Multivariate Polynomials via libSINGULAR
This module implements specialized and optimized implementations for multivariate polynomials over many coefficient rings, via a shared library interface to SINGULAR. In particular, the following coefficient rings are supported by this implementation:
AUTHORS:
The libSINGULAR interface was implemented by
TODO:
EXAMPLES:
We show how to construct various multivariate polynomial rings:
sage: P.<x,y,z> = QQ[]
sage: P
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: f = 27/113 * x^2 + y*z + 1/2; f
27/113*x^2 + y*z + 1/2
sage: P.term_order()
Degree reverse lexicographic term order
sage: P = PolynomialRing(GF(127),3,names='abc', order='lex')
sage: P
Multivariate Polynomial Ring in a, b, c over Finite Field of size 127
sage: a,b,c = P.gens()
sage: f = 57 * a^2*b + 43 * c + 1; f
57*a^2*b + 43*c + 1
sage: P.term_order()
Lexicographic term order
sage: z = QQ['z'].0
sage: K.<s> = NumberField(z^2 - 2)
sage: P.<x,y> = PolynomialRing(K, 2)
sage: 1/2*s*x^2 + 3/4*s
(1/2*s)*x^2 + (3/4*s)
sage: P.<x,y,z> = ZZ[]; P
Multivariate Polynomial Ring in x, y, z over Integer Ring
sage: P.<x,y,z> = Zmod(2^10)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 1024
sage: P.<x,y,z> = Zmod(3^10)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 59049
sage: P.<x,y,z> = Zmod(2^100)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 1267650600228229401496703205376
sage: P.<x,y,z> = Zmod(2521352)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 2521352
sage: type(P)
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>
sage: P.<x,y,z> = Zmod(25213521351515232)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 25213521351515232
sage: type(P)
<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_with_category'>
We construct the Frobenius morphism on over
:
sage: R.<x,y,z> = PolynomialRing(GF(5), 3)
sage: frob = R.hom([x^5, y^5, z^5])
sage: frob(x^2 + 2*y - z^4)
-z^20 + x^10 + 2*y^5
sage: frob((x + 2*y)^3)
x^15 + x^10*y^5 + 2*x^5*y^10 - 2*y^15
sage: (x^5 + 2*y^5)^3
x^15 + x^10*y^5 + 2*x^5*y^10 - 2*y^15
We make a polynomial ring in one variable over a polynomial ring in two variables:
sage: R.<x, y> = PolynomialRing(QQ, 2)
sage: S.<t> = PowerSeriesRing(R)
sage: t*(x+y)
(x + y)*t
TESTS:
sage: P.<x,y,z> = QQ[]
sage: loads(dumps(P)) == P
True
sage: loads(dumps(x)) == x
True
sage: P.<x,y,z> = GF(2^8,'a')[]
sage: loads(dumps(P)) == P
True
sage: loads(dumps(x)) == x
True
sage: P.<x,y,z> = GF(127)[]
sage: loads(dumps(P)) == P
True
sage: loads(dumps(x)) == x
True
sage: P.<x,y,z> = GF(127)[]
sage: loads(dumps(P)) == P
True
sage: loads(dumps(x)) == x
True
sage: Rt.<t> = PolynomialRing(QQ,1)
sage: p = 1+t
sage: R.<u,v> = PolynomialRing(QQ, 2)
sage: p(u/v)
(u + v)/v
Check if #6160 is fixed:
sage: x=var('x')
sage: K.<j> = NumberField(x-1728)
sage: R.<b,c> = K[]
sage: b-j*c
b - 1728*c
Bases: sage.rings.polynomial.multi_polynomial_ring_generic.MPolynomialRing_generic
Construct a multivariate polynomial ring subject to the following conditions:
INPUT:
QQ or absolute number field)
n - number of variables (must be at least 1)
names - names of ring variables, may be string of list/tuple
order - term order (default: degrevlex)
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: P
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: f = 27/113 * x^2 + y*z + 1/2; f
27/113*x^2 + y*z + 1/2
sage: P.term_order()
Degree reverse lexicographic term order
sage: P = PolynomialRing(GF(127),3,names='abc', order='lex')
sage: P
Multivariate Polynomial Ring in a, b, c over Finite Field of size 127
sage: a,b,c = P.gens()
sage: f = 57 * a^2*b + 43 * c + 1; f
57*a^2*b + 43*c + 1
sage: P.term_order()
Lexicographic term order
sage: z = QQ['z'].0
sage: K.<s> = NumberField(z^2 - 2)
sage: P.<x,y> = PolynomialRing(K, 2)
sage: 1/2*s*x^2 + 3/4*s
(1/2*s)*x^2 + (3/4*s)
sage: P.<x,y,z> = ZZ[]; P
Multivariate Polynomial Ring in x, y, z over Integer Ring
sage: P.<x,y,z> = Zmod(2^10)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 1024
sage: P.<x,y,z> = Zmod(3^10)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 59049
sage: P.<x,y,z> = Zmod(2^100)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 1267650600228229401496703205376
sage: P.<x,y,z> = Zmod(2521352)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 2521352
sage: type(P)
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular'>
sage: P.<x,y,z> = Zmod(25213521351515232)[]; P
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 25213521351515232
sage: type(P)
<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_with_category'>
sage: P.<x,y,z> = PolynomialRing(Integers(2^32),order='lex')
sage: P(2^32-1)
4294967295
TEST:
Make sure that a faster coercion map from the base ring is used; see trac ticket #9944:
sage: R.<x,y> = PolynomialRing(ZZ)
sage: R.coerce_map_from(R.base_ring())
Polynomial base injection morphism:
From: Integer Ring
To: Multivariate Polynomial Ring in x, y over Integer Ring
alias of MPolynomial_libsingular
Returns the n-th generator of this multivariate polynomial ring.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: P.gen(),P.gen(1)
(x, y)
sage: P = PolynomialRing(GF(127),1000,'x')
sage: P.gen(500)
x500
sage: P.<SAGE,SINGULAR> = QQ[] # weird names
sage: P.gen(1)
SINGULAR
Create an ideal in this polynomial ring.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: sage.rings.ideal.Katsura(P)
Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y) of Multivariate Polynomial Ring in x, y, z over Rational Field
sage: P.ideal([x + 2*y + 2*z-1, 2*x*y + 2*y*z-y, x^2 + 2*y^2 + 2*z^2-x])
Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 + 2*y^2 + 2*z^2 - x) of Multivariate Polynomial Ring in x, y, z over Rational Field
Return a list of all monomials that divide t.
Coefficients are ignored.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: P.monomial_all_divisors(x^2*z^3)
[x, x^2, z, x*z, x^2*z, z^2, x*z^2, x^2*z^2, z^3, x*z^3, x^2*z^3]
ALGORITHM: addwithcarry idea by Toon Segers
Return False if a does not divide b and True otherwise.
Coefficients are ignored.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: P.monomial_divides(x*y*z, x^3*y^2*z^4)
True
sage: P.monomial_divides(x^3*y^2*z^4, x*y*z)
False
TESTS:
sage: P.<x,y,z> = QQ[]
sage: P.monomial_divides(P(1), P(0))
True
sage: P.monomial_divides(P(1), x)
True
LCM for monomials. Coefficients are ignored.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: P.monomial_lcm(3/2*x*y,x)
x*y
TESTS:
sage: R.<x,y,z> = QQ[]
sage: P.<x,y,z> = QQ[]
sage: P.monomial_lcm(x*y,R.gen())
x*y
sage: P.monomial_lcm(P(3/2),P(2/3))
1
sage: P.monomial_lcm(x,P(1))
x
Return True if h and g are pairwise prime. Both are treated as monomials.
Coefficients are ignored.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: P.monomial_pairwise_prime(x^2*z^3, y^4)
True
sage: P.monomial_pairwise_prime(1/2*x^3*y^2, 3/4*y^3)
False
TESTS:
sage: Q.<x,y,z> = QQ[]
sage: P.<x,y,z> = QQ[]
sage: P.monomial_pairwise_prime(x^2*z^3, Q('y^4'))
True
sage: P.monomial_pairwise_prime(1/2*x^3*y^2, Q(0))
True
sage: P.monomial_pairwise_prime(P(1/2),x)
False
Return f/g, where both f and`` g are treated as monomials.
Coefficients are ignored by default.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: P.monomial_quotient(3/2*x*y,x)
y
sage: P.monomial_quotient(3/2*x*y,x,coeff=True)
3/2*y
Note, that behaves different if coeff=True:
sage: P.monomial_quotient(2*x,3*x)
1
sage: P.<x,y> = PolynomialRing(ZZ)
sage: P.monomial_quotient(2*x,3*x,coeff=True)
Traceback (most recent call last):
...
ArithmeticError: Cannot divide these coefficients.
TESTS:
sage: R.<x,y,z> = QQ[]
sage: P.<x,y,z> = QQ[]
sage: P.monomial_quotient(x*y,x)
y
sage: P.monomial_quotient(x*y,R.gen())
y
sage: P.monomial_quotient(P(0),P(1))
0
sage: P.monomial_quotient(P(1),P(0))
Traceback (most recent call last):
...
ZeroDivisionError
sage: P.monomial_quotient(P(3/2),P(2/3), coeff=True)
9/4
sage: P.monomial_quotient(x,y) # Note the wrong result
x*y^1048575*z^1048575 # 64-bit
x*y^65535*z^65535 # 32-bit
sage: P.monomial_quotient(x,P(1))
x
Warning
Assumes that the head term of f is a multiple of the head term of g and return the multiplicant m. If this rule is violated, funny things may happen.
Try to find a g in G where g.lm() divides f. If found (flt,g) is returned, (0,0) otherwise, where flt is f/g.lm().
It is assumed that G is iterable and contains only elements in this polynomial ring.
Coefficients are ignored.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: f = x*y^2
sage: G = [ 3/2*x^3 + y^2 + 1/2, 1/4*x*y + 2/7, 1/2 ]
sage: P.monomial_reduce(f,G)
(y, 1/4*x*y + 2/7)
TESTS:
sage: P.<x,y,z> = QQ[]
sage: f = x*y^2
sage: G = [ 3/2*x^3 + y^2 + 1/2, 1/4*x*y + 2/7, 1/2 ]
sage: P.monomial_reduce(P(0),G)
(0, 0)
sage: P.monomial_reduce(f,[P(0)])
(0, 0)
Returns the number of variables in this multivariate polynomial ring.
EXAMPLES:
sage: P.<x,y> = QQ[]
sage: P.ngens()
2
sage: k.<a> = GF(2^16)
sage: P = PolynomialRing(k,1000,'x')
sage: P.ngens()
1000
Bases: sage.rings.polynomial.multi_polynomial.MPolynomial
A multivariate polynomial implemented using libSINGULAR.
Return self + m*q, where m must be a monomial and q a polynomial.
INPUT:
EXAMPLES:
sage: P.<x,y,z>=PolynomialRing(QQ,3)
sage: x.add_m_mul_q(y,z)
y*z + x
TESTS:
sage: R.<x,y,z>=PolynomialRing(QQ,3)
sage: P.<x,y,z>=PolynomialRing(QQ,3)
sage: P(0).add_m_mul_q(P(0),P(1))
0
sage: x.add_m_mul_q(R.gen(),R.gen(1))
x*y + x
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:
Note
For coefficients of specific monomials, look at monomial_coefficient().
EXAMPLES:
sage: R.<x,y> = QQ[]
sage: f=x*y+y+5
sage: f.coefficient({x:0,y:1})
1
sage: f.coefficient({x:0})
y + 5
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
Be aware that this may not be what you think! 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> = GF(389)[]
sage: f=x*y+5
sage: c=f.coefficient({x:0,y:0}); c
5
sage: parent(c)
Multivariate Polynomial Ring in x, y over Finite Field of size 389
AUTHOR:
Return the nonzero coefficients of this polynomial in a list. The returned list is decreasingly ordered by the term ordering of the parent.
EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ, 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, order='lex')
sage: f=23*x^6*y^7 + x^3*y+6*x^7*z
sage: f.coefficients()
[6, 23, 1]
AUTHOR:
Return the constant coefficient of this multivariate polynomial.
EXAMPLES:
sage: P.<x, y> = QQ[]
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
Return the maximal degree of this polynomial in x, where x must be one of the generators for the parent of this polynomial.
INPUT:
EXAMPLES:
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
TESTS:
sage: P.<x, y> = QQ[]
sage: P(0).degree(x)
-1
sage: P(1).degree(x)
0
With a matrix term ordering, the grading of the generators is determined by the first row of the matrix. This affects the behavior of degree() when no variable is specified. To evaluate the degree with a standard grading, use the optional argument std_grading=True.
sage: tord = TermOrder(matrix([3,0,1,1,1,0,1,0,0])) sage: R.<x,y,z> = PolynomialRing(QQ,’x’,3,order=tord) sage: (x^3*y+x*z^4).degree() 9 sage: (x^3*y+x*z^4).degree(std_grading=True) 5 sage: x.degree(x), y.degree(y), z.degree(z) (1, 1, 1)
The following example is inspired by trac 11652:
sage: R.<p,q,t> = ZZ[]
sage: poly = p+q^2+t^3
sage: poly = poly.polynomial(t)[0]
sage: poly
q^2 + p
There is no canonical coercion from R to the parent of poly, so this doesn’t work:
sage: poly.degree(q)
Traceback (most recent call last):
...
TypeError: argument must canonically coerce to parent
Using a non-canonical coercion does work, but we require this to be done explicitly, since it can lead to confusing results if done automatically:
sage: poly.degree(poly.parent()(q))
2
sage: poly.degree(poly.parent()(p))
1
sage: T.<x,y> = ZZ[]
sage: poly.degree(poly.parent()(x)) # noncanonical coercions can be confusing
1
The argument to degree has to be a generator:
sage: pp = poly.parent().gen(0)
sage: poly.degree(pp)
1
sage: poly.degree(pp+1)
Traceback (most recent call last):
...
TypeError: argument must be a generator
Canonical coercions are used:
sage: S = ZZ['p,q']
sage: poly.degree(S.0)
1
sage: poly.degree(S.1)
2
Returns a tuple with the maximal degree of each variable in this polynomial. The list of degrees is ordered by the order of the generators.
EXAMPLES:
sage: R.<y0,y1,y2> = PolynomialRing(QQ,3)
sage: q = 3*y0*y1*y1*y2; q
3*y0*y1^2*y2
sage: q.degrees()
(1, 2, 1)
sage: (q + y0^5).degrees()
(5, 2, 1)
Return a dictionary representing self. This dictionary is in the same format as the generic MPolynomial: The dictionary consists of ETuple:coefficient pairs.
EXAMPLES:
sage: R.<x,y,z> = QQ[]
sage: f=2*x*y^3*z^2 + 1/7*x^2 + 2/3
sage: f.dict()
{(0, 0, 0): 2/3, (1, 3, 2): 2, (2, 0, 0): 1/7}
Returns the discriminant of self with respect to the given variable.
INPUT:
- variable - The variable with respect to which we compute
the discriminant
OUTPUT:
- An element of the base ring of the polynomial ring.
EXAMPLES:
sage: R.<x,y,z>=QQ[]
sage: f=4*x*y^2 + 1/4*x*y*z + 3/2*x*z^2 - 1/2*z^2
sage: f.discriminant(x)
1
sage: f.discriminant(y)
-383/16*x^2*z^2 + 8*x*z^2
sage: f.discriminant(z)
-383/16*x^2*y^2 + 8*x*y^2
Note that, unlike the univariate case, the result lives in the same ring as the polynomial:
sage: R.<x,y>=QQ[]
sage: f=x^5*y+3*x^2*y^2-2*x+y-1
sage: f.discriminant(y)
x^10 + 2*x^5 + 24*x^3 + 12*x^2 + 1
sage: f.polynomial(y).discriminant()
x^10 + 2*x^5 + 24*x^3 + 12*x^2 + 1
sage: f.discriminant(y).parent()==f.polynomial(y).discriminant().parent()
False
Return the exponents of the monomials appearing in this polynomial.
INPUT:
otherwise returns a list of tuples
EXAMPLES:
sage: R.<a,b,c> = QQ[]
sage: f = a^3 + b + 2*b^2
sage: f.exponents()
[(3, 0, 0), (0, 2, 0), (0, 1, 0)]
sage: f.exponents(as_ETuples=False)
[(3, 0, 0), (0, 2, 0), (0, 1, 0)]
Return the factorization of this polynomial.
INPUT:
EXAMPLES:
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, but over a finite field of order 9.:
sage: K.<a> = GF(3^2)
sage: R.<x, y> = K[]
sage: f = (x^3 + 2*a*y^2*x) * (x^2 + x + 1); f
x^5 + (-a)*x^3*y^2 + x^4 + (-a)*x^2*y^2 + x^3 + (-a)*x*y^2
sage: F = f.factor()
sage: F
((-a)) * x * (x - 1)^2 * ((-a + 1)*x^2 + y^2)
sage: f - F
0
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()
2
Another example:
sage: R.<x,y,z> = GF(32003)[]
sage: f = 9*(x-1)^2*(y+z)
sage: f.factor()
(9) * (y + z) * (x - 1)^2
sage: R.<x,w,v,u> = QQ['x','w','v','u']
sage: p = (4*v^4*u^2 - 16*v^2*u^4 + 16*u^6 - 4*v^4*u + 8*v^2*u^3 + v^4)
sage: p.factor()
(-2*v^2*u + 4*u^3 + v^2)^2
sage: R.<a,b,c,d> = QQ[]
sage: f = (-2) * (a - d) * (-a + b) * (b - d) * (a - c) * (b - c) * (c - d)
sage: F = f.factor(); F
(-2) * (c - d) * (-b + c) * (b - d) * (-a + c) * (-a + b) * (a - d)
sage: F[0][0]
c - d
sage: F.unit()
-2
Constant elements are factorized in the base rings.
sage: P.<x,y> = ZZ[]
sage: P(2^3*7).factor()
2^3 * 7
Factorization for finite prime fields with characteristic
is not supported either.
sage: q = 1073741789
sage: T.<aa, bb> = PolynomialRing(GF(q))
sage: f = aa^2 + 12124343*bb*aa + 32434598*bb^2
sage: f.factor()
Traceback (most recent call last):
...
NotImplementedError: Factorization of multivariate polynomials over prime fields with characteristic > 2^29 is not implemented.
Finally, factorization over the integers is not supported.
sage: P.<x,y> = PolynomialRing(ZZ)
sage: f = (3*x + 4)*(5*x - 2)
sage: f.factor()
Traceback (most recent call last):
...
NotImplementedError: Factorization of multivariate polynomials over non-fields is not implemented.
TESTS:
This shows that trac ticket #10270 is fixed:
sage: R.<x,y,z> = GF(3)[]
sage: f = x^2*z^2+x*y*z-y^2
sage: f.factor()
x^2*z^2 + x*y*z - y^2
This checks that trac ticket #11838 is fixed:
sage: K = GF(4,'a')
sage: a = K.gens()[0]
sage: R.<x,y> = K[]
sage: p=x^8*y^3 + x^2*y^9 + a*x^9 + a*x*y^4
sage: q=y^11 + (a)*y^10 + (a + 1)*x*y^3
sage: f = p*q
sage: f.factor()
x * y^3 * (y^8 + (a)*y^7 + (a + 1)*x) * (x^7*y^3 + x*y^9 + (a)*x^8 + (a)*y^4)
We test several examples which were known to return wrong results in the past (see trac ticket #10902):
sage: R.<x,y> = GF(2)[]
sage: p = x^3*y^7 + x^2*y^6 + x^2*y^3
sage: q = x^3*y^5
sage: f = p*q
sage: p.factor()*q.factor()
x^5 * y^8 * (x*y^4 + y^3 + 1)
sage: f.factor()
x^5 * y^8 * (x*y^4 + y^3 + 1)
sage: f.factor().expand() == f
True
sage: R.<x,y> = GF(2)[]
sage: p=x^8 + y^8; q=x^2*y^4 + x
sage: f=p*q
sage: lf = f.factor()
sage: f-lf
0
sage: R.<x,y> = GF(3)[]
sage: p = -x*y^9 + x
sage: q = -x^8*y^2
sage: f = p*q
sage: f
x^9*y^11 - x^9*y^2
sage: f.factor()
y^2 * (y - 1)^9 * x^9
sage: f - f.factor()
0
sage: R.<x,y> = GF(5)[]
sage: p=x^27*y^9 + x^32*y^3 + 2*x^20*y^10 - x^4*y^24 - 2*x^17*y
sage: q=-2*x^10*y^24 + x^9*y^24 - 2*x^3*y^30
sage: f=p*q; f-f.factor()
0
sage: R.<x,y> = GF(7)[]
sage: p=-3*x^47*y^24
sage: q=-3*x^47*y^37 - 3*x^24*y^49 + 2*x^56*y^8 + 3*x^29*y^15 - x^2*y^33
sage: f=p*q
sage: f-f.factor()
0
The following examples used to give a Segmentation Fault, see trac ticket #12918 and trac ticket #13129:
sage: R.<x,y> = GF(2)[]
sage: f = x^6 + x^5 + y^5 + y^4
sage: f.factor()
x^6 + x^5 + y^5 + y^4
sage: f = x^16*y + x^10*y + x^9*y + x^6*y + x^5 + x*y + y^2
sage: f.factor()
x^16*y + x^10*y + x^9*y + x^6*y + x^5 + x*y + y^2
Test trac ticket #12928:
sage: R.<x,y> = GF(2)[]
sage: p = x^2 + y^2 + x + 1
sage: q = x^4 + x^2*y^2 + y^4 + x*y^2 + x^2 + y^2 + 1
sage: factor(p*q)
(x^2 + y^2 + x + 1) * (x^4 + x^2*y^2 + y^4 + x*y^2 + x^2 + y^2 + 1)
Check that trac ticket #13770 is fixed:
sage: U.<y,t> = GF(2)[]
sage: f = y*t^8 + y^5*t^2 + y*t^6 + t^7 + y^6 + y^5*t + y^2*t^4 + y^2*t^2 + y^2*t + t^3 + y^2 + t^2
sage: l = f.factor()
sage: l[0][0]==t^2 + y + t + 1 or l[1][0]==t^2 + y + t + 1
True
The following used to sometimes take a very long time or get stuck, see trac ticket #12846. These 100 iterations should take less than 1 second:
sage: K.<a> = GF(4)
sage: R.<x,y> = K[]
sage: f = (a + 1)*x^145*y^84 + (a + 1)*x^205*y^17 + x^32*y^112 + x^92*y^45
sage: for i in range(100):
... assert len(f.factor()) == 4
Return the greatest common divisor of self and right.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: f = (x*y*z)^6 - 1
sage: g = (x*y*z)^4 - 1
sage: f.gcd(g)
x^2*y^2*z^2 - 1
sage: GCD([x^3 - 3*x + 2, x^4 - 1, x^6 -1])
x - 1
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
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
TESTS:
sage: Q.<x,y,z> = QQ[]
sage: P.<x,y,z> = QQ[]
sage: P(0).gcd(Q(0))
0
sage: x.gcd(1)
1
sage: k.<a> = GF(9)
sage: R.<x,y> = PolynomialRing(k)
sage: f = R.change_ring(GF(3)).gen()
sage: g = x+y
sage: g.gcd(f)
1
sage: x.gcd(R.change_ring(GF(3)).gen())
x
Return a list of partial derivatives of this polynomial, ordered by the variables of the parent.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ,3)
sage: f= x*y + 1
sage: f.gradient()
[y, x, 0]
Integrates this polynomial with respect to the provided variable.
One requires that is contained in the ring.
INPUT:
EXAMPLES:
sage: R.<x, y> = PolynomialRing(QQ, 2)
sage: f = 3*x^3*y^2 + 5*y^2 + 3*x + 2
sage: f.integral(x)
3/4*x^4*y^2 + 5*x*y^2 + 3/2*x^2 + 2*x
sage: f.integral(y)
x^3*y^3 + 5/3*y^3 + 3*x*y + 2*y
Check that trac ticket #15896 is solved:
sage: s = x+y
sage: s.integral(x)+x
1/2*x^2 + x*y + x
sage: s.integral(x)*s
1/2*x^3 + 3/2*x^2*y + x*y^2
TESTS:
sage: z, w = polygen(QQ, 'z, w')
sage: f.integral(z)
Traceback (most recent call last):
...
TypeError: the variable is not in the same ring as self
sage: f.integral(y**2)
Traceback (most recent call last):
...
TypeError: not a variable in the same ring as self
sage: x,y = polygen(ZZ,'x,y')
sage: y.integral(x)
Traceback (most recent call last):
...
TypeError: the ring must contain the rational numbers
Return the inverse of this polynomial if it is a unit.
EXAMPLES:
sage: R.<x,y> = QQ[]
sage: x.inverse_of_unit()
Traceback (most recent call last):
...
ArithmeticError: Element is not a unit.
sage: R(1/2).inverse_of_unit()
2
Return True if this polynomial is constant.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(GF(127))
sage: x.is_constant()
False
sage: P(1).is_constant()
True
Return True if this polynomial is homogeneous.
EXAMPLES:
sage: P.<x,y> = PolynomialRing(RationalField(), 2)
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
Return True if this polynomial is a monomial. A monomial is defined to be a product of generators with coefficient 1.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: x.is_monomial()
True
sage: (2*x).is_monomial()
False
sage: (x*y).is_monomial()
True
sage: (x*y + x).is_monomial()
False
Return True if this polynomial is square free.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: f= x^2 + 2*x*y + 1/2*z
sage: f.is_squarefree()
True
sage: h = f^2
sage: h.is_squarefree()
False
Return True if self is a unit.
EXAMPLES:
sage: R.<x,y> = QQ[]
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()
True
sage: R.<x,y> = ZZ[]
sage: R(1).is_unit()
True
sage: R(2).is_unit()
False
Return True if self is a univariate polynomial, that is if self contains only one variable.
EXAMPLES:
sage: P.<x,y,z> = GF(2)[]
sage: f = x^2 + 1
sage: f.is_univariate()
True
sage: f = y*x^2 + 1
sage: f.is_univariate()
False
sage: f = P(0)
sage: f.is_univariate()
True
Return True if this polynomial is zero.
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQ)
sage: x.is_zero()
False
sage: (x-x).is_zero()
True
Leading coefficient of this polynomial with respect to the term order of self.parent().
EXAMPLES:
sage: R.<x,y,z>=PolynomialRing(GF(7),3,order='lex')
sage: f = 3*x^1*y^2 + 2*y^3*z^4
sage: f.lc()
3
sage: f = 5*x^3*y^2*z^4 + 4*x^3*y^2*z^1
sage: f.lc()
5
Return the least common multiple of self and .
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: p = (x+y)*(y+z)
sage: q = (z^4+2)*(y+z)
sage: lcm(p,q)
x*y*z^4 + y^2*z^4 + x*z^5 + y*z^5 + 2*x*y + 2*y^2 + 2*x*z + 2*y*z
sage: P.<x,y,z> = ZZ[]
sage: p = 2*(x+y)*(y+z)
sage: q = 3*(z^4+2)*(y+z)
sage: lcm(p,q)
6*x*y*z^4 + 6*y^2*z^4 + 6*x*z^5 + 6*y*z^5 + 12*x*y + 12*y^2 + 12*x*z + 12*y*z
sage: r.<x,y> = PolynomialRing(GF(2**8, 'a'), 2)
sage: a = r.base_ring().0
sage: f = (a^2+a)*x^2*y + (a^4+a^3+a)*y + a^5
sage: f.lcm(x^4)
(a^2 + a)*x^6*y + (a^4 + a^3 + a)*x^4*y + (a^5)*x^4
sage: w = var('w')
sage: r.<x,y> = PolynomialRing(NumberField(w^4 + 1, 'a'), 2)
sage: a = r.base_ring().0
sage: f = (a^2+a)*x^2*y + (a^4+a^3+a)*y + a^5
sage: f.lcm(x^4)
(a^2 + a)*x^6*y + (a^3 + a - 1)*x^4*y + (-a)*x^4
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.
A ValueError exception is raised if g (== self) does not belong to I.
EXAMPLES:
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
Check that trac ticket #13671 is fixed:
sage: R.<x1,x2> = QQ[]
sage: I = R.ideal(x2**2 + x1 - 2, x1**2 - 1)
sage: f = I.gen(0) + x2*I.gen(1)
sage: f.lift(I)
[1, x2]
sage: (f+1).lift(I)
Traceback (most recent call last):
...
ValueError: polynomial is not in the ideal
sage: f.lift(I)
[1, x2]
TESTS:
Check that trac ticket #13714 is fixed:
sage: R.<x1,x2> = QQ[]
sage: I = R.ideal(x2**2 + x1 - 2, x1**2 - 1)
sage: R.one().lift(I)
Traceback (most recent call last):
...
ValueError: polynomial is not in the ideal
sage: foo = I.complete_primary_decomposition() # indirect doctest
sage: foo[0][0]
Ideal (x2 - 1, x1 - 1) of Multivariate Polynomial Ring in x1, x2 over Rational Field
Returns the lead monomial of self with respect to the term order of self.parent(). In Sage a monomial is a product of variables in some power without a coefficient.
EXAMPLES:
sage: R.<x,y,z>=PolynomialRing(GF(7),3,order='lex')
sage: f = x^1*y^2 + y^3*z^4
sage: f.lm()
x*y^2
sage: f = x^3*y^2*z^4 + x^3*y^2*z^1
sage: f.lm()
x^3*y^2*z^4
sage: R.<x,y,z>=PolynomialRing(QQ,3,order='deglex')
sage: f = x^1*y^2*z^3 + x^3*y^2*z^0
sage: f.lm()
x*y^2*z^3
sage: f = x^1*y^2*z^4 + x^1*y^1*z^5
sage: f.lm()
x*y^2*z^4
sage: R.<x,y,z>=PolynomialRing(GF(127),3,order='degrevlex')
sage: f = x^1*y^5*z^2 + x^4*y^1*z^3
sage: f.lm()
x*y^5*z^2
sage: f = x^4*y^7*z^1 + x^4*y^2*z^3
sage: f.lm()
x^4*y^7*z
Leading term of this polynomial. In Sage a term is a product of variables in some power and a coefficient.
EXAMPLES:
sage: R.<x,y,z>=PolynomialRing(GF(7),3,order='lex')
sage: f = 3*x^1*y^2 + 2*y^3*z^4
sage: f.lt()
3*x*y^2
sage: f = 5*x^3*y^2*z^4 + 4*x^3*y^2*z^1
sage: f.lt()
-2*x^3*y^2*z^4
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:
EXAMPLES:
sage: P.<x,y> = QQ[]
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()
Rational Field
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
Return the list of monomials in self. The returned list is decreasingly ordered by the term ordering of self.parent().
EXAMPLES:
sage: P.<x,y,z> = QQ[]
sage: f = x + 3/2*y*z^2 + 2/3
sage: f.monomials()
[y*z^2, x, 1]
sage: f = P(3/2)
sage: f.monomials()
[1]
TESTS:
sage: P.<x,y,z> = QQ[]
sage: f = x
sage: f.monomials()
[x]
Check if trac ticket #12706 is fixed:
sage: f = P(0)
sage: f.monomials()
[]
Check if trac ticket #7152 is fixed:
sage: x=var('x')
sage: K.<rho> = NumberField(x**2 + 1)
sage: R.<x,y> = QQ[]
sage: p = rho*x
sage: q = x
sage: p.monomials()
[x]
sage: q.monomials()
[x]
sage: p.monomials()
[x]
Return a numerator of self computed as self * self.denominator()
If the base_field of self is the Rational Field then the numerator is a polynomial whose base_ring is the Integer Ring, this is done for compatibility to the univariate case.
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*y + 1
sage: f.numerator()
x^3 + 17*y + 1
sage: f == f.numerator()
True
Next we compute the numerator of a polynomial with rational coefficients.
sage: R.<x,y> = PolynomialRing(QQ)
sage: f = (1/17)*x^19 - (2/3)*y + 1/3; f
1/17*x^19 - 2/3*y + 1/3
sage: f.numerator()
3*x^19 - 34*y + 17
sage: f == f.numerator()
False
sage: f.numerator().base_ring()
Integer Ring
We check that the computation of numerator and denominator is valid.
sage: K=QQ['x,y']
sage: f=K.random_element()
sage: f.numerator() / f.denominator() == f
True
The following tests against a bug that has been fixed in trac ticket #11780:
sage: P.<foo,bar> = ZZ[]
sage: Q.<foo,bar> = QQ[]
sage: f = Q.random_element()
sage: f.numerator().parent() is P
True
Return the number variables in this polynomial.
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(GF(127))
sage: f = x*y + z
sage: f.nvariables()
3
sage: f = x + y
sage: f.nvariables()
2
Returns quotient and remainder of self and right.
EXAMPLES:
sage: R.<x,y> = QQ[]
sage: f = y*x^2 + x + 1
sage: f.quo_rem(x)
(x*y + 1, 1)
sage: f.quo_rem(y)
(x^2, x + 1)
sage: R.<x,y> = ZZ[]
sage: f = 2*y*x^2 + x + 1
sage: f.quo_rem(x)
(2*x*y + 1, 1)
sage: f.quo_rem(y)
(2*x^2, x + 1)
sage: f.quo_rem(3*x)
(0, 2*x^2*y + x + 1)
TESTS:
sage: R.<x,y> = QQ[]
sage: R(0).quo_rem(R(1))
(0, 0)
sage: R(1).quo_rem(R(0))
Traceback (most recent call last):
...
ZeroDivisionError
Return the normal form of self w.r.t. I, i.e. return the remainder of this polynomial with respect to the polynomials in I. If the polynomial set/list I is not a (strong) Groebner basis the result is not canonical.
A strong Groebner basis G of I implies that for every leading term t of I there exists an element g of G, such that the leading term of g divides t.
INPUT:
EXAMPLES:
sage: P.<x,y,z> = QQ[]
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
is also supported.
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])
3*x
Compute the resultant of this polynomial and the first argument with respect to the variable given as the second argument.
If a second argument is not provide the first variable of the parent is chosen.
INPUT:
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQ,2)
sage: a = x+y
sage: b = x^3-y^3
sage: c = a.resultant(b); c
-2*y^3
sage: d = a.resultant(b,y); d
2*x^3
The SINGULAR example:
sage: R.<x,y,z> = PolynomialRing(GF(32003),3)
sage: f = 3 * (x+2)^3 + y
sage: g = x+y+z
sage: f.resultant(g,x)
3*y^3 + 9*y^2*z + 9*y*z^2 + 3*z^3 - 18*y^2 - 36*y*z - 18*z^2 + 35*y + 36*z - 24
Resultants are also supported over the Integers:
sage: R.<x,y,a,b,u>=PolynomialRing(ZZ, 5, order='lex')
sage: r = (x^4*y^2+x^2*y-y).resultant(x*y-y*a-x*b+a*b+u,x)
sage: r
y^6*a^4 - 4*y^5*a^4*b - 4*y^5*a^3*u + y^5*a^2 - y^5 + 6*y^4*a^4*b^2 + 12*y^4*a^3*b*u - 4*y^4*a^2*b + 6*y^4*a^2*u^2 - 2*y^4*a*u + 4*y^4*b - 4*y^3*a^4*b^3 - 12*y^3*a^3*b^2*u + 6*y^3*a^2*b^2 - 12*y^3*a^2*b*u^2 + 6*y^3*a*b*u - 4*y^3*a*u^3 - 6*y^3*b^2 + y^3*u^2 + y^2*a^4*b^4 + 4*y^2*a^3*b^3*u - 4*y^2*a^2*b^3 + 6*y^2*a^2*b^2*u^2 - 6*y^2*a*b^2*u + 4*y^2*a*b*u^3 + 4*y^2*b^3 - 2*y^2*b*u^2 + y^2*u^4 + y*a^2*b^4 + 2*y*a*b^3*u - y*b^4 + y*b^2*u^2
TESTS:
sage: P.<x,y> = PolynomialRing(QQ, order='degrevlex')
sage: a = x+y
sage: b = x^3-y^3
sage: c = a.resultant(b); c
-2*y^3
sage: d = a.resultant(b,y); d
2*x^3
sage: P.<x,y> = PolynomialRing(ZZ,2)
sage: f = x+y
sage: g=y^2+x
sage: f.resultant(g,y)
x^2 + x
Return self - m*q, where m must be a monomial and q a polynomial.
INPUT:
EXAMPLES:
sage: P.<x,y,z>=PolynomialRing(QQ,3)
sage: x.sub_m_mul_q(y,z)
-y*z + x
TESTS:
sage: Q.<x,y,z>=PolynomialRing(QQ,3)
sage: P.<x,y,z>=PolynomialRing(QQ,3)
sage: P(0).sub_m_mul_q(P(0),P(1))
0
sage: x.sub_m_mul_q(Q.gen(1),Q.gen(2))
-y*z + x
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 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, but should be much faster if only few variables are to be fixed.
INPUT:
EXAMPLES:
sage: R.<x,y> = QQ[]
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
sage: f.subs(x=5)
25*y^2 + y + 30
sage: P.<x,y,z> = PolynomialRing(GF(2),3)
sage: f = x + y + 1
sage: f.subs({x:y+1})
0
sage: f.subs(x=y)
1
sage: f.subs(x=x)
x + y + 1
sage: f.subs({x:z})
y + z + 1
sage: f.subs(x=z+1)
y + z
sage: f.subs(x=1/y)
(y^2 + y + 1)/y
sage: f.subs({x:1/y})
(y^2 + y + 1)/y
The parameters are subsituted in order and without side effects:
sage: R.<x,y>=QQ[]
sage: g=x+y
sage: g.subs({x:x+1,y:x*y})
x*y + x + 1
sage: g.subs({x:x+1}).subs({y:x*y})
x*y + x + 1
sage: g.subs({y:x*y}).subs({x:x+1})
x*y + x + y + 1
sage: R.<x,y> = QQ[]
sage: f = x + 2*y
sage: f.subs(x=y,y=x)
2*x + y
TESTS:
sage: P.<x,y,z> = QQ[]
sage: f = y
sage: f.subs({y:x}).subs({x:z})
z
We test that we change the ring even if there is nothing to do:
sage: P = QQ['x,y']
sage: x = var('x')
sage: parent(P.zero_element() / x)
Symbolic Ring
We are catching overflows:
sage: R.<x,y> = QQ[]
sage: n=1000; f = x^n
sage: try:
....: f.subs(x = x^n)
....: print "no overflow"
....: except OverflowError:
....: print "overflow"
overflow # 32-bit
x^1000000 # 64-bit
no overflow # 64-bit
sage: n=100000;
sage: try:
....: f = x^n
....: f.subs(x = x^n)
....: print "no overflow"
....: except OverflowError:
....: print "overflow"
overflow
Return the total degree of self, which is the maximum degree of all monomials in self.
EXAMPLES:
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
TESTS:
sage: R.<x,y,z> = QQ[]
sage: R(0).total_degree()
-1
sage: R(1).total_degree()
0
With a matrix term ordering, the grading changes. To evaluate the total degree using the standard grading, use the optional argument``std_grading=True``.
sage: tord=TermOrder(matrix([3,0,1,1,1,0,1,0,0])) sage: R.<x,y,z> = PolynomialRing(QQ,’x’,3,order=tord) sage: (x^2*y).total_degree() 6 sage: (x^2*y).total_degree(std_grading=True) 3
Returns a univariate polynomial associated to this multivariate polynomial.
INPUT:
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 R is provided.
EXAMPLES:
sage: R.<x, y> = QQ[]
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*y^2 - 2*y + 305
sage: g.univariate_polynomial(PolynomialRing(QQ,'z'))
700*z^2 - 2*z + 305
Here’s an example with a constant multivariate polynomial:
sage: g = R(1)
sage: h = g.univariate_polynomial(); h
1
sage: h.parent()
Univariate Polynomial Ring in x over Rational Field
Return the i-th variable occurring in self. The index i is the index in self.variables().
EXAMPLES:
sage: P.<x,y,z> = GF(2)[]
sage: f = x*z^2 + z + 1
sage: f.variables()
(x, z)
sage: f.variable(1)
z
Return a tuple of all variables occurring in self.
EXAMPLES:
sage: P.<x,y,z> = GF(2)[]
sage: f = x*z^2 + z + 1
sage: f.variables()
(x, z)
inverse function for MPolynomialRing_libsingular.__reduce__
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQ)
sage: loads(dumps(P)) is P # indirect doctest
True
Deserialize an MPolynomial_libsingular object
INPUT:
EXAMPLES:
sage: P.<x,y> = PolynomialRing(QQ)
sage: loads(dumps(x)) == x # indirect doctest
True