Polynomials

Polynomial powers

How do I compute modular polynomial powers in Sage?

To compute \(x^{2006} \pmod {x^3 + 7}\) in \(GF(97)[x]\), we create the quotient ring \(GF(97)[x]/(x^3+7)\), and compute \(x^{2006}\) in it. As a matter of Sage notation, we must distinguish between the \(x\) in \(GF(97)[x]\) and the corresponding element (which we denote by \(a\)) in the quotient ring \(GF(97)[x]/(x^3+7)\).

sage: R = PolynomialRing(GF(97),'x')
sage: x = R.gen()
sage: S = R.quotient(x^3 + 7, 'a')
sage: a = S.gen()
sage: S
Univariate Quotient Polynomial Ring in a over
Finite Field of size 97 with modulus x^3 + 7
sage: a^2006
4*a^2

Another approach to this:

sage: R = PolynomialRing(GF(97),'x')
sage: x = R.gen()
sage: S = R.quotient(x^3 + 7, 'a')
sage: a = S.gen()
sage: a^20062006
80*a
sage: print gap.eval("R:= PolynomialRing( GF(97))")
GF(97)[x_1]
sage: print gap.eval("i:= IndeterminatesOfPolynomialRing(R)")
[ x_1 ]
sage: gap.eval("x:= i[1];; f:= x;;")
''
sage: print gap.eval("PowerMod( R, x, 20062006, x^3+7 );")
Z(97)^41*x_1
sage: print gap.eval("PowerMod( R, x, 20062006, x^3+7 );")
Z(97)^41*x_1
sage: print gap.eval("PowerMod( R, x, 2006200620062006, x^3+7 );")
Z(97)^4*x_1^2
sage: a^2006200620062006
43*a^2
sage: print gap.eval("PowerMod( R, x, 2006200620062006, x^3+7 );")
Z(97)^4*x_1^2
sage: print gap.eval("Int(Z(97)^4)")
43

Factorization

You can factor a polynomial using Sage.

Using Sage to factor a univariate polynomial is a matter of applying the method factor to the PolynomialRingElement object f. In fact, this method actually calls Pari, so the computation is fairly fast.

sage: x = PolynomialRing(RationalField(), 'x').gen()
sage: f = (x^3 - 1)^2-(x^2-1)^2
sage: f.factor()
(x - 1)^2 * x^2 * (x^2 + 2*x + 2)

Using the Singular interface, Sage also factors multivariate polynomials.

sage: x, y = PolynomialRing(RationalField(), 2, ['x','y']).gens()
sage: f =  9*y^6 - 9*x^2*y^5 - 18*x^3*y^4 - 9*x^5*y^4 + 9*x^6*y^2 + 9*x^7*y^3\
...   + 18*x^8*y^2 - 9*x^11
sage: f.factor()
(9) * (-x^5 + y^2) * (x^6 - 2*x^3*y^2 - x^2*y^3 + y^4)

Polynomial GCD’s

This example illustrates single variable polynomial GCD’s:

sage: x = PolynomialRing(RationalField(), 'x').gen()
sage: f = 3*x^3 + x
sage: g = 9*x*(x+1)
sage: f.gcd(g)
x

This example illustrates multivariate polynomial GCD’s:

sage: R = PolynomialRing(RationalField(),3, ['x','y','z'], 'lex')
sage: x,y,z = PolynomialRing(RationalField(),3, ['x','y','z'], 'lex').gens()
sage: f = 3*x^2*(x+y)
sage: g = 9*x*(y^2 - x^2)
sage: f.gcd(g)
x^2 + x*y

Here’s another way to do this:

sage: R2 = singular.ring(0, '(x,y,z)', 'lp')
sage: a = singular.new('3x2*(x+y)')
sage: b = singular.new('9x*(y2-x2)')
sage: g = a.gcd(b)
sage: g
x^2+x*y

This example illustrates univariate polynomial GCD’s via the GAP interface.

sage: R = gap.PolynomialRing(gap.GF(2)); R
PolynomialRing( GF(2), ["x_1"] )
sage: i = R.IndeterminatesOfPolynomialRing(); i
[ x_1 ]
sage: x_1 = i[1]
sage: f = (x_1^3 - x_1 + 1)*(x_1 + x_1^2); f
x_1^5+x_1^4+x_1^3+x_1
sage: g = (x_1^3 - x_1 + 1)*(x_1 + 1); g
x_1^4+x_1^3+x_1^2+Z(2)^0
sage: f.Gcd(g)
x_1^4+x_1^3+x_1^2+Z(2)^0

We can, of course, do the same computation in , which uses the NTL library (which does huge polynomial gcd’s over finite fields very quickly).

sage: x = PolynomialRing(GF(2), 'x').gen()
sage: f = (x^3 - x + 1)*(x + x^2); f
x^5 + x^4 + x^3 + x
sage: g = (x^3 - x + 1)*(x + 1)
sage: f.gcd(g)
x^4 + x^3 + x^2 + 1

Roots of polynomials

Sage can compute roots of a univariant polynomial.

sage: x = PolynomialRing(RationalField(), 'x').gen()
sage: f = x^3 - 1
sage: f.roots()
[(1, 1)]
sage: f = (x^3 - 1)^2
sage: f.roots()
[(1, 2)]
sage: x = PolynomialRing(CyclotomicField(3), 'x').gen()
sage: f = x^3 - 1
sage: f.roots()
[(1, 1), (zeta3, 1), (-zeta3 - 1, 1)]

The first of the pair is the root, the second of the pair is its multiplicity.

There are some situations where GAP does find the roots of a univariate polynomial but GAP does not do this generally. (The roots must generate either a finite field or a subfield of a cyclotomic field.) However, there is a GAP package called RadiRoot, which must be installed into ‘s installation of GAP, which does help to do this for polynomials with rational coefficients (radiroot itself requires other packages to be installed; please see its webpage for more details). The Factors command actually has an option which allows you to increase the groundfield so that a factorization actually returns the roots. Please see the examples given in section 64.10 “Polynomial Factorization” of the GAP Reference Manual for more details.

Evaluation of multivariate functions

You can evaluate polynomials in Sage as usual by substituting in points:

sage: x = PolynomialRing(RationalField(), 3, 'x').gens()
sage: f = x[0] + x[1] - 2*x[1]*x[2]
sage: f
-2*x1*x2 + x0 + x1
sage: f(1,2,0)
3
sage: f(1,2,5)
-17

This also will work with rational functions:

sage: h = f /(x[1] + x[2])
sage: h
(-2*x1*x2 + x0 + x1)/(x1 + x2)
sage: h(1,2,3)
-9/5

Sage also performs symbolic manipulation:

sage: var('x,y,z')
(x, y, z)
sage: f = (x + 3*y + x^2*y)^3; f
(x^2*y + x + 3*y)^3
sage: f(x=1,y=2,z=3)
729
sage: f.expand()
x^6*y^3 + 3*x^5*y^2 + 9*x^4*y^3 + 3*x^4*y + 18*x^3*y^2 +
27*x^2*y^3 +
x^3 + 9*x^2*y + 27*x*y^2 + 27*y^3
sage: f(x = 5/z)
(3*y + 25*y/z^2 + 5/z)^3
sage: g = f.subs(x = 5/z); g
(3*y + 25*y/z^2 + 5/z)^3
sage: h = g.rational_simplify(); h
(27*y^3*z^6 + 135*y^2*z^5 + 225*(3*y^3 + y)*z^4 + 125*(18*y^2 + 1)*z^3 +
15625*y^3 + 9375*y^2*z + 1875*(3*y^3 + y)*z^2)/z^6

Roots of multivariate polynomials

Sage (using the interface to Singular) can solve multivariate polynomial equations in some situations (they assume that the solutions form a zero-dimensional variety) using Gröbner bases. Here is a simple example:

sage: R = PolynomialRing(QQ, 2, 'ab', order='lp')
sage: a,b = R.gens()
sage: I = (a^2-b^2-3, a-2*b)*R
sage: B = I.groebner_basis(); B
[a - 2*b, b^2 - 1]

So \(b=\pm 1\) and \(a=2b\).

Gröbner bases

This computation uses Singular behind the scenes to compute the Gröbner basis.

sage: R = PolynomialRing(QQ, 4, 'abcd', order='lp')
sage: a,b,c,d = R.gens()
sage: I = (a+b+c+d, a*b+a*d+b*c+c*d, a*b*c+a*b*d+a*c*d+b*c*d, a*b*c*d-1)*R; I
Ideal (a + b + c + d, a*b + a*d + b*c + c*d, a*b*c + a*b*d + a*c*d + b*c*d,
a*b*c*d - 1) of Multivariate Polynomial Ring in a, b, c, d over Rational Field
sage: B = I.groebner_basis(); B
[a + b + c + d,
 b^2 + 2*b*d + d^2,
 b*c - b*d + c^2*d^4 + c*d - 2*d^2,
 b*d^4 - b + d^5 - d,
 c^3*d^2 + c^2*d^3 - c - d,
 c^2*d^6 - c^2*d^2 - d^4 + 1]

You can work with multiple rings without having to switch back and forth like in Singular. For example,

sage: a,b,c = QQ['a,b,c'].gens()
sage: X,Y = GF(7)['X,Y'].gens()
sage: I = ideal(a, b^2, b^3+c^3)
sage: J = ideal(X^10 + Y^10)

sage: I.minimal_associated_primes ()
[Ideal (c, b, a) of Multivariate Polynomial Ring in a, b, c over Rational Field]

sage: J.minimal_associated_primes ()     # slightly random output
[Ideal (Y^4 + 3*X*Y^3 + 4*X^2*Y^2 + 4*X^3*Y + X^4) of Multivariate Polynomial
Ring in X, Y over Finite Field of size 7,
 Ideal (Y^4 + 4*X*Y^3 + 4*X^2*Y^2 + 3*X^3*Y + X^4) of Multivariate Polynomial
Ring in X, Y over Finite Field of size 7,
 Ideal (Y^2 + X^2) of Multivariate Polynomial Ring in X, Y over Finite Field
of size 7]

All the real work is done by Singular.

Sage also includes gfan which provides other fast algorithms for computing Gröbner bases. See the section on “Gröbner fans” in the Reference Manual for more details.