Mathematical Sciences: Arithmetic of Polynomials and Theta Functions

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The Principal Investigator will continue his work on complex powers, both local and global, and, in addition, continue his work on the so-called Igusa local zeta functions and zeta distributions. He will concentrate on the prehomogeneous case, which is interesting in itself and from which one can formulate sharp conjectures. The theory of theta functions will also be looked at. The author studies special kinds of zeta functions which have importance in the arithmetic theory of algebraic geometry. Determining the polynomials associated to the local factors will receive special attention. Also discussed will be the convergence, analytic continuation and distribution of the poles of these functions.