URS AND URSIMS FOR P-ADIC MEROMORPHIC FUNCTIONS INSIDE A DISC

Let $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We show that the $p$-adic main Nevanlinna Theorem holds for meromorphic functions inside an ‘open’ disc in $K$. Let $P_{n,c}$ be the Frank–Reinders’s polynomial$$ (n-1)(n-2)X^n-2n(n-2)X^{n-1}+ n(n-1)X^{n-2}-c\qq (c\neq0,\ c\neq1,\ c\neq2) $$and let $S_{n,c}$ be the set of its $n$ distinct zeros. For every $n\geq 7$, we show that $S_{n,c}$ is an $n$-points unique range set (counting multiplicities) for unbounded analytic functions inside an ‘open disc’, and for every $n\geq10$, we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for the same set of functions. Similar results are obtained for meromorphic functions whose characteristic function is unbounded: we obtain unique range sets ignoring multiplicities of $17$ points. A better result is obtained for an analytic or a meromorphic function $f$ when its derivative is ‘small’ comparatively to $f$. In particular, for every $n\geq5$ we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for unbounded analytic functions with small derivative. Actually, in each case, results also apply to pairs of analytic functions when just one of them is supposed unbounded. The method we use is based upon the $p$-adic Nevanlinna Theory, and Frank–Reinders’s and Fujimoto’s methods used for meromorphic functions in $\mathbb{C}$. Among other results, we show that the set of functions having a bounded characteristic function is just the field of fractions of the ring of bounded analytic functions in the disc.AMS 2000 Mathematics subject classification: Primary 12H25. Secondary 12J25; 46S10

The Depth Efficacy of Unbounded: Characteristic Finite Field Arithmetic

We introduce an arithmetic model of parallel computation. The basic operations are ½ and Š gates over finite fields. Functions computed are unary and increasing input size is modelled by shifting the arithmetic base to a larger field. When only finite fields of bounded characteristic are used, then the above model is fully general for parallel computations in that size and depth of optimal arithmetic solutions are polynomially related to size and depth of general (boolean) solutions. In the case of finite fields of unbounded characteristic, we prove that the existence of a fast parallel (boolean) solution to the problem of powering an integer modulo a prime (and powering a polynomial modulo an irreducible polynomial) in combination with the existence of a fast parallel (arithmetic) solution for the problem of computing a single canonical function, f<em>(x)</em>, in the prime fields, guarantees the full generality of the finite field model of computation. We prove that the function f<em>(x)</em>, has a fast parallel arithmetic solution for any ''shallow'' class of primes, i.e. primes <em>p</em> such that any prime power divisor <em>q</em> of <em>p</em> -1 is bounded in value by a polynomial in log <em>p</em>.