Date Available
3-22-2011
Year of Publication
2010
Degree Name
Doctor of Philosophy (PhD)
Document Type
Dissertation
College
Arts and Sciences
Department
Mathematics
First Advisor
Dr. David Lee
Abstract
The Poincaré series, Py(f) of a polynomial f was first introduced by Borevich and Shafarevich in [BS66], where they conjectured, that the series is always rational. Denef and Igusa independently proved this conjecture. However it is still of interest to explicitly compute the Poincaré series in special cases. In this direction several people looked at diagonal polynomials with restrictions on the coefficients or the exponents and computed its Poincaré series. However in this dissertation we consider a general diagonal polynomial without any restrictions and explicitly compute its Poincaré series, thus extending results of Goldman, Wang and Han. In a separate chapter some new results are also presented that give a criterion for an element to be an mth power in a complete discrete valuation ring.
Recommended Citation
Deb, Dibyajyoti, "DIAGONAL FORMS AND THE RATIONALITY OF THE POINCARÉ SERIES" (2010). University of Kentucky Doctoral Dissertations. 25.
https://uknowledge.uky.edu/gradschool_diss/25
