Usage:
<l_tangentcone I J
Compute the tangent cone, or the associated graded algebra,
of a ring R/I, where R is the localization of a polynomial ring S at the
origin (ie. the localization at the homogeneous maximal ideal).
Parameters:
I = 1xn matrix over a polynomial ring S.
Output values:
J = a standard basis for the ideal defining the tangent cone.
J is a 1xr matrix whose entries are homogeneous.
By definition, S/J = R/m + m/m^2 + m^2/m^3 + ...,
where "m" is the homogeneous maximal ideal of R or S.
J is computed by homogenizing w.r.t. a new variable, say "t",
and computing a standard basis of I using the monomial order which
eliminates "t", refined by graded reverse lexicographic order.
After dehomogenizing each element of this standard basis, a standard
The leading form of lowest degree of each element of this standard
basis, after setting "t" to one, forms a standard basis for the
ideal J.
This standard basis J can be used to determine various information
about the ring R/I:
1. the codimension of I:
<l_tangentcone I J
codim J
2. the multiplicity of R/I:
<l_tangentcone I J
degree J
3. ideal of initial monomials:
<l_tangentcone I J
in J in'I