28.7 Multivariate Polynomials

Module: sage.rings.polynomial.multi_polynomial_element

Multivariate Polynomials

Author Log:

We verify Lagrange's four squares identity:

sage: R.<a0,a1,a2,a3,b0,b1,b2,b3> = ZZ[]
sage: (a0^2 + a1^2 + a2^2 + a3^2)*(b0^2 + b1^2 + b2^2 + b3^2) == (a0*b0 - a1*b1 - a2*b2 - a3*b3)^2 + (a0*b1 + a1*b0 + a2*b3 - a3*b2)^2 + (a0*b2 - a1*b3 + a2*b0 + a3*b1)^2 + (a0*b3 + a1*b2 - a2*b1 + a3*b0)^2
True

Module-level Functions

degree_lowest_rational_function( r, x)

Input:

r
- a multivariate rational function
x
- a multivariate polynomial ring generator x

Output:
integer
- the degree of r in x and its "leading" (in the x-adic sense) coefficient.

NOTES: This function is dependent on the ordering of a python dict. Thus, it isn't really mathematically well-defined. I think that it should made a method of the FractionFieldElement class and rewritten.

sage: R1 = PolynomialRing(FiniteField(5), 3, names = ["a","b","c"])
sage: F = FractionField(R1)
sage: a,b,c = R1.gens()
sage: f = 3*a*b^2*c^3+4*a*b*c
sage: g = a^2*b*c^2+2*a^2*b^4*c^7

Consider the quotient $ f/g = \frac{4 + 3 bc^{2}}{ac + 2 ab^{3}c^{6}}$ (note the cancellation).

sage: r = f/g; r
(-2*b*c^2 - 1)/(2*a*b^3*c^6 + a*c)
sage: degree_lowest_rational_function(r,a)
(-1, 3)
sage: degree_lowest_rational_function(r,b)
(0, 4)
sage: degree_lowest_rational_function(r,c)
(-1, 4)

is_MPolynomial( x)

Class: MPolynomial_element

class MPolynomial_element
MPolynomial_element( self, parent, x)

Functions: change_ring,$ \,$ element

Special Functions: __call__,$ \,$ __cmp__,$ \,$ __init__,$ \,$ __rpow__,$ \,$ _add_,$ \,$ _div_,$ \,$ _im_gens_,$ \,$ _lmul_,$ \,$ _mul_,$ \,$ _repr_,$ \,$ _rmul_,$ \,$ _sub_

__call__( self)

Evaluate this multi-variate polynomial at $ x$ , where $ x$ is either the tuple of values to substitute in, or one can use functional notation $ f(a_0,a_1,a_2, \ldots)$ to evaluate $ f$ with the ith variable replaced by $ a_i$ .

sage: R.<x,y> = QQ[]
sage: f = x^2 + y^2
sage: f(1,2)
5
sage: f((1,2))
5

sage: x = PolynomialRing(QQ,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

Author: David Kohel, 2005-09-27

__cmp__( self, right)

Compares right to self with respect to the term order of self.parent().

sage: R.<x,y,z>=PolynomialRing(GF(7),3,order='lex')
sage: x^1*y^2 > y^3*z^4
True
sage: x^3*y^2*z^4 < x^3*y^2*z^1
False

sage: R.<x,y,z>=PolynomialRing(QQ,3,order='deglex')
sage: x^1*y^2*z^3 > x^3*y^2*z^0
True
sage: x^1*y^2*z^4 < x^1*y^1*z^5
False

sage: R.<x,y,z>=PolynomialRing(ZZ,3,order='degrevlex')
sage: x^1*y^5*z^2 > x^4*y^1*z^3
True
sage: x^4*y^7*z^1 < x^4*y^2*z^3
False

_div_( self, right)

sage: R.<x,y> = QQ['x,y']
sage: f = (x + y)/3
sage: f.parent()
Multivariate Polynomial Ring in x, y over Rational Field

If we do the same over $ \mathbf{Z}$ the result is the same as multiplying by 1/3 (i.e. base extension).

sage: R.<x,y> = ZZ[]
sage: f = (x + y)/3      
sage: f.parent()
Multivariate Polynomial Ring in x, y over Rational Field
sage: f = (x + y) * 1/3      
sage: f.parent()
Multivariate Polynomial Ring in x, y over Rational Field

But we get a true fraction field if the denominator is not in the fration field of the basering.

sage: f = x/y
sage: f.parent()
Fraction Field of Multivariate Polynomial Ring in x, y over Integer Ring

_im_gens_( self, codomain, im_gens)

sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: f = R.hom([y,x], R)
sage: f(x^2 + 3*y^5)
3*x^5 + y^2

_lmul_( self, a)

Left Scalar Multiplication

Note that it is not really possible to do a meaningful example since sage mpoly rings refuse to have non-commutative bases.

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

_rmul_( self, a)

Right Scalar Multiplication

Note that it is not really possible to do a meaningful example since sage mpoly rings refuse to have non-commutative bases.

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

Class: MPolynomial_macaulay2_repr

class MPolynomial_macaulay2_repr
Multivariate polynomials that are representable in Macaulay2.

Special Functions: _macaulay2_

_macaulay2_( self, [macaulay2=Macaulay2])

Return corresponding Macaulay2 polynomial.

sage: R.<x,y> = GF(7)[]
sage: f = (x^3 + 2*y^2*x)^7; f
x^21 + 2*x^7*y^14
sage: macaulay2(R)                      # optional
ZZ/7 [x, y, MonomialOrder => GRevLex, MonomialSize => 16]
sage: h = f._macaulay2_(); print h      # optional
 21     7 14
x   + 2x y            
sage: R(h)                              # optional
x^21 + 2*x^7*y^14
sage: R(h^20) == f^20                   # optional
True

Class: MPolynomial_polydict

class MPolynomial_polydict
Multivariate polynomials implemented in pure python using polydicts.
MPolynomial_polydict( self, parent, x)

sage: R, x = PolynomialRing(QQ, 10, 'x').objgens()
sage: x
(x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)
sage: loads(dumps(x)) == x
True

Functions: coefficient,$ \,$ constant_coefficient,$ \,$ content,$ \,$ degree,$ \,$ degrees,$ \,$ dict,$ \,$ exponents,$ \,$ factor,$ \,$ gcd,$ \,$ inverse_of_unit,$ \,$ is_constant,$ \,$ is_generator,$ \,$ is_homogeneous,$ \,$ is_monomial,$ \,$ is_unit,$ \,$ is_univariate,$ \,$ lc,$ \,$ lift,$ \,$ lm,$ \,$ lt,$ \,$ monomial_coefficient,$ \,$ monomials,$ \,$ newton_polytope,$ \,$ nvariables,$ \,$ quo_rem,$ \,$ reduce,$ \,$ subs,$ \,$ total_degree,$ \,$ univariate_polynomial,$ \,$ variable,$ \,$ variables

coefficient( self, degrees)

Return the coefficient of the variables with the degrees specified in the python dictionary degrees. Mathematically, this is the coefficient in the base ring adjoined by the variables of this ring not listed in degrees. However, the result has the same parent as this polynomial.

This function contrasts with the function monomial_coefficient which returns the coefficient in the base ring of a monomial.

Input:

degrees
- Can be any of:
- a dictionary of degree restrictions
- a list of degree restrictions (with None in the unrestricted variables)
- a monomial (very fast, but not as flexible)

Output: element of the parent of self

SEE ALSO: For coefficients of specific monomials, look at efmonomial_coefficient.

sage: R.<x, y> = ZZ[]
sage: f = 2 * x * y
sage: c = f.coefficient({x:1,y:1}); c
2
sage: c.parent()
Multivariate Polynomial Ring in x, y over Integer Ring
sage: c in PolynomialRing(IntegerRing(), 2, names = ['x','y'])
True
sage: f = y^2 - x^9 - 7*x + 5*x*y
sage: f.coefficient({y:1})
5*x
sage: f.coefficient({y:0})
-x^9 - 7*x
sage: f.coefficient({x:0,y:0})
0
sage: f=(1+y+y^2)*(1+x+x^2)
sage: f.coefficient({x:0})
y^2 + y + 1
sage: f.coefficient([0,None])
y^2 + y + 1
sage: f.coefficient(x)
y^2 + y + 1
sage: # Be aware that this may not be what you think!
sage: # The physical appearance of the variable x is deceiving -- particularly if the exponent would be a variable.
sage: f.coefficient(x^0) # outputs the full polynomial
x^2*y^2 + x^2*y + x*y^2 + x^2 + x*y + y^2 + x + y + 1

sage: R.<x,y> = RR[]
sage: f=x*y+5
sage: c=f.coefficient({x:0,y:0}); c
5.00000000000000
sage: parent(c)
Multivariate Polynomial Ring in x, y over Real Field with 53 bits of
precision

Author: Joel B. Mohler (2007.10.31)

constant_coefficient( self)

Return the constant coefficient of this multivariate polynomial.

sage: R.<x,y> = ZZ[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.constant_coefficient()
5
sage: f = 3*x^2 
sage: f.constant_coefficient()
0

content( self)

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

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

degree( self, [x=None])

Return the degree of self in x, where x must be one of the generators for the parent of self.

Input:

x
- multivariate polynomial (a generator of the parent of self) If x is not specified (or is None), return the total degree, which is the maximum degree of any monomial.

Output: integer

sage: R.<x,y> = QQ[]
sage: f = y^2 - x^9 - x
sage: f.degree(x)
9
sage: f.degree(y)
2
sage: (y^10*x - 7*x^2*y^5 + 5*x^3).degree(x)
3
sage: (y^10*x - 7*x^2*y^5 + 5*x^3).degree(y)
10

degrees( self)

Returns a list (precisely - an ETuple) with the degree of each variable in this polynomial. The list of degrees is, of course, ordered by the order of the generators.

sage: R.<x,y,z>=PolynomialRing(ZZ)
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.degrees()
(2, 2, 0)
sage: f = x^2+z^2
sage: f.degrees()
(2, 0, 2)
sage: f.total_degree()  # this simply illustrates that total degree is not the sum of the degrees
2
sage: R.<x,y,z,u>=PolynomialRing(ZZ)
sage: f=(1-x)*(1+y+z+x^3)^5
sage: f.degrees()
(16, 5, 5, 0)
sage: R(0).degrees()
(0, 0, 0, 0)

dict( self)

Return underlying dictioniary with keys the exponents and values the coefficients of this polynomial.

exponents( self)

Return the exponents of the monomials appearing in self.

sage: R.<a,b,c> = PolynomialRing(QQ, 3)
sage: f = a^3 + b + 2*b^2
sage: f.exponents()
[(3, 0, 0), (0, 2, 0), (0, 1, 0)]

factor( self)

Compute the irreducible factorization of this polynomial.

ALGORITHM: Use Singular.

sage: R.<x, y> = QQ[]
sage: f = (x^3 + 2*y^2*x) * (x^2 + x + 1); f
x^5 + 2*x^3*y^2 + x^4 + 2*x^2*y^2 + x^3 + 2*x*y^2
sage: F = f.factor()
sage: F
x * (x^2 + x + 1) * (x^2 + 2*y^2)

Next we factor the same polynomial, but over the finite field of order $ 3$ .

sage: R.<x, y> = GF(3)[]
sage: f = (x^3 + 2*y^2*x) * (x^2 + x + 1); f
x^5 - x^3*y^2 + x^4 - x^2*y^2 + x^3 - x*y^2
sage: F = f.factor()
sage: F # order is somewhat random
(-1) * x * (-x + y) * (x + y) * (x - 1)^2

Next we factor a polynomial over a number field.

sage: p = var('p')
sage: K.<s> = NumberField(p^3-2)
sage: KXY.<x,y> = K[]
sage: factor(x^3 - 2*y^3)
(x + (-s)*y) * (x^2 + s*x*y + s^2*y^2)
sage: k = (x^3-2*y^3)^5*(x+s*y)^2*(2/3 + s^2)
sage: k.factor()
(s^2 + 2/3) * (x + s*y)^2 * (x + (-s)*y)^5 * (x^2 + s*x*y + s^2*y^2)^5

This shows that ticket #2780 is fixed, i.e. that the unit part of the factorization is set correctly:

sage: x = var('x')
sage: K.<a> = NumberField(x^2 + 1)
sage: R.<y, z> = PolynomialRing(K)
sage: f = 2*y^2 + 2*z^2
sage: F = f.factor(); F.unit_part()
2

gcd( self, f)

Compute the greatest common divisor of this polynomial and f.

ALGORITHM: Use Singular.

sage: R.<x,y> = QQ[]
sage: f = (x^3 + 2*y^2*x)^2
sage: g = x^2*y^2
sage: f.gcd(g)
x^2

This also works correctly over ZZ:

sage: R.<x,y> = ZZ[]  
sage: gcd(2*x,4*x)
2*x
sage: gcd(2*x,4*x)
2*x
sage: gcd(9*x*y*(x^2-y^2), 15*x*y^2*(x^2+y^2))
3*x*y

We compute a gcd over a finite field.

sage: F.<u> = GF(31^2)
sage: R.<x,y,z> = F[]
sage: p = x^3 + (1+u)*y^3 + z^3
sage: q = p^3 * (x - y + z*u)
sage: gcd(p,q)
x^3 + (u + 1)*y^3 + z^3
sage: gcd(p,q)  # yes, twice -- tests that singular ring is properly set.
x^3 + (u + 1)*y^3 + z^3

We compute a gcd over a number field:

sage: x = polygen(QQ)
sage: F.<u> = NumberField(x^3 - 2)
sage: R.<x,y,z> = F[]
sage: p = x^3 + (1+u)*y^3 + z^3
sage: q = p^3 * (x - y + z*u)
sage: gcd(p,q)
x^3 + (u + 1)*y^3 + z^3

is_constant( self)

True if polynomial is constant, and False otherwise.

sage: R.<x,y> = ZZ[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.is_constant()
False
sage: g = 10*x^0
sage: g.is_constant()
True

is_generator( self)

Returns True if self is a generator of it's parent.

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

is_homogeneous( self)

Return True if self is a homogeneous polynomial.

sage: R.<x,y> = QQ[]
sage: (x+y).is_homogeneous()
True
sage: (x.parent()(0)).is_homogeneous()
True
sage: (x+y^2).is_homogeneous()
False
sage: (x^2 + y^2).is_homogeneous()
True
sage: (x^2 + y^2*x).is_homogeneous()
False
sage: (x^2*y + y^2*x).is_homogeneous()
True

is_monomial( self)

Returns True if self is a monomial. Here, we define a monomial as a product of variables with a coefficient (possibly not equal to 1).

sage: R.<x,y>=ZZ[]
sage: x.is_monomial()
True
sage: (x+2*y).is_monomial()
False
sage: (2*x).is_monomial()
True

is_unit( self)

Return True if self is a unit.

sage: R.<x,y> = ZZ[]
sage: (x+y).is_unit()
False
sage: R(0).is_unit()
False
sage: R(-1).is_unit()
True
sage: R(-1 + x).is_unit()
False
sage: R(2).is_unit()
False

is_univariate( self)

Returns True if this multivariate polynomial is univariate and False otherwise.

sage: R.<x,y> = QQ[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.is_univariate()
False
sage: g = f.subs({x:10}); g
700*y^2 - 2*y + 305
sage: g.is_univariate()
True
sage: f = x^0
sage: f.is_univariate()
True

lc( self)

Returns the leading coefficent of self i.e., self.coefficient(self.lm())

sage: R.<x,y,z>=ZZ[]
sage: f=3*x^2-y^2-x*y
sage: f.lc()
3

lift( self, 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

ALGORITHM: Use Singular.

sage: A.<x,y> = PolynomialRing(QQ,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

lm( self)

Returns the lead monomial of self with respect to the term order of self.parent().

sage: R.<x,y,z>=PolynomialRing(GF(7),3,order='lex')
sage: (x^1*y^2 + y^3*z^4).lm()
x*y^2
sage: (x^3*y^2*z^4 + x^3*y^2*z^1).lm()
x^3*y^2*z^4

sage: R.<x,y,z>=PolynomialRing(QQ,3,order='deglex')
sage: (x^1*y^2*z^3 + x^3*y^2*z^0).lm()
x*y^2*z^3
sage: (x^1*y^2*z^4 + x^1*y^1*z^5).lm()
x*y^2*z^4

sage: R.<x,y,z>=PolynomialRing(ZZ,3,order='degrevlex')
sage: (x^1*y^5*z^2 + x^4*y^1*z^3).lm()
x*y^5*z^2
sage: (x^4*y^7*z^1 + x^4*y^2*z^3).lm()
x^4*y^7*z

lt( self)

Returns the leading term of self i.e., self.lc()*self.lm(). The notion of "leading term" depends on the ordering defined in the parent ring.

sage: R.<x,y,z>=PolynomialRing(ZZ)
sage: f=3*x^2-y^2-x*y
sage: f.lt()
3*x^2
sage: R.<x,y,z>=PolynomialRing(ZZ,order="invlex")
sage: f=3*x^2-y^2-x*y
sage: f.lt()
-y^2

monomial_coefficient( self, mon)

Return the coefficient in the base ring of the monomial mon in self, where mon must have the same parent as self.

This function contrasts with the function coefficient which returns the coefficient of a monomial viewing this polynomial in a polynomial ring over a base ring having fewer variables.

Input:

mon
- a monomial

Output: coefficient in base ring

SEE ALSO: For coefficients in a base ring of fewer variables, look at efcoefficient.

The parent of the return is a member of the base ring.

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

The parent of the return is a member of the base ring.

sage: f = 2 * x * y
sage: c = f.monomial_coefficient(x*y); c
2
sage: c.parent()
Integer Ring

sage: f = y^2 + y^2*x - x^9 - 7*x + 5*x*y
sage: f.monomial_coefficient(y^2)
1
sage: f.monomial_coefficient(x*y)
5
sage: f.monomial_coefficient(x^9)
-1
sage: f.monomial_coefficient(x^10)
0

sage: var('a')
a
sage: K.<a> = NumberField(a^2+a+1)
sage: P.<x,y> = K[]
sage: f=(a*x-1)*((a+1)*y-1); f
-x*y + (-a)*x + (-a - 1)*y + 1
sage: f.monomial_coefficient(x)
-a

monomials( self)

Returns the list of monomials in self. The returned list is decreasingly ordered by the term ordering of self.parent().

Output: list of MPolynomials representing Monomials

sage: R.<x,y> = ZZ[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.monomials()
[x^2*y^2, x^2, y, 1]

sage: R.<fx,fy,gx,gy> = ZZ[]
sage: F = ((fx*gy - fy*gx)^3)
sage: F
-fy^3*gx^3 + 3*fx*fy^2*gx^2*gy - 3*fx^2*fy*gx*gy^2 + fx^3*gy^3
sage: F.monomials()
[fy^3*gx^3, fx*fy^2*gx^2*gy, fx^2*fy*gx*gy^2, fx^3*gy^3]
sage: F.coefficients()
[-1, 3, -3, 1]
sage: sum(map(mul,zip(F.coefficients(),F.monomials()))) == F
True

newton_polytope( self)

Return the Newton polytope of this polynomial.

You should have the optional polymake package installed.

sage: R.<x,y> = PolynomialRing(QQ,2)
sage: f = 1 + x*y + x^3 + y^3
sage: P = f.newton_polytope()
sage: P
Convex hull of points [[1, 0, 0], [1, 0, 3], [1, 1, 1], [1, 3, 0]]
sage: P.facets()
[(0, 1, 0), (3, -1, -1), (0, 0, 1)]
sage: P.is_simple()
True

nvariables( self)

Number of variables in this polynomial

sage: R.<x,y> = ZZ[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.nvariables ()
2
sage: g = f.subs({x:10}); g
700*y^2 - 2*y + 305
sage: g.nvariables ()
1

quo_rem( self, right)

Returns quotient and remainder of self and right.

ALGORITHM: Use Singular.

reduce( self, I)

Reduce this polynomial by the the polynomials in I.

Input:

I
- a list of polynomials or an ideal

sage: P.<x,y,z> = ZZ[]
sage: f1 = -2 * x^2 + x^3
sage: f2 = -2 * y + x* y
sage: f3 = -x^2 + y^2
sage: F = Ideal([f1,f2,f3])
sage: g = x*y - 3*x*y^2
sage: g.reduce(F)
6*y^2 - 2*y
sage: g.reduce(F.gens())
6*y^2 - 2*y

sage: f = 3*x
sage: f.reduce([2*x,y])
x

subs( self, [fixed=None])

Fixes some given variables in a given multivariate polynomial and returns the changed multivariate polynomials. The polynomial itself is not affected. The variable,value pairs for fixing are to be provided as a dictionary of the form variable:value.

This is a special case of evaluating the polynomial with some of the variables constants and the others the original variables.

Input:

fixed
- (optional) dictionary of inputs
**kw
- named parameters

Output: new MPolynomial

sage: R.<x,y> = ZZ[]
sage: f = x^2 + y + x^2*y^2 + 5
sage: f((5,y))
25*y^2 + y + 30
sage: f.subs({x:5})
25*y^2 + y + 30

total_degree( self)

Return the total degree of self, which is the maximum degree of any monomial in self.

sage: R.<x,y,z> = QQ[]
sage: f=2*x*y^3*z^2
sage: f.total_degree()
6
sage: f=4*x^2*y^2*z^3
sage: f.total_degree()
7
sage: f=99*x^6*y^3*z^9
sage: f.total_degree()
18
sage: f=x*y^3*z^6+3*x^2
sage: f.total_degree()
10
sage: f=z^3+8*x^4*y^5*z
sage: f.total_degree()
10
sage: f=z^9+10*x^4+y^8*x^2
sage: f.total_degree()
10

univariate_polynomial( self, [R=None])

Returns a univariate polynomial associated to this multivariate polynomial.

Input:

R
- (default: None) PolynomialRing

If this polynomial is not in at most one variable, then a ValueError exception is raised. This is checked using the is_univariate() method. The new Polynomial is over the same base ring as the given MPolynomial and in the variable 'x' if no ring 'ring' is provided.

sage: R.<x,y> = ZZ[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.univariate_polynomial()
Traceback (most recent call last):
...
TypeError: polynomial must involve at most one variable
sage: g = f.subs({x:10}); g
700*y^2 - 2*y + 305
sage: g.univariate_polynomial ()
700*x^2 - 2*x + 305
sage: g.univariate_polynomial(PolynomialRing(QQ,'z'))
700*z^2 - 2*z + 305

variable( self, i)

Returns $ i$ -th variable occuring in this polynomial.

sage: R.<x,y> = ZZ[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.variable(0)
x
sage: f.variable(1)
y

variables( self)

Returns the list of variables occuring in this polynomial.

sage: R.<x,y> = ZZ[]
sage: f = 3*x^2 - 2*y + 7*x^2*y^2 + 5
sage: f.variables()
[x, y]
sage: g = f.subs({x:10}); g
700*y^2 - 2*y + 305
sage: g.variables()
[y]

Special Functions: __eq__,$ \,$ __floordiv__,$ \,$ __getitem__,$ \,$ __init__,$ \,$ __iter__,$ \,$ __ne__,$ \,$ __neg__,$ \,$ _derivative,$ \,$ _homogenize,$ \,$ _latex_,$ \,$ _repr_,$ \,$ _repr_with_changed_varnames

__eq__( self, right)

__floordiv__( self, right)

Quotient of division of self by other. This is denoted //.

NOTES: It's not clear to me that this is well-defined if self is not exactly divisible by other.

sage: R.<x,y>=ZZ[]
sage: 2*x*y//y
2*x
sage: 2*x//y
0
sage: 2*x//4
0
sage: type(0//y)
<class 'sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict
'>

__getitem__( self, x)

Input:

x
- a tuple or, in case of a single-variable MPolynomial ring x can also be an integer.

sage: R.<x, y> = PolynomialRing(QQ, 2)
sage: f = -10*x^3*y + 17*x*y
sage: f[3,1]
-10
sage: f[1,1]
17
sage: f[0,1]
0

sage: R.<x> = PolynomialRing(GF(7),1); R
Multivariate Polynomial Ring in x over Finite Field of size 7
sage: f = 5*x^2 + 3; f
-2*x^2 + 3
sage: f[2]
5

__iter__( self)

Facilitates iterating over the monomials of self, returning tuples of the form (coeff, mon) for each non-zero monomial.

sage: R = ZZ['t']
sage: P.<x,y,z> = PolynomialRing(R,3)
sage: f = 3*x^3*y + 16*x + 7
sage: [(c,m) for c,m in f]
[(3, x^3*y), (16, x), (7, 1)]
sage: f = P.random_element(10,10)
sage: sum(c*m for c,m in f) == f
True

__ne__( self, right)

__neg__( self)

sage: R.<x,y>=ZZ[]
sage: -x
-x
sage: -(y-1)
-y + 1

_derivative( self, [var=None])

Differentiates self with respect to variable var.

If var is not one of the generators of this ring, _derivative(var) is called recursively on each coefficient of this polynomial.

SEE ALSO: self.derivative()

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)._derivative(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

_homogenize( self, var)

Return self if self is homogeneous. Otherwise return a homogenized polynomial constructed by modifying the degree of the variable with index var.

Input:

var
- an integer indicating which variable to use to homogenize (0 <= var < parent(self).ngens())

Output: a multivariate polynomial

sage: P.<x,y> = ZZ[]
sage: f = x^2 + y + 1 + 5*x*y^1
sage: g = f.homogenize('z'); g # indirect doctest
x^2 + 5*x*y + y*z + z^2
sage: g.parent()
Multivariate Polynomial Ring in x, y, z over Integer Ring

SEE: self.homogenize

_latex_( self)

sage: R.<x,y>=ZZ[]
sage: latex(-x^2-y+1)
-x^{2} - y + 1
sage: K.<I>=QuadraticField(-1)
sage: R.<x,y>=K[]
sage: latex(-I*y+I*x^2)
I x^{2} + (-I) y

_repr_( self)

sage: R.<x,y>=ZZ[]
sage: repr(-x^2-y+1)  # indirect doc-test
'-x^2 - y + 1'
sage: K.<I>=QuadraticField(-1)
sage: R.<x,y>=K[]
sage: repr(-I*y-x^2)  # indirect doc-test
'-x^2 + (-I)*y'

_repr_with_changed_varnames( self, varnames)

sage: R.<x,y>=ZZ[]
sage: f=-x^2-y+1
sage: f._repr_with_changed_varnames(['jack','jill'])
'-jack^2 - jill + 1'

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