Base class for elements of multivariate polynomial rings

class sage.rings.polynomial.multi_polynomial.MPolynomial

Bases: sage.structure.element.CommutativeRingElement

args()

Returns the named of the arguments of self, in the order they are accepted from call.

EXAMPLES:

sage: R.<x,y> = ZZ[]
sage: x.args()
(x, y)

change_ring(R)

Return a copy of this polynomial but with coefficients in R, if at all possible.

INPUT:

• R – a ring

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: f = x^3 + 3/5*y + 1
sage: f.change_ring(GF(7))
x^3 + 2*y + 1

sage: R.<x,y> = GF(9,'a')[]
sage: (x+2*y).change_ring(GF(3))
x - y

coefficients()

Return the nonzero coefficients of this polynomial in a list. The returned list is decreasingly ordered by the term ordering of self.parent(), i.e. the list of coefficients matches the list of monomials returned by monomials().

EXAMPLES:

sage: R.<x,y,z> = PolynomialRing(QQ,3,order='degrevlex')
sage: f=23*x^6*y^7 + x^3*y+6*x^7*z
sage: f.coefficients()
[23, 6, 1]
sage: R.<x,y,z> = PolynomialRing(QQ,3,order='lex')
sage: f=23*x^6*y^7 + x^3*y+6*x^7*z
sage: f.coefficients()
[6, 23, 1]

Test the same stuff with base ring – different implementation:

sage: R.<x,y,z> = PolynomialRing(ZZ,3,order='degrevlex')
sage: f=23*x^6*y^7 + x^3*y+6*x^7*z
sage: f.coefficients()
[23, 6, 1]
sage: R.<x,y,z> = PolynomialRing(ZZ,3,order='lex')
sage: f=23*x^6*y^7 + x^3*y+6*x^7*z
sage: f.coefficients()
[6, 23, 1]

AUTHOR:

• Didier Deshommes

content()

Returns the content of this polynomial. Here, we define content as the gcd of the coefficients in the base ring.

EXAMPLES:

sage: R.<x,y> = ZZ[]
sage: f = 4*x+6*y
sage: f.content()
2
sage: f.content().parent()
Integer Ring

TESTS:

sage: R.<x,y> = QQ[]
sage: f = 4*x+6*y
sage: f.content()
1

denominator()

Return a denominator of self.

First, the lcm of the denominators of the entries of self is computed and returned. If this computation fails, the unit of the parent of self is returned.

Note that some subclases may implement its own denominator function.

Warning

This is not the denominator of the rational function defined by self, which would always be 1 since self is a polynomial.

EXAMPLES:

First we compute the denominator of a polynomial with integer coefficients, which is of course 1.

sage: R.<x,y> = ZZ[]
sage: f = x^3 + 17*y + x + y
sage: f.denominator()
1

Next we compute the denominator of a polynomial over a number field.

sage: R.<x,y> = NumberField(symbolic_expression(x^2+3)  ,'a')['x,y']
sage: f = (1/17)*x^19 + (1/6)*y - (2/3)*x + 1/3; f
1/17*x^19 - 2/3*x + 1/6*y + 1/3
sage: f.denominator()
102

Finally, we try to compute the denominator of a polynomial with coefficients in the real numbers, which is a ring whose elements do not have a denominator method.

sage: R.<a,b,c> = RR[]
sage: f = a + b + RR('0.3'); f
a + b + 0.300000000000000
sage: f.denominator()
1.00000000000000

derivative(*args)

The formal derivative of this polynomial, with respect to variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

See also

_derivative()

EXAMPLES:

Polynomials implemented via Singular:

sage: R.<x, y> = PolynomialRing(FiniteField(5))
sage: f = x^3*y^5 + x^7*y
sage: type(f)
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
sage: f.derivative(x)
2*x^6*y - 2*x^2*y^5
sage: f.derivative(y)
x^7

Generic multivariate polynomials:

sage: R.<t> = PowerSeriesRing(QQ)
sage: S.<x, y> = PolynomialRing(R)
sage: f = (t^2 + O(t^3))*x^2*y^3 + (37*t^4 + O(t^5))*x^3
sage: type(f)
<class 'sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict'>
sage: f.derivative(x)   # with respect to x
(2*t^2 + O(t^3))*x*y^3 + (111*t^4 + O(t^5))*x^2
sage: f.derivative(y)   # with respect to y
(3*t^2 + O(t^3))*x^2*y^2
sage: f.derivative(t)   # with respect to t (recurses into base ring)
(2*t + O(t^2))*x^2*y^3 + (148*t^3 + O(t^4))*x^3
sage: f.derivative(x, y) # with respect to x and then y
(6*t^2 + O(t^3))*x*y^2
sage: f.derivative(y, 3) # with respect to y three times
(6*t^2 + O(t^3))*x^2
sage: f.derivative()    # can't figure out the variable
...
ValueError: must specify which variable to differentiate with respect to

Polynomials over the symbolic ring (just for fun....):

sage: x = var("x")
sage: S.<u, v> = PolynomialRing(SR)
sage: f = u*v*x
sage: f.derivative(x) == u*v
True
sage: f.derivative(u) == v*x
True

gradient()

Return a list of partial derivatives of this polynomial, ordered by the variables of self.parent().

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(ZZ,3)
sage: f = x*y + 1
sage: f.gradient()
[y, x, 0]

homogenize(var='h')

Return self if self is homogeneous. Otherwise return a homogenized polynomial for self. If a string is given, return a polynomial in one more variable named after the string such that setting that variable equal to 1 yields self. This variable is added to the end of the variables. If a variable in self.parent() is given, this variable is used to homogenize the polynomial. If an integer is given, the variable with this index is used for homogenization.

INPUT:

• var – either a variable name, variable index or a variable (default: ‘h’).

OUTPUT:

a multivariate polynomial

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ,2)
sage: f = x^2 + y + 1 + 5*x*y^10
sage: g = f.homogenize('z'); g
5*x*y^10 + x^2*z^9 + y*z^10 + z^11
sage: g.parent()
Multivariate Polynomial Ring in x, y, z over Rational Field

sage: f.homogenize(x)
2*x^11 + x^10*y + 5*x*y^10

sage: f.homogenize(0)
2*x^11 + x^10*y + 5*x*y^10

sage: x, y = Zmod(3)['x', 'y'].gens()
sage: (x + x^2).homogenize(y)
x^2 + x*y

sage: x, y = Zmod(3)['x', 'y'].gens()
sage: (x + x^2).homogenize(y).parent()
Multivariate Polynomial Ring in x, y over Ring of integers modulo 3

sage: x, y = GF(3)['x', 'y'].gens()
sage: (x + x^2).homogenize(y)
x^2 + x*y

sage: x, y = GF(3)['x', 'y'].gens()
sage: (x + x^2).homogenize(y).parent()
Multivariate Polynomial Ring in x, y over Finite Field of size 3

TESTS:

sage: R = PolynomialRing(QQ, 'x', 5)
sage: p = R.random_element()
sage: q1 = p.homogenize()
sage: q2 = p.homogenize()
sage: q1.parent() is q2.parent()
True

is_generator()

Returns True if this polynomial is a generator of its parent.

EXAMPLES:

sage: R.<x,y>=ZZ[]
sage: x.is_generator()
True
sage: (x+y-y).is_generator()
True
sage: (x*y).is_generator()
False
sage: R.<x,y>=QQ[]
sage: x.is_generator()
True
sage: (x+y-y).is_generator()
True
sage: (x*y).is_generator()
False

is_homogeneous()

Return True if self is a homogeneous polynomial.

TESTS:

sage: from sage.rings.polynomial.multi_polynomial import MPolynomial
sage: P.<x, y> = PolynomialRing(QQ, 2)
sage: MPolynomial.is_homogeneous(x+y) 
True 
sage: MPolynomial.is_homogeneous(P(0)) 
True 
sage: MPolynomial.is_homogeneous(x+y^2) 
False 
sage: MPolynomial.is_homogeneous(x^2 + y^2) 
True 
sage: MPolynomial.is_homogeneous(x^2 + y^2*x) 
False 
sage: MPolynomial.is_homogeneous(x^2*y + y^2*x) 
True 

Note

This is a generic implementation which is likely overridden by subclasses.

jacobian_ideal()

Return the Jacobian ideal of the polynomial self.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: f = x^3 + y^3 + z^3
sage: f.jacobian_ideal()
Ideal (3*x^2, 3*y^2, 3*z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field

map_coefficients(f)

Returns a new element of self.parent() obtained by applying the function f to all of the coefficients of self.

EXAMPLES:

sage: k.<a> = GF(9); R.<x,y> = k[];  f = x*a + 2*x^3*y*a + a
sage: f.map_coefficients(lambda a : a + 1)
(-a + 1)*x^3*y + (a + 1)*x + (a + 1)

newton_polytope()

Return the Newton polytope of this polynomial.

You should have the optional polymake package installed.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: f = 1 + x*y + x^3 + y^3
sage: P = f.newton_polytope()
sage: P
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 3 vertices.
sage: P.is_simple()
True

TESTS:

sage: R.<x,y> = QQ[]
sage: R(0).newton_polytope()
The empty polyhedron in QQ^0.
sage: R(1).newton_polytope()
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex.

numerator()

Return a numerator of self computed as self * self.denominator()

Note that some subclases may implement its own numerator function.

Warning

This is not the numerator of the rational function defined by self, which would always be self since self is a polynomial.

EXAMPLES:

First we compute the numerator of a polynomial with integer coefficients, which is of course self.

sage: R.<x, y> = ZZ[]
sage: f = x^3 + 17*x + y + 1
sage: f.numerator()
x^3 + 17*x + y + 1
sage: f == f.numerator()
True

Next we compute the numerator of a polynomial over a number field.

sage: R.<x,y> = NumberField(symbolic_expression(x^2+3)  ,'a')['x,y']
sage: f = (1/17)*y^19 - (2/3)*x + 1/3; f
1/17*y^19 - 2/3*x + 1/3
sage: f.numerator()
3*y^19 - 34*x + 17
sage: f == f.numerator()
False

We try to compute the numerator of a polynomial with coefficients in the finite field of 3 elements.

sage: K.<x,y,z> = GF(3)['x, y, z']
sage: f = 2*x*z + 2*z^2 + 2*y + 1; f
-x*z - z^2 - y + 1
sage: f.numerator()
-x*z - z^2 - y + 1

We check that the computation the numerator and denominator are valid

sage: K=NumberField(symbolic_expression('x^3+2'),'a')['x']['s,t']
sage: f=K.random_element()
sage: f.numerator() / f.denominator() == f
True
sage: R=RR['x,y,z']
sage: f=R.random_element()
sage: f.numerator() / f.denominator() == f
True

polynomial(var)

Let var be one of the variables of the parent of self. This returns self viewed as a univariate polynomial in var over the polynomial ring generated by all the other variables of the parent.

EXAMPLES:

sage: R.<x,w,z> = QQ[]
sage: f = x^3 + 3*w*x + w^5 + (17*w^3)*x + z^5
sage: f.polynomial(x)
x^3 + (17*w^3 + 3*w)*x + w^5 + z^5
sage: parent(f.polynomial(x))
Univariate Polynomial Ring in x over Multivariate Polynomial Ring in w, z over Rational Field

sage: f.polynomial(w)
w^5 + 17*x*w^3 + 3*x*w + z^5 + x^3
sage: f.polynomial(z)
z^5 + w^5 + 17*x*w^3 + x^3 + 3*x*w
sage: R.<x,w,z,k> = ZZ[]
sage: f = x^3 + 3*w*x + w^5 + (17*w^3)*x + z^5 +x*w*z*k + 5
sage: f.polynomial(x)
x^3 + (17*w^3 + w*z*k + 3*w)*x + w^5 + z^5 + 5
sage: f.polynomial(w)
w^5 + 17*x*w^3 + (x*z*k + 3*x)*w + z^5 + x^3 + 5
sage: f.polynomial(z)
z^5 + x*w*k*z + w^5 + 17*x*w^3 + x^3 + 3*x*w + 5
sage: f.polynomial(k)
x*w*z*k + w^5 + z^5 + 17*x*w^3 + x^3 + 3*x*w + 5
sage: R.<x,y>=GF(5)[]
sage: f=x^2+x+y
sage: f.polynomial(x)
x^2 + x + y
sage: f.polynomial(y)
y + x^2 + x

truncate(var, n)

Returns a new multivariate polynomial obtained from self by deleting all terms that involve the given variable to a power at least n.

sage.rings.polynomial.multi_polynomial.is_MPolynomial(x)