Module: sage.rings.polynomial_ring
Author Log:
Module-level Functions
base_ring, [name=False], [sparse=None], [names=None], [order=False], [macaulay2=None]) |
Return a univariate or multivariate polynomial ring.
INPUT: base_ring -- the base ring name -- (str) the name of the generator sparse -- (bool; default: False) whether or not elements are represented using sparse methods; note that multivariate polynomials are always sparse names -- names of the generators (for multivariate poly) order -- term order of ring macaulay2 (bool; default: False) -- whether or not to use Macaulay2 (multivariate only)
sage: PolynomialRing(ZZ) Univariate Polynomial Ring in x over Integer Ring sage: PolynomialRing(ZZ, 'y') Univariate Polynomial Ring in y over Integer Ring sage: PolynomialRing(PolynomialRing(QQ,'z'), 'y') Univariate Polynomial Ring in y over Univariate Polynomial Ring in z over Rational Field sage: PolynomialRing(QQ, name='abc') Univariate Polynomial Ring in abc over Rational Field sage: PolynomialRing(QQ, name='abc', sparse=True) Sparse Univariate Polynomial Ring in abc over Rational Field sage: PolynomialRing(QQ, 3, sparse=True) Polynomial Ring in x0, x1, x2 over Rational Field sage: PolynomialRing(QQ, 3, macaulay2=True) Polynomial Ring in x0, x1, x2 over Rational Field
x) |
base_ring, [name=x]) |
Class: PolynomialRing_dense_mod_n
self, base_ring, [name=x]) |
Special Functions: __call__,
_ntl_set_modulus
Class: PolynomialRing_dense_mod_p
self, base_ring, [name=x]) |
Special Functions: __call__
Class: PolynomialRing_field
self, base_ring, [name=False], [sparse=x]) |
Class: PolynomialRing_generic
self, base_ring, [name=False], [sparse=None]) |
sage: R, x = Q['x'].objgen() sage: R(-1) + R(1) 0 sage: (x - Q('2/3'))*(x**2 - 8*x + 16) x^3 - 26/3*x^2 + 64/3*x - 32/3
Functions: base_extend,
base_ring,
characteristic,
cyclotomic_polynomial,
gen,
is_field,
is_sparse,
krull_dimension,
ngens,
parameter,
quotient_by_principal_ideal,
random_element
self, n) |
The nth cyclotomic polynomial.
sage: R = Q['x'] sage: R.cyclotomic_polynomial(8) x^4 + 1 sage: R.cyclotomic_polynomial(12) x^4 - x^2 + 1 sage: S = PolynomialRing(FiniteField(7)) sage: S.cyclotomic_polynomial(12) x^4 + 6*x^2 + 1
self, [n=0]) |
If this is R[x], return x.
self, f, [names=None]) |
Return the quotient of this polynomial ring by the principal
ideal generated by
.
self, degree, [bound=0]) |
Return a random polynomial.
INPUT: degree -- an int bound -- an int (default: 0, which tries to spread choice across ring, if implemented) OUTPUT: Polynomial -- A polynomial such that the coefficient of x^i, for i up to degree, are coercisions to the base ring of random integers between -bound and bound.
Special Functions: __call__,
__cmp__,
__reduce__,
__repr__,
_coerce_,
_is_valid_homomorphism_,
_latex_,
_PolynomialRing_generic__set_polynomial_class
Class: PolynomialRing_integral_domain
self, base_ring, [name=False], [sparse=x]) |