syngular.field

class syngular.field.Field(*args)

A class representing number fields.

property I

property characteristic

property digits

epsilon(shape=(1,))

property i

property is_algebraically_closed

property j

property name

property one

random(shape=(1,))

random_element(*args, **kwargs)

random_square()

set(*args): (name, characteristic, digits)

property singular_notation

sqrt(val)

property tollerance

property zero

property ε

syngular.ring

class syngular.ring.Ring(field, variables, ordering)

property field

property ordering: Monomial ordering

random_point(field, seed=None): Returns a random numerical point in the given field on the zero ideal of the ring.

test_valid_ring()

unit_ideal(): Returns the unit ideal ⟨1⟩ in the ring.

univariate_slice(field, extra_approximate_constraints=(), indepSet=None, seed=None, verbose=False)

property variables

zero_ideal(): Returns the zero ideal ⟨0⟩ in the ring.

syngular.qring

syngular.qring.QRing: alias of QuotientRing

class syngular.qring.QuotientRing(ring, ideal)

test_valid_qring()

syngular.ideal

class syngular.ideal.Ideal(ring, generators)

property codim

property codims

delete_cached_properties()

property dim

property dims

eliminate(var_range)

property generators

generators_eval(**kwargs)

get_groebner_basis(reduced=False, algorithm='slimgb')

property groebner_basis

guess_indep_set(): Guesses an independent set, you can provide codim_upper_bound attribute to help.

property indepSet

property indepSets

static intersection(*args): Intersection of Ideals - wrapper around & operator for chained intersection.

property is_unit_ideal

property leadGBmonomials: Gives the leading monomials of the Groebner basis polynomials.

property minbase

property primary_decomposition

property radical: Returns the radical of the ideal.

reduce(other): Remainder of division, i.e. reduction.

property reduced_groebner_basis

saturation(other): Saturation of ideals (self : other^∞), returns both saturation ideal and saturation index.

saturation_index(other): Saturation of ideals (self : other^∞), returns only the saturation index.

squash()

squoosh()

test_valid_ideal()

to_full_ring()

to_qring(other)

syngular.ideal.monomial_to_exponents(variables, monomial): Converts a monomial in the variables of a polynomial ring into a numpy.array of exponents.

syngular.ideal.reduce(poly, ideal)

syngular.variety

class syngular.variety.Variety_of_Ideal

point_on_variety(field, base_point={}, directions=None, valuations=(), indepSet='guess', seed=None, verbose=False, directions_analytic_check=False): Generate a representative point on or close to the variety associated to this ideal. The point is ‘valuations’ away from the exact variety, in the directions specified by ‘directions’. If ‘directions’ are not provided, pick the first n=codim simplest generators from ‘self’. If the ideal is not prime, an irreducible branch will be picked at random. If independent sets are too hard to compute, they will be guessed, you can provide codim_upper_bound to help.

syngular.variety.check_solutions(equations, root_dicts, field): Checks that all solutions in root_dicts solve the equations.

syngular.variety.lex_groebner_solve(equations, prime=None): Returns the variety corresponding to a given zero dimensional ideal in lexicographic groebner basis form. The variety take the form of a list of dictionaries for the possible values of the variables.

syngular.variety.retry_to_find_root(max_tries=100)

syngular.variety.univariate_finite_field_solver(equation, root_dict, prime): Returns all possible solutions of ‘equation’ over a finite field of cardinality ‘prime’. If already satisfied returns True, if no solution exists returns False.

syngular.variety.univariate_floating_point_solver(equation, root_dict): Returns all possible solutions of ‘equation’ over arbitrary precision complex numbers.

syngular.variety.update_point_dict(base_point_dict, new_vals_dict, field)

syngular.variety.update_root_dict(symbol, solutions, root_dict): Given solutions and root_dict returns updated root_dicts.

Module contents

class syngular.Field(*args)

A class representing number fields.

property I

property characteristic

property digits

epsilon(shape=(1,))

property i

property is_algebraically_closed

property j

property name

property one

random(shape=(1,))

random_element(*args, **kwargs)

random_square()

set(*args): (name, characteristic, digits)

property singular_notation

sqrt(val)

property tollerance

property zero

property ε

class syngular.Ideal(ring, generators)

property codim

property codims

delete_cached_properties()

property dim

property dims

eliminate(var_range)

property generators

generators_eval(**kwargs)

get_groebner_basis(reduced=False, algorithm='slimgb')

property groebner_basis

guess_indep_set(): Guesses an independent set, you can provide codim_upper_bound attribute to help.

property indepSet

property indepSets

static intersection(*args): Intersection of Ideals - wrapper around & operator for chained intersection.

property is_unit_ideal

property leadGBmonomials: Gives the leading monomials of the Groebner basis polynomials.

property minbase

property primary_decomposition

property radical: Returns the radical of the ideal.

reduce(other): Remainder of division, i.e. reduction.

property reduced_groebner_basis

saturation(other): Saturation of ideals (self : other^∞), returns both saturation ideal and saturation index.

saturation_index(other): Saturation of ideals (self : other^∞), returns only the saturation index.

squash()

squoosh()

test_valid_ideal()

to_full_ring()

to_qring(other)

class syngular.Monomial(*_)

A FrozenMultiset representation of a Monomial. Positive integer multiplicities represent powers.

as_exps_list(ring): Converts the monomial into an array of exponents w.r.t. variables in ring.

property exps

property invs

subs(values_dict)

tolist()

property variables

class syngular.Polynomial(coeffs_and_monomials, field)

Generalization of the concept of Multiset where multiplicities are in an arbitrary Field and the elements are Monomials.

property coeffs

property coeffs_and_monomials

property field

property lead_monomial

property lead_term

property lexps

property linvs

property monomials

rationalise()

reduce(): Merges equal monomials

subs(base_point, field=None)

property variables

syngular.QRing: alias of QuotientRing

class syngular.QuotientRing(ring, ideal)

test_valid_qring()

class syngular.Ring(field, variables, ordering)

property field

property ordering: Monomial ordering

random_point(field, seed=None): Returns a random numerical point in the given field on the zero ideal of the ring.

test_valid_ring()

unit_ideal(): Returns the unit ideal ⟨1⟩ in the ring.

univariate_slice(field, extra_approximate_constraints=(), indepSet=None, seed=None, verbose=False)

property variables

zero_ideal(): Returns the zero ideal ⟨0⟩ in the ring.

class syngular.RingPoint(ring, field, seed=None, val=None)

Represents a numerical or semi-numerical point on a variety within the space defiend by the ring. Generalizes the idea of a phase space point from particle physics.

copy() → a shallow copy of D

singular_variety(directions_or_ideal=None, valuations=(), seed=None, verbose=False)

subs(myDict)

univariate_slice(extra_approximate_constraints=(), indepSet=None, seed=None, verbose=False)

class syngular.RingPoints(*args)

property field

exception syngular.SingularException

class syngular.TemporarySetting(module_or_module_name, setting_name, new_value)

syngular.get_singular_version(raise_error=True)