Base class for elements of multivariate polynomial rings

Base class for elements of multivariate polynomial rings

class sage.rings.polynomial.multi_polynomial.MPolynomial

Bases: sage.structure.element.CommutativeRingElement

INPUT:

  • parent - a SageObject
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 sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular.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 \(\ZZ\) – 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:

Since trac ticket #10771, the gcd in QQ restricts to the gcd in ZZ:

sage: R.<x,y> = QQ[]
sage: f = 4*x+6*y
sage: f.content(); f.content().parent()
2
Rational Field
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

Check that the denominator is an element over the base whenever the base has no denominator function. This closes #9063

sage: R.<a,b,c> = GF(5)[]
sage: x = R(0)
sage: x.denominator()
1
sage: type(x.denominator())
<type 'sage.rings.finite_rings.integer_mod.IntegerMod_int'>
sage: type(a.denominator())
<type 'sage.rings.finite_rings.integer_mod.IntegerMod_int'>
sage: from sage.rings.polynomial.multi_polynomial_element import MPolynomial
sage: isinstance(a / b, MPolynomial)
False
sage: isinstance(a.numerator() / a.denominator(), MPolynomial)
True
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
Traceback (most recent call last):
...
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
inverse_mod(I)

Returns an inverse of self modulo the polynomial ideal \(I\), namely a multivariate polynomial \(f\) such that self * f - 1 belongs to \(I\).

INPUT:
  • I – an ideal of the polynomial ring in which self lives

OUTPUT:

  • a multivariate polynomial representing the inverse of f modulo I

EXAMPLES:

sage: R.<x1,x2> = QQ[]
sage: I = R.ideal(x2**2 + x1 - 2, x1**2 - 1)
sage: f = x1 + 3*x2^2; g = f.inverse_mod(I); g
1/16*x1 + 3/16
sage: (f*g).reduce(I)
1

Test a non-invertible element:

sage: R.<x1,x2> = QQ[]
sage: I = R.ideal(x2**2 + x1 - 2, x1**2 - 1)
sage: f = x1 + x2
sage: f.inverse_mod(I)
Traceback (most recent call last):
...
ArithmeticError: element is non-invertible
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
lift(I)

given an ideal I = (f_1,...,f_r) and some g (== self) in I, find s_1,...,s_r such that g = s_1 f_1 + ... + s_r f_r.

EXAMPLE:

sage: A.<x,y> = PolynomialRing(CC,2,order='degrevlex')
sage: I = A.ideal([x^10 + x^9*y^2, y^8 - x^2*y^7 ])
sage: f = x*y^13 + y^12
sage: M = f.lift(I)
sage: M
[y^7, x^7*y^2 + x^8 + x^5*y^3 + x^6*y + x^3*y^4 + x^4*y^2 + x*y^5 + x^2*y^3 + y^4]
sage: sum( map( mul , zip( M, I.gens() ) ) ) == f
True
map_coefficients(f, new_base_ring=None)

Returns the polynomial obtained by applying f to the non-zero coefficients of self.

If f is a sage.categories.map.Map, then the resulting polynomial will be defined over the codomain of f. Otherwise, the resulting polynomial will be over the same ring as self. Set new_base_ring to override this behaviour.

INPUT:

  • f – a callable that will be applied to the coefficients of self.
  • new_base_ring (optional) – if given, the resulting polynomial will be defined over this ring.

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)

Examples with different base ring:

sage: R.<r> = GF(9); S.<s> = GF(81)
sage: h = Hom(R,S)[0]; h
Ring morphism:
  From: Finite Field in r of size 3^2
  To:   Finite Field in s of size 3^4
  Defn: r |--> 2*s^3 + 2*s^2 + 1
sage: T.<X,Y> = R[]
sage: f = r*X+Y
sage: g = f.map_coefficients(h); g
(-s^3 - s^2 + 1)*X + Y
sage: g.parent()
Multivariate Polynomial Ring in X, Y over Finite Field in s of size 3^4
sage: h = lambda x: x.trace()
sage: g = f.map_coefficients(h); g
X - Y
sage: g.parent()
Multivariate Polynomial Ring in X, Y over Finite Field in r of size 3^2
sage: g = f.map_coefficients(h, new_base_ring=GF(3)); g
X - Y
sage: g.parent()
Multivariate Polynomial Ring in X, Y over Finite Field of size 3
newton_polytope()

Return the Newton polytope of this polynomial.

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 ZZ^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 ZZ^0
sage: R(1).newton_polytope()
A 0-dimensional polyhedron in ZZ^2 defined as the convex hull of 1 vertex
sage: R(x^2+y^2).newton_polytope().integral_points()
((0, 2), (1, 1), (2, 0))
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
sylvester_matrix(right, variable=None)

Given two nonzero polynomials self and right, returns the Sylvester matrix of the polynomials with respect to a given variable.

Note that the Sylvester matrix is not defined if one of the polynomials is zero.

INPUT:

  • self , right: multivariate polynomials
  • variable: optional, compute the Sylvester matrix with respect to this variable. If variable is not provided, the first variable of the polynomial ring is used.

OUTPUT:

  • The Sylvester matrix of self and right.

EXAMPLES:

sage: R.<x, y> = PolynomialRing(ZZ)
sage: f = (y + 1)*x + 3*x**2
sage: g = (y + 2)*x + 4*x**2
sage: M = f.sylvester_matrix(g, x)
sage: print M
[    3 y + 1     0     0]
[    0     3 y + 1     0]
[    4 y + 2     0     0]
[    0     4 y + 2     0]

If the polynomials share a non-constant common factor then the determinant of the Sylvester matrix will be zero:

sage: M.determinant()
0

sage: f.sylvester_matrix(1 + g, x).determinant()
y^2 - y + 7

If both polynomials are of positive degree with respect to variable, the determinant of the Sylvester matrix is the resultant:

sage: f = R.random_element(4)
sage: g = R.random_element(4)
sage: f.sylvester_matrix(g, x).determinant() == f.resultant(g, x)
True

TEST:

The variable is optional:

sage: f = x + y
sage: g = x + y
sage: f.sylvester_matrix(g)
[1 y]
[1 y]

Polynomials must be defined over compatible base rings:

sage: K.<x, y> = QQ[]
sage: f = x + y
sage: L.<x, y> = ZZ[]
sage: g = x + y
sage: R.<x, y> = GF(25, 'a')[]
sage: h = x + y
sage: f.sylvester_matrix(g, 'x')
[1 y]
[1 y]
sage: g.sylvester_matrix(h, 'x')
[1 y]
[1 y]
sage: f.sylvester_matrix(h, 'x')
Traceback (most recent call last):
...
TypeError: no common canonical parent for objects with parents: 'Multivariate Polynomial Ring in x, y over Rational Field' and 'Multivariate Polynomial Ring in x, y over Finite Field in a of size 5^2'
sage: K.<x, y, z> = QQ[]
sage: f = x + y
sage: L.<x, z> = QQ[]
sage: g = x + z
sage: f.sylvester_matrix(g)
[1 y]
[1 z]

Corner cases:

sage: K.<x ,y>=QQ[]
sage: f = x^2+1
sage: g = K(0)
sage: f.sylvester_matrix(g)
Traceback (most recent call last):
...
ValueError: The Sylvester matrix is not defined for zero polynomials
sage: g.sylvester_matrix(f)
Traceback (most recent call last):
...
ValueError: The Sylvester matrix is not defined for zero polynomials
sage: g.sylvester_matrix(g)
Traceback (most recent call last):
...
ValueError: The Sylvester matrix is not defined for zero polynomials
sage: K(3).sylvester_matrix(x^2)
[3 0]
[0 3]
sage: K(3).sylvester_matrix(K(4))
[]
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)

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