A new generating function for counting zeroes of polynomials

04/26/2016 - 12:15pm
04/26/2016 - 1:10pm
Daniel Katz (Cal State Northridge)

Suppose that we want to count the zeroes of a multivariable polynomial f whose coefficients come from the ring of integers modulo a power of a prime p. It is helpful to think of the polynomial f as having coefficients in the integers, and then can we count zeroes of f modulo each power of the prime p. The Igusa local zeta function Z_f is a generating function that organizes all these zero counts, and its poles tell us about the p-divisibility of the counts.

We devise a new method for calculating the Igusa local zeta function that involves a new kind of generating function G_f. Our new generating function is a projective limit of a family of generating functions, and contains more data than the Igusa local zeta function. The new generating function G_f resides in an algebra whose structure is naturally compatible with operations on the underlying polynomials, thus facilitating calculation of local zeta functions and helping us find zero counts of polynomials over finite fields and rings.

This is joint work with Raemeon A. Cowan and Lauren M. White.

Millikan 2099 (Pomona College)