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radical -- the radical of an ideal

Synopsis

Description

If I is an ideal in an affine ring (i.e. a quotient of a polynomial ring over a field), and if the characteristic of this field is large enough (see below), then this routine yields the radical of the ideal I.

The method used is the Eisenbud-Huneke-Vasconcelos algorithm. See their paper in Inventiones Mathematicae, 1993, for more details on the algorithm.

The algorithms used generally require that the characteristic of the ground field is larger than the degree of each primary component. In practice, this means that if the characteristic is something like 32003, rather than, for example, 5, the methods used will produce the radical of I. Of course, you may do the computation over QQ, but it will often run much slower. In general, this routine still needs to be tuned for speed.

Computes the radical of I using the Eisenbud-Huneke-Vasconcelos algorithm. If I is a monomial ideal, a faster "combinatorial" algorithm is used.

i1 : R=QQ[x,y]

o1 = R

o1 : PolynomialRing
i2 : I=ideal((x^2+1)^2*y, y+1)

             4      2
o2 = ideal (x y + 2x y + y, y + 1)

o2 : Ideal of R
i3 : radical I

                        2
o3 = ideal (- y - 1, - x  - 1)

o3 : Ideal of R

For another example, see component example.

Caveat

The current implementation requires that the characteristic of the ground field is either zero or a "large" prime (unless I is a monomial ideal).

See also

Ways to use radical :