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In this paper we study a class of countable and discrete subsets of a Euclidean space that are “self-similar” with respect to a finite set of (affine) similarities. Any such set can be interpreted as having a fractal structure. We introduce a zeta function for these sets, and derive basic analytic properties of this “fractal” zeta function. Motivating examples that come from combinatorial geometry...
A question of Igusa (from 1978) inquires about the singular behavior of the singular series, determined by a polynomial mapping $$P:K^n \to K^m ,m \leqslant n$$ , where K is a local field of characteristic zero. This paper describes in geometric terms the singularities of the singular series for two classes of polynomial maps $$P = \left( {P_2{1} ,P_2{2} } \right):K^n \to K^2{2} $$ . The...
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