FiniteFieldPolynomial
FiniteFieldPolynomial()A Symbolica polynomial over finite fields.
Methods
| Name | Description |
|---|---|
| adjoin | Adjoin the coefficient ring of this polynomial R[a] with b, whose minimal polynomial |
| coefficient_list | Get the coefficient list, optionally in the variables vars. |
| contains | Check if the polynomial contains the given variable. |
| degree | Get the degree of the polynomial in var. |
| derivative | Take a derivative in x. |
| evaluate | Evaluate the polynomial at the given values. |
| 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 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. |
| nterms | Get the number of terms. |
| parse | Parse a polynomial with integer coefficients from a string. |
| primitive | Get the primitive part of the polynomial, i.e., the polynomial |
| quot_rem | Divide self by rhs, returning the quotient and remainder. |
| reduce | Completely reduce the polynomial w.r.t the polynomials gs. |
| reorder | Set a new variable ordering for the polynomial. |
| replace | Replace the variable x with a polynomial v. |
| resultant | Compute the resultant of two polynomials with respect to the variable var. |
| simplify_algebraic_number | Find the minimal polynomial for the algebraic number represented by this polynomial |
| to_expression | Convert the polynomial to an expression. |
| to_galois_field | Convert the polynomial to a Galois field defined by the minimal polynomial minimal_poly. |
| to_integer_polynomial | Convert the finite field polynomial to a polynomial with integer coefficients. |
| to_latex | Convert the polynomial into a LaTeX string. |
| to_polynomial | Convert a Galois field polynomial to a simple finite field polynomial. |
adjoin
FiniteFieldPolynomial.adjoin(b, new_symbol=None)Adjoin the coefficient ring of this polynomial R[a] with b, whose minimal polynomial is R[a][b] and form R[b]. Also return the new representation of a and b.
b must be irreducible over R and R[a]; this is not checked.
If new_symbol is provided, the variable of the new extension will be renamed to it. Otherwise, the variable of the new extension will be the same as that of b.
coefficient_list
FiniteFieldPolynomial.coefficient_list(vars=None)Get the coefficient list, optionally in the variables vars.
Examples
>>> from symbolica import Expression
>>> 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
FiniteFieldPolynomial.contains(var)Check if the polynomial contains the given variable.
degree
FiniteFieldPolynomial.degree(var)Get the degree of the polynomial in var.
derivative
FiniteFieldPolynomial.derivative(x)Take a derivative in x.
Examples
>>> from symbolica import Expression
>>> x = S('x')
>>> p = E('x^2+2').to_polynomial()
>>> print(p.derivative(x))evaluate
FiniteFieldPolynomial.evaluate(values)Evaluate the polynomial at the given values.
Examples
>>> from symbolica import *
>>> x, y = S('x', 'y')
>>> p = E('x*y+2*x+x^2').to_polynomial(modulus=5)
>>> print(p.evaluate([2, 3]))
4extended_gcd
FiniteFieldPolynomial.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(modulus=5).extended_gcd(E('x^2+2').to_polynomial(modulus=5))yields (1, 3+4*x, 3+x).
factor
FiniteFieldPolynomial.factor()Factorize the polynomial.
Examples
>>> from symbolica import Expression
>>> 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('\t({})^{}'.format(f, exp))factor_square_free
FiniteFieldPolynomial.factor_square_free()Compute the square-free factorization of the polynomial.
Examples
>>> from symbolica import Expression
>>> 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('\t({})^{}'.format(f, exp))format
FiniteFieldPolynomial.format(
mode=Ellipsis,
terms_on_new_line=False,
color_top_level_sum=True,
color_builtin_symbols=True,
print_ring=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,
precision=None,
show_namespaces=False,
include_attributes=False,
max_terms=None,
custom_print_mode=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))gcd
FiniteFieldPolynomial.gcd(*rhs)Compute the greatest common divisor (GCD) of two or more polynomials.
get_minimal_polynomial
FiniteFieldPolynomial.get_minimal_polynomial()Get the minimal polynomial of the algebraic extension.
get_variables
FiniteFieldPolynomial.get_variables()Get the list of variables in the internal ordering of the polynomial.
groebner_basis
FiniteFieldPolynomial.groebner_basis(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.
integrate
FiniteFieldPolynomial.integrate(x)Integrate the polynomial in x.
Examples
>>> from symbolica import Expression
>>> x = S('x')
>>> p = E('x^2+2').to_polynomial()
>>> print(p.integrate(x))lcoeff
FiniteFieldPolynomial.lcoeff()Get the leading coefficient.
Examples
>>> from symbolica import Expression
>>> p = E('3x^2+6x+9').to_polynomial().lcoeff()
>>> print(p)Yields 3.
nterms
FiniteFieldPolynomial.nterms()Get the number of terms.
parse
FiniteFieldPolynomial.parse(arg, vars, prime, 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'], 5)Raises
| Name | Type | Description |
|---|---|---|
| ValueError | If the input is not a valid Symbolica polynomial. |
primitive
FiniteFieldPolynomial.primitive()Get the primitive part of the polynomial, i.e., the polynomial with a leading coefficient of 1.
Examples
>>> from symbolica import Expression
>>> p = E('3x^2+6x+9').to_polynomial().primitive()
>>> print(p)Yields x^2+2*x+3.
quot_rem
FiniteFieldPolynomial.quot_rem(rhs)Divide self by rhs, returning the quotient and remainder.
reduce
FiniteFieldPolynomial.reduce(system, grevlex=True)Completely reduce the polynomial w.r.t the polynomials gs. For example reducing f=y^2+x by g=[x] yields y^2.
reorder
FiniteFieldPolynomial.reorder(order)Set a new variable ordering for the polynomial. This can be used to introduce new variables as well.
replace
FiniteFieldPolynomial.replace(x, v)Replace the variable x with a polynomial v.
Examples
>>> from symbolica import Expression
>>> x = S('x')
>>> p = E('x*y+2*x+x^2').to_polynomial()
>>> r = E('y+1').to_polynomial())
>>> p.replace(x, r)resultant
FiniteFieldPolynomial.resultant(rhs, var)Compute the resultant of two polynomials with respect to the variable var.
simplify_algebraic_number
FiniteFieldPolynomial.simplify_algebraic_number(minimal_poly)Find the minimal polynomial for the algebraic number represented by this polynomial expressed in the number field defined by minimal_poly.
Examples
>>> from symbolica import *
>>> (min_poly, rep2, rep23) = P('a^2-2').adjoin(P('b^2-3'))
>>> rep2.simplify_algebraic_number(min_poly)Yields b^2-2.
to_expression
FiniteFieldPolynomial.to_expression()Convert the polynomial to an expression.
to_galois_field
FiniteFieldPolynomial.to_galois_field(minimal_poly)Convert the polynomial to a Galois field defined by the minimal polynomial minimal_poly.
to_integer_polynomial
FiniteFieldPolynomial.to_integer_polynomial(symmetric_representation=True)Convert the finite field polynomial to a polynomial with integer coefficients.
to_latex
FiniteFieldPolynomial.to_latex()Convert the polynomial into a LaTeX string.
to_polynomial
FiniteFieldPolynomial.to_polynomial()Convert a Galois field polynomial to a simple finite field polynomial.