Univariate Polynomials and Polynomial Rings#
Sage’s architecture for polynomials ‘under the hood’ is complex, interfacing to a variety of C/C++ libraries for polynomials over specific rings. In practice, the user rarely has to worry about which backend is being used.
The hierarchy of class inheritance is somewhat confusing, since most of the polynomial element classes are implemented as Cython extension types rather than pure Python classes and thus can only inherit from a single base class, whereas others have multiple bases.
- Univariate Polynomial Rings
PolynomialRing_cdvf
PolynomialRing_cdvr
PolynomialRing_commutative
PolynomialRing_dense_finite_field
PolynomialRing_dense_mod_n
PolynomialRing_dense_mod_p
PolynomialRing_dense_padic_field_capped_relative
PolynomialRing_dense_padic_field_generic
PolynomialRing_dense_padic_ring_capped_absolute
PolynomialRing_dense_padic_ring_capped_relative
PolynomialRing_dense_padic_ring_fixed_mod
PolynomialRing_dense_padic_ring_generic
PolynomialRing_field
PolynomialRing_general
PolynomialRing_integral_domain
is_PolynomialRing()
polygen()
polygens()
- Ring homomorphisms from a polynomial ring to another ring
- Univariate polynomial base class
- Univariate Polynomials over domains and fields
Polynomial_generic_cdv
Polynomial_generic_cdvf
Polynomial_generic_cdvr
Polynomial_generic_dense_cdv
Polynomial_generic_dense_cdvf
Polynomial_generic_dense_cdvr
Polynomial_generic_dense_field
Polynomial_generic_domain
Polynomial_generic_field
Polynomial_generic_sparse
Polynomial_generic_sparse_cdv
Polynomial_generic_sparse_cdvf
Polynomial_generic_sparse_cdvr
Polynomial_generic_sparse_field
- Univariate Polynomials over GF(2) via NTL’s GF2X
- Univariate polynomials over number fields.
- Dense univariate polynomials over \(\ZZ\), implemented using FLINT
- Dense univariate polynomials over \(\ZZ\), implemented using NTL.
- Univariate polynomials over \(\QQ\) implemented via FLINT
- Dense univariate polynomials over \(\ZZ/n\ZZ\), implemented using FLINT
- Dense univariate polynomials over \(\ZZ/n\ZZ\), implemented using NTL
- Dense univariate polynomials over \(\RR\), implemented using MPFR
- Polynomial Interfaces to Singular
- Base class for generic \(p\)-adic polynomials
- p-adic Capped Relative Dense Polynomials
- p-adic Flat Polynomials
- Univariate Polynomials over GF(p^e) via NTL’s ZZ_pEX
- Isolate Real Roots of Real Polynomials
PrecisionError
bernstein_down()
bernstein_expand()
bernstein_polynomial_factory
bernstein_polynomial_factory_ar
bernstein_polynomial_factory_intlist
bernstein_polynomial_factory_ratlist
bernstein_up()
bitsize_doctest()
cl_maximum_root()
cl_maximum_root_first_lambda()
cl_maximum_root_local_max()
context
de_casteljau_doublevec()
de_casteljau_intvec()
degree_reduction_next_size()
dprod_imatrow_vec()
get_realfield_rndu()
interval_bernstein_polynomial
interval_bernstein_polynomial_float
interval_bernstein_polynomial_integer
intvec_to_doublevec()
island
linear_map
max_abs_doublevec()
max_bitsize_intvec_doctest()
maximum_root_first_lambda()
maximum_root_local_max()
min_max_delta_intvec()
min_max_diff_doublevec()
min_max_diff_intvec()
mk_context()
mk_ibpf()
mk_ibpi()
ocean
precompute_degree_reduction_cache()
pseudoinverse()
rational_root_bounds()
real_roots()
relative_bounds()
reverse_intvec()
root_bounds()
rr_gap
scale_intvec_var()
split_for_targets()
subsample_vec_doctest()
taylor_shift1_intvec()
to_bernstein()
to_bernstein_warp()
warp_map
wordsize_rational()
- Isolate Complex Roots of Polynomials
- Refine polynomial roots using Newton–Raphson
- Ideals in Univariate Polynomial Rings
- Quotients of Univariate Polynomial Rings
- Elements of Quotients of Univariate Polynomial Rings
- Polynomial Compilers
- Polynomial multiplication by Kronecker substitution