) sage: S. = PolynomialRing(QQ) sage: x == y False sage: R == S False sage: R(y) x sage: R(y^2 - 17) x^2 - 17
Python
from sage.all import * R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1) S = PolynomialRing(QQ, names=('y',)); (y,) = S._first_ngens(1) x == y False R == S False R(y) x R(y**Integer(2) - Integer(17)) x^2 - 17
The ring is determined by the variable. Note that making another ring with variable called x does not return a different ring.
Sage
sage: R = PolynomialRing(QQ, "x") sage: T = PolynomialRing(QQ, "x") sage: R == T True sage: R is T True sage: R.0 == T.0 True
Python
from sage.all import * R = PolynomialRing(QQ, "x") T = PolynomialRing(QQ, "x") R == T True R is T True R.gen(0) == T.gen(0) True
Sage also has support for power series and Laurent series rings over any base ring. In the following example, we create an element of \(\GF{7}[[T]]\) and divide to create an element of\(\GF{7}((T))\).
Sage
sage: R. = PowerSeriesRing(GF(7)); R Power Series Ring in T over Finite Field of size 7 sage: f = T + 3T^2 + T^3 + O(T^4) sage: f^3 T^3 + 2T^4 + 2*T^5 + O(T^6) sage: 1/f T^-1 + 4 + T + O(T^2) sage: parent(1/f) Laurent Series Ring in T over Finite Field of size 7
Python
from sage.all import * R = PowerSeriesRing(GF(Integer(7)), names=('T',)); (T,) = R._first_ngens(1); R Power Series Ring in T over Finite Field of size 7 f = T + Integer(3)TInteger(2) + TInteger(3) + O(TInteger(4)) fInteger(3) T^3 + 2T^4 + 2*T^5 + O(T^6) Integer(1)/f T^-1 + 4 + T + O(T^2) parent(Integer(1)/f) Laurent Series Ring in T over Finite Field of size 7
You can also create power series rings using a double-brackets shorthand:
Sage
sage: GF(7)[['T']] Power Series Ring in T over Finite Field of size 7
Python
from sage.all import * GF(Integer(7))[['T']] Power Series Ring in T over Finite Field of size 7
Multivariate Polynomials¶
To work with polynomials of several variables, we declare the polynomial ring and variables first.
Sage
sage: R = PolynomialRing(GF(5),3,"z") # here, 3 = number of variables sage: R Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
Python
from sage.all import * R = PolynomialRing(GF(Integer(5)),Integer(3),"z") # here, 3 = number of variables R Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
Just as for defining univariate polynomial rings, there are alternative ways:
Sage
sage: GF(5)['z0, z1, z2'] Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5 sage: R.<z0,z1,z2> = GF(5)[]; R Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
Python
from sage.all import * GF(Integer(5))['z0, z1, z2'] Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5 R = GF(Integer(5))['z0, z1, z2']; (z0, z1, z2,) = R._first_ngens(3); R Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
Also, if you want the variable names to be single letters then you can use the following shorthand:
Sage
sage: PolynomialRing(GF(5), 3, 'xyz') Multivariate Polynomial Ring in x, y, z over Finite Field of size 5
Python
from sage.all import * PolynomialRing(GF(Integer(5)), Integer(3), 'xyz') Multivariate Polynomial Ring in x, y, z over Finite Field of size 5
Next let’s do some arithmetic.
Sage
sage: z = GF(5)['z0, z1, z2'].gens() sage: z (z0, z1, z2) sage: (z[0]+z[1]+z[2])^2 z0^2 + 2z0z1 + z1^2 + 2z0z2 + 2z1z2 + z2^2
Python
from sage.all import * z = GF(Integer(5))['z0, z1, z2'].gens() z (z0, z1, z2) (z[Integer(0)]+z[Integer(1)]+z[Integer(2)])*Integer(2) z0^2 + 2z0z1 + z1^2 + 2z0z2 + 2z1*z2 + z2^2
You can also use more mathematical notation to construct a polynomial ring.
Sage
sage: R = GF(5)['x,y,z'] sage: x,y,z = R.gens() sage: QQ['x'] Univariate Polynomial Ring in x over Rational Field sage: QQ['x,y'].gens() (x, y) sage: QQ['x'].objgens() (Univariate Polynomial Ring in x over Rational Field, (x,))
Python
from sage.all import * R = GF(Integer(5))['x,y,z'] x,y,z = R.gens() QQ['x'] Univariate Polynomial Ring in x over Rational Field QQ['x,y'].gens() (x, y) QQ['x'].objgens() (Univariate Polynomial Ring in x over Rational Field, (x,))
Multivariate polynomials are implemented in Sage using Python dictionaries and the “distributive representation” of a polynomial. Sage makes some use of Singular [Si], e.g., for computation of gcd’s and Gröbner basis of ideals.
Sage
sage: R, (x, y) = PolynomialRing(RationalField(), 2, 'xy').objgens() sage: f = (x^3 + 2y^2x)^2 sage: g = x^2*y^2 sage: f.gcd(g) x^2
Python
from sage.all import * R, (x, y) = PolynomialRing(RationalField(), Integer(2), 'xy').objgens() f = (x**Integer(3) + Integer(2)*y**Integer(2)*x)Integer(2) g = xInteger(2)*y**Integer(2) f.gcd(g) x^2
Next we create the ideal \((f,g)\) generated by \(f\)and \(g\), by simply multiplying (f,g) by R (we could also write ideal([f,g]) or ideal(f,g)).
Sage
sage: I = (f, g)R; I Ideal (x^6 + 4x^4y^2 + 4x^2y^4, x^2y^2) of Multivariate Polynomial Ring in x, y over Rational Field sage: B = I.groebner_basis(); B [x^6, x^2*y^2] sage: x^2 in I False
Python
from sage.all import * I = (f, g)R; I Ideal (x^6 + 4x^4y^2 + 4x^2y^4, x^2y^2) of Multivariate Polynomial Ring in x, y over Rational Field B = I.groebner_basis(); B [x^6, x^2*y^2] x**Integer(2) in I False
Incidentally, the Gröbner basis above is not a list but an immutable sequence. This means that it has a universe, parent, and cannot be changed (which is good because changing the basis would break other routines that use the Gröbner basis).
Sage
sage: B.universe() Multivariate Polynomial Ring in x, y over Rational Field sage: B[1] = x Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead.
Python
from sage.all import * B.universe() Multivariate Polynomial Ring in x, y over Rational Field B[Integer(1)] = x Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead.
Some (read: not as much as we would like) commutative algebra is available in Sage, implemented via Singular. For example, we can compute the primary decomposition and associated primes of\(I\):
Sage
sage: I.primary_decomposition() [Ideal (x^2) of Multivariate Polynomial Ring in x, y over Rational Field, Ideal (y^2, x^6) of Multivariate Polynomial Ring in x, y over Rational Field] sage: I.associated_primes() [Ideal (x) of Multivariate Polynomial Ring in x, y over Rational Field, Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field]
Python
from sage.all import * I.primary_decomposition() [Ideal (x^2) of Multivariate Polynomial Ring in x, y over Rational Field, Ideal (y^2, x^6) of Multivariate Polynomial Ring in x, y over Rational Field] I.associated_primes() [Ideal (x) of Multivariate Polynomial Ring in x, y over Rational Field, Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field]