Examples

In [1]:f=x(0)+x(1)+x(2)

In [2]:ambient_space=(x(0)*x(1)*x(2)).divisors()

In [3]:ambient_space
Out[3]:{{x(0),x(1),x(2)}, {x(0),x(1)}, {x(0),x(2)}, {x(0)}, {x(1),x(2)}, {x(1)}, {x(2)}, {}}

In [4]:f.zerosIn(ambient_space)
Out[4]:{{x(0),x(1)}, {x(0),x(2)}, {x(1),x(2)}, {}}

In [5]:S=BooleSet([x(0),x(1)*x(2)])

In [6]:f.zerosIn(S)
Out[6]:{{x(1),x(2)}}

A example of the second kind, where only the full ambient space can be considered count lex_groebner_basis_points

In [1]:S=BooleSet([x(0),x(1)*x(2)])

In [2]:from polybori.interpolate import *              

In [3]:lex_groebner_basis_points(S,x(0)*x(1)*x(2))
Out[3]:[x(0) + x(2) + 1, x(1) + x(2)]

In [4]:lex_groebner_basis_points(S,x(0)*x(1)*x(2)*x(3))
Out[4]:[x(0) + x(2) + 1, x(1) + x(2), x(3)]

This function calculates the reduced lexicographical Gröbner basis of the vanishing ideal of $ S$ . Here the ambient space matters, as an additional component would mean, that the corresponding entries are zero, so we would get an additional generator for the ideal $ x_3$ .



2009-09-10