Polynomial

Polynomial()

A Symbolica polynomial with rational coefficients.

Methods

Name	Description
approximate_roots	Approximate all complex roots of a univariate polynomial, given a maximal number of iterations
coefficient_list	Get the coefficient list, optionally in the variables `xs`.
content	Get the content, i.e., the GCD of the coefficients.
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`
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 polynomials.
get_var_list	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`.
interpolate	Perform Newton interpolation in the variable `x` given the sample points
isolate_roots	Isolate the real roots of the polynomial. The result is a list of intervals with rational bounds that contain exactly one root,
nterms	Get the number of terms in the polynomial.
parse	Parse a polynomial with integer coefficients from a string.
quot_rem	Divide `self` by `rhs`, returning the quotient and remainder.
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_finite_field	Convert the coefficients of the polynomial to a finite field with prime `prime`.
to_latex	Convert the polynomial into a LaTeX string.

approximate_roots

Polynomial.approximate_roots(max_iterations, tolerance)

Approximate all complex roots of a univariate polynomial, given a maximal number of iterations and a given tolerance. Returns the roots and their multiplicity.

Examples

>>> p = E('x^10+9x^7+4x^3+2x+1').to_polynomial()
>>> for (r, m) in p.approximate_roots(1000, 1e-10):
>>>     print(r, m)

coefficient_list

Polynomial.coefficient_list(xs)

Get the coefficient list, optionally in the variables xs.

Examples

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

content

Polynomial.content()

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

Examples

>>> from symbolica import Expression
>>> p = Expression.parse('3x^2+6x+9').to_polynomial()
>>> print(p.content())

derivative

Polynomial.derivative(x)

Take a derivative in x.

Examples

>>> from symbolica import Expression
>>> x = Expression.symbol('x')
>>> p = Expression.parse('x^2+2').to_polynomial()
>>> print(p.derivative(x))

extended_gcd

Polynomial.extended_gcd(rhs)

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).

Examples

>>> from symbolica import *
>>> E('(1+x)(20+x)').to_polynomial().extended_gcd(E('x^2+2').to_polynomial())

yields (1, 1/67-7/402*x, 47/134+7/402*x).

factor

Polynomial.factor()

Factorize the polynomial.

Examples

>>> from symbolica import Expression
>>> p = Expression.parse('(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

Polynomial.factor_square_free()

Compute the square-free factorization of the polynomial.

Examples

>>> from symbolica import Expression
>>> p = Expression.parse('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

Polynomial.format(
    terms_on_new_line=False,
    color_top_level_sum=True,
    color_builtin_symbols=True,
    print_finite_field=True,
    symmetric_representation_for_finite_field=False,
    explicit_rational_polynomial=False,
    number_thousands_separator=None,
    multiplication_operator='*',
    double_star_for_exponentiation=False,
    square_brackets_for_function=False,
    num_exp_as_superscript=True,
    latex=False,
    precision=None,
    show_namespaces=False,
    max_terms=None,
)

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

Examples

>>> p = FiniteFieldPolynomial.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

Polynomial.gcd(rhs)

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

get_var_list

Polynomial.get_var_list()

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

groebner_basis

Polynomial.groebner_basis(_cls, system, grevlex=True, print_stats=False)

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.

Examples

>>> basis = Polynomial.groebner_basis(
>>>     [Expression.parse("a b c d - 1").to_polynomial(),
>>>     Expression.parse("a b c + a b d + a c d + b c d").to_polynomial(),
>>>     Expression.parse("a b + b c + a d + c d").to_polynomial(),
>>>     Expression.parse("a + b + c + d").to_polynomial()],
>>>     grevlex=True,
>>>     print_stats=True
>>> )
>>> for p in basis:
>>>     print(p)

integrate

Polynomial.integrate(x)

Integrate the polynomial in x.

Examples

>>> from symbolica import Expression
>>> x = Expression.symbol('x')
>>> p = Expression.parse('x^2+2').to_polynomial()
>>> print(p.integrate(x))

interpolate

Polynomial.interpolate(_cls, x, sample_points, values)

Perform Newton interpolation in the variable x given the sample points sample_points and the values values.

Examples

>>> x, y = S('x', 'y')
>>> a = Polynomial.interpolate(
>>>         x, [4, 5], [(y**2+5).to_polynomial(), (y**3).to_polynomial()])
>>> print(a)

yields 25-5*x+5*y^2-y^2*x-4*y^3+y^3*x.

isolate_roots

Polynomial.isolate_roots(refine=None)

Isolate the real roots of the polynomial. The result is a list of intervals with rational bounds that contain exactly one root, and the multiplicity of that root. Optionally, the intervals can be refined to a given precision.

Examples

>>> from symbolica import Expression
>>> p = Expression.parse('2016+5808*x+5452*x^2+1178*x^3+-753*x^4+-232*x^5+41*x^6').to_polynomial()
>>> for a, b, n in p.isolate_roots():
>>>     print('({},{}): {}'.format(a, b, n))

yields

(-56/45,-77/62): 1
(-98/79,-119/96): 1
(-119/96,-21/17): 1
(-7/6,0): 1
(0,6): 1
(6,12): 1

nterms

Polynomial.nterms()

Get the number of terms in the polynomial.

parse

Polynomial.parse(_cls, input, vars, default_namespace='python')

Parse a polynomial with integer coefficients from a string. The input must be written in an expanded format and a list of all the variables must be provided.

If these requirements are too strict, use Expression.to_polynomial() or RationalPolynomial.parse() instead.

Examples

>>> e = Polynomial.parse('3*x^2+y+y*4', ['x', 'y'])

Raises

Name	Type	Description
	ValueError	If the input is not a valid Symbolica polynomial.

quot_rem

Polynomial.quot_rem(rhs)

Divide self by rhs, returning the quotient and remainder.

replace

Polynomial.replace(x, v)

Replace the variable x with a polynomial v.

Examples

>>> from symbolica import Expression
>>> x = Expression.symbol('x')
>>> p = Expression.parse('x*y+2*x+x^2').to_polynomial()
>>> r = Expression.parse('y+1').to_polynomial())
>>> p.replace(x, r)

resultant

Polynomial.resultant(rhs, var)

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

to_expression

Polynomial.to_expression()

Convert the polynomial to an expression.

Examples

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

to_finite_field

Polynomial.to_finite_field(prime)

Convert the coefficients of the polynomial to a finite field with prime prime.

to_latex

Polynomial.to_latex()

Convert the polynomial into a LaTeX string.