Igusa zeta function
In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, "modulo" "p", "p", "p", and so on. Definition. For a prime number "p" let "K" be a p-adic field, i.e. formula_1, "R" the valuation ring and "P" the maximal ideal. For formula_2 we denote by formula_3 the valuation of "z", formula_4, and formula_5 for a uniformizing parameter π of "R". Furthermore let formula_6 be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let formula_7 be a character of formula_8. In this  situation one associates to a non-constant polynomial formula_9 the Igusa zeta function where formula_11 and "dx" is Haar measure so normalized that formula_12 has measure 1. Igusa's theorem. showed that formula_13 is a rational function in formula_14. The proof uses Heisuke Hironaka's theorem about the resolution of singularities. Later, an entirely different proof was given by Jan Denef using p-adic cell decomposition. Little is known, however, about explicit formulas. (There are some results about Igusa zeta functions of Fermat varieties.) Congruences modulo powers of formula_15. Henceforth we take formula_16 to be the characteristic function of formula_12  and formula_7 to be the trivial character. Let formula_19 denote the number of solutions of the congruence Then the Igusa zeta function is closely related to the Poincaré series by