NumberFieldPolynomial

NumberFieldPolynomial

NumberFieldPolynomial()

A Symbolica polynomial with rational coefficients.

Methods

Name	Description
__add__	Add two polynomials self and rhs, returning the result.
__contains__	Check if the polynomial contains the given variable.
__copy__	Copy the polynomial.
__eq__	Check if two polynomials are equal.
__floordiv__	Divide the polynomial self by rhs, rounding down, returning the result.
__ge__	Check if the polynomial is greater than or equal to an integer.
__gt__	Check if the polynomial is greater than an integer.
__le__	Check if the polynomial is less than or equal to an integer.
__lt__	Check if the polynomial is less than an integer.
__mod__	Compute the remainder of the division of self by rhs.
__mul__	Multiply two polynomials self and rhs, returning the result.
__ne__	Check if two polynomials are not equal.
__neg__	Negate the polynomial.
__pow__	Raise the polynomial to the power of exp, returning the result.
__radd__	Add two polynomials self and rhs, returning the result.
__rmul__	Multiply two polynomials self and rhs, returning the result.
__rsub__	Subtract polynomials self from rhs, returning the result.
__str__	Print the polynomial in a human-readable format.
__sub__	Subtract polynomials rhs from self, returning the result.
__truediv__	Divide the polynomial self by rhs if possible, returning the result.
coefficient_list	Get the coefficient list, optionally in the variables xs.
contains	Check if the polynomial contains the given variable.
content	Get the content, i.e., the GCD of the coefficients.
degree	Get the degree of the polynomial in var.
derivative	Take a derivative in x.
extended_gcd	Compute the extended GCD of two polynomials, yielding the GCD and the Bezout coefficients s and t such that self * s + rhs * t = gcd(self, rhs).
factor	Factorize the polynomial.
factor_square_free	Compute the square-free factorization of the polynomial.
format	Convert the polynomial into a human-readable string, with tunable settings.
gcd	Compute the greatest common divisor (GCD) of two or more polynomials.
get_minimal_polynomial	Get the minimal polynomial of the algebraic extension.
get_variables	Get the list of variables in the internal ordering of the polynomial.
groebner_basis	Compute the Groebner basis of a polynomial system
integrate	Integrate the polynomial in x.
lcoeff	Get the leading coefficient.
monic	Get the monic part of the polynomial, i.e., the polynomial divided by its leading coefficient.
nterms	Get the number of terms in the polynomial.
primitive	Get the primitive part of the polynomial, i.e., the polynomial with the content removed.
quot_rem	Divide self by rhs, returning the quotient and remainder.
reduce	Completely reduce the polynomial w.r.t the polynomials gs
reorder	Reorder the polynomial in-place to use the given variable order.
replace	Replace the variable x with a polynomial v.
resultant	Compute the resultant of two polynomials with respect to the variable var.
to_expression	Convert the polynomial to an expression.
to_latex	Convert the polynomial into a LaTeX string.
to_polynomial	Convert the number field polynomial to a rational polynomial.

__add__

NumberFieldPolynomial.__add__(rhs: NumberFieldPolynomial | int) -> NumberFieldPolynomial

Add two polynomials self and rhs, returning the result.

__contains__

NumberFieldPolynomial.__contains__(var: Expression) -> bool

Related type: Expression

Check if the polynomial contains the given variable.

__copy__

NumberFieldPolynomial.__copy__() -> NumberFieldPolynomial

Copy the polynomial.

__eq__

NumberFieldPolynomial.__eq__(rhs: Polynomial | int) -> bool

Related type: Polynomial

Check if two polynomials are equal.

__floordiv__

NumberFieldPolynomial.__floordiv__(rhs: Polynomial) -> Polynomial

Related type: Polynomial

Divide the polynomial self by rhs, rounding down, returning the result.

__ge__

NumberFieldPolynomial.__ge__(rhs: int) -> bool

Check if the polynomial is greater than or equal to an integer.

__gt__

NumberFieldPolynomial.__gt__(rhs: int) -> bool

Check if the polynomial is greater than an integer.

__le__

NumberFieldPolynomial.__le__(rhs: int) -> bool

Check if the polynomial is less than or equal to an integer.

__lt__

NumberFieldPolynomial.__lt__(rhs: int) -> bool

Check if the polynomial is less than an integer.

__mod__

NumberFieldPolynomial.__mod__(rhs: NumberFieldPolynomial) -> NumberFieldPolynomial

Compute the remainder of the division of self by rhs.

__mul__

NumberFieldPolynomial.__mul__(rhs: NumberFieldPolynomial | int) -> NumberFieldPolynomial

Multiply two polynomials self and rhs, returning the result.

__ne__

NumberFieldPolynomial.__ne__(rhs: Polynomial | int) -> bool

Related type: Polynomial

Check if two polynomials are not equal.

__neg__

NumberFieldPolynomial.__neg__() -> NumberFieldPolynomial

Negate the polynomial.

__pow__

NumberFieldPolynomial.__pow__(exp: int) -> NumberFieldPolynomial

Raise the polynomial to the power of exp, returning the result.

__radd__

NumberFieldPolynomial.__radd__(rhs: NumberFieldPolynomial | int) -> NumberFieldPolynomial

Add two polynomials self and rhs, returning the result.

__rmul__

NumberFieldPolynomial.__rmul__(rhs: NumberFieldPolynomial | int) -> NumberFieldPolynomial

Multiply two polynomials self and rhs, returning the result.

__rsub__

NumberFieldPolynomial.__rsub__(rhs: NumberFieldPolynomial | int) -> NumberFieldPolynomial

Subtract polynomials self from rhs, returning the result.

__str__

NumberFieldPolynomial.__str__() -> str

Print the polynomial in a human-readable format.

__sub__

NumberFieldPolynomial.__sub__(rhs: NumberFieldPolynomial | int) -> NumberFieldPolynomial

Subtract polynomials rhs from self, returning the result.

__truediv__

NumberFieldPolynomial.__truediv__(rhs: NumberFieldPolynomial) -> NumberFieldPolynomial

Divide the polynomial self by rhs if possible, returning the result.

coefficient_list

NumberFieldPolynomial.coefficient_list(xs: Optional[Expression | Sequence[Expression]] = None) -> list[Tuple[list[int], NumberFieldPolynomial]]

Related type: Expression

Get the coefficient list, optionally in the variables xs.

Examples

from symbolica import *
x = S('x')
p = E('x*y+2*x+x^2').to_polynomial()
for n, pp in p.coefficient_list(x):
    print(n, pp)

contains

NumberFieldPolynomial.contains(var: Expression) -> bool

Related type: Expression

Check if the polynomial contains the given variable.

content

NumberFieldPolynomial.content() -> NumberFieldPolynomial

Get the content, i.e., the GCD of the coefficients.

Examples

from symbolica import *
p = E('3x^2+6x+9').to_polynomial()
print(p.content())

degree

NumberFieldPolynomial.degree(var: Expression) -> int

Related type: Expression

Get the degree of the polynomial in var.

derivative

NumberFieldPolynomial.derivative(x: Expression) -> NumberFieldPolynomial

Related type: Expression

Take a derivative in x.

Examples

from symbolica import *
x = S('x')
p = E('x^2+2').to_polynomial()
print(p.derivative(x))

extended_gcd

NumberFieldPolynomial.extended_gcd(rhs: NumberFieldPolynomial) -> Tuple[NumberFieldPolynomial, NumberFieldPolynomial, NumberFieldPolynomial]

Compute the extended GCD of two polynomials, yielding the GCD and the Bezout coefficients s and t such that self * s + rhs * t = gcd(self, rhs).

factor

NumberFieldPolynomial.factor() -> list[Tuple[NumberFieldPolynomial, int]]

Factorize the polynomial.

Examples

from symbolica import *
p = E('(x+1)(x+2)(x+3)(x+4)(x+5)(x^2+6)(x^3+7)(x+8)(x^4+9)(x^5+x+10)').expand().to_polynomial()
print('Factorization of {}:'.format(p))
for f, exp in p.factor():
    print(' ({})^{}'.format(f, exp))

factor_square_free

NumberFieldPolynomial.factor_square_free() -> list[Tuple[NumberFieldPolynomial, int]]

Compute the square-free factorization of the polynomial.

Examples

from symbolica import *
p = E('3*(2*x^2+y)(x^3+y)^2(1+4*y)^2(1+x)').expand().to_polynomial()
print('Square-free factorization of {}:'.format(p))
for f, exp in p.factor_square_free():
    print(' ({})^{}'.format(f, exp))

format

NumberFieldPolynomial.format(
    mode: PrintMode = PrintMode.Symbolica,
    max_line_length: int | None = 80,
    indentation: int = 4,
    fill_indented_lines: bool = True,
    terms_on_new_line: bool = False,
    color_top_level_sum: bool = True,
    color_builtin_symbols: bool = True,
    bracket_level_colors: Sequence[int] | None = [244, 25, 97, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60],
    print_ring: bool = True,
    symmetric_representation_for_finite_field: bool = False,
    explicit_rational_polynomial: bool = False,
    number_thousands_separator: Optional[str] = None,
    multiplication_operator: str = '*',
    double_star_for_exponentiation: bool = False,
    square_brackets_for_function: bool = False,
    function_brackets: tuple[str, str] = ('(', ')'),
    num_exp_as_superscript: bool = True,
    precision: Optional[int] = None,
    show_namespaces: bool = False,
    hide_namespace: Optional[str] = None,
    include_attributes: bool = False,
    max_terms: Optional[int] = None,
    custom_print_mode: Optional[int] = None,
) -> str

Related type: PrintMode

Convert the polynomial into a human-readable string, with tunable settings.

Examples

p = FiniteFieldNumberFieldPolynomial.parse("3*x^2+2*x+7*x^3", ['x'], 11)
print(p.format(symmetric_representation_for_finite_field=True))

Yields z³⁴+x^(x+2)+y⁴+f(x,x²)+128_378_127_123 z^(2/3) w² x⁻¹ y⁻¹+3/5.

gcd

NumberFieldPolynomial.gcd(*rhs: NumberFieldPolynomial) -> NumberFieldPolynomial

Compute the greatest common divisor (GCD) of two or more polynomials.

get_minimal_polynomial

NumberFieldPolynomial.get_minimal_polynomial() -> Polynomial

Get the minimal polynomial of the algebraic extension.

get_variables

NumberFieldPolynomial.get_variables() -> Sequence[Expression]

Get the list of variables in the internal ordering of the polynomial.

groebner_basis

NumberFieldPolynomial.groebner_basis(
    system: list[NumberFieldPolynomial],
    grevlex: bool = True,
    print_stats: bool = False,
) -> list[NumberFieldPolynomial]

Compute the Groebner basis of a polynomial system.

If grevlex=True, reverse graded lexicographical ordering is used, otherwise the ordering is lexicographical.

If print_stats=True intermediate statistics will be printed.

integrate

NumberFieldPolynomial.integrate(x: Expression) -> NumberFieldPolynomial

Related type: Expression

Integrate the polynomial in x.

Examples

from symbolica import *
x = S('x')
p = E('x^2+2').to_polynomial()
print(p.integrate(x))

lcoeff

NumberFieldPolynomial.lcoeff() -> NumberFieldPolynomial

Get the leading coefficient.

Examples

from symbolica import Expression as E
p = E('3x^2+6x+9').to_polynomial().lcoeff()
print(p)

Yields 3.

monic

NumberFieldPolynomial.monic() -> NumberFieldPolynomial

Get the monic part of the polynomial, i.e., the polynomial divided by its leading coefficient.

Examples

from symbolica import Expression as E
p = E('6x^2+3x+9').to_polynomial()
print(p.monic())

Yields x^2+1/2*x+2/3.

nterms

NumberFieldPolynomial.nterms() -> int

Get the number of terms in the polynomial.

primitive

NumberFieldPolynomial.primitive() -> NumberFieldPolynomial

Get the primitive part of the polynomial, i.e., the polynomial with the content removed.

Examples

from symbolica import Expression as E
p = E('3x^2+6x+9').to_polynomial()
print(p.primitive())

Yields x^2+2*x+3.

quot_rem

NumberFieldPolynomial.quot_rem(rhs: NumberFieldPolynomial) -> Tuple[NumberFieldPolynomial, NumberFieldPolynomial]

Divide self by rhs, returning the quotient and remainder.

reduce

NumberFieldPolynomial.reduce(
    gs: Sequence[Polynomial],
    grevlex: bool = True,
) -> NumberFieldPolynomial

Related type: Polynomial

Completely reduce the polynomial w.r.t the polynomials gs.

If grevlex=True, reverse graded lexicographical ordering is used, otherwise the ordering is lexicographical.

Examples

E('y^2+x').to_polynomial().reduce([E('x').to_polynomial()])

yields y^2

reorder

NumberFieldPolynomial.reorder(vars: Sequence[Expression]) -> None

Related type: Expression

Reorder the polynomial in-place to use the given variable order.

replace

NumberFieldPolynomial.replace(
    x: Expression,
    v: NumberFieldPolynomial | int,
) -> NumberFieldPolynomial

Related type: Expression

Replace the variable x with a polynomial v.

Examples

from symbolica import *
x = S('x')
p = E('x*y+2*x+x^2').to_polynomial()
r = E('y+1').to_polynomial())
p.replace(x, r)

resultant

NumberFieldPolynomial.resultant(
    rhs: NumberFieldPolynomial,
    var: Expression,
) -> NumberFieldPolynomial

Related type: Expression

Compute the resultant of two polynomials with respect to the variable var.

to_expression

NumberFieldPolynomial.to_expression() -> Expression

Convert the polynomial to an expression.

Examples

from symbolica import *
e = E('x*y+2*x+x^2')
p = e.to_polynomial()
print((e - p.to_expression()).expand())

to_latex

NumberFieldPolynomial.to_latex() -> str

Convert the polynomial into a LaTeX string.

to_polynomial

NumberFieldPolynomial.to_polynomial() -> Polynomial

Convert the number field polynomial to a rational polynomial.