The Segre zeta function of an ideal

Paolo Aluffi

We define a power series associated with a homogeneous ideal in a
polynomial ring, encoding information on the Segre classes defined by
extensions of the ideal in projective spaces of arbitrarily high
dimension. We prove that this power series is rational, with poles
corresponding to generators of the ideal, and with numerator of
bounded degree and with nonnegative coefficients. We also prove that
this `Segre zeta function' only depends on the integral closure of the
ideal.

The results follow from good functoriality properties of the `shadows'
of rational equivalence classes of projective bundles. More precise
results can be given if all homogeneous generators have the same
degree, and for monomial ideals.

In certain cases, the general description of the Segre zeta function
given here leads to substantial improvements in the speed of
algorithms for the computation of Segre classes. We also compute the
projective ranks of a nonsingular variety in terms of the
corresponding zeta function, and we discuss the Segre zeta function of
a local complete intersection of low codimension in projective space.