Effective model theory is the subject that analyzes the typical notions and results of model theory to determine their effective content and counterparts. The subject has been developed both in the former Soviet Union and in the west with various names (recursive model theory, constructive model theory,

this paper we are interested in those structures in which the basic computations can be performed by Turing machines.

In this paper we show how group theoretic information can be used to derive a set of necessary conditions on the coefficients of L(y) for L(y) = 0 to have a liouvillian solution. The method is used to derive (and improve in one case) the necessary conditions of the Kovacic algorithm and to derive an explicit set of necessary conditions for third order differential equations. 1 Introduction In our previous work [20], [21], we have shown how group theoretic techniques can be used to develop effective algorithms to calculate Galois groups of second and third order homogeneous linear differential equations and to decide questions about the algebraic nature of the solutions of such equations (e.g., solvability in terms of liouvillian functions or in terms of linear differential equations of lower order). -- A weaker version of these results were announcened in Liouvillian Solutions of Third Order Linear Differential Equations: New Bounds and Necessary Conditions, Proceedings of the 1992 ...

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Abstract. We show that Mazur’s conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers Z in the rational numbers Q, i.e., there is no diophantine set D in some cartesian power Q i such that there exist two binary relations S, P on D whose graphs are diophantine in Q 3i (via the inclusion D 3 ⊂ Q 3i), and such that for two specific elements d0, d1 ∈ D the structure (D, S, P, d0, d1) is a model for integer arithmetic (Z,+, ·,0, 1). Using a construction of Pheidas, we give a counterexample to the analogue of Mazur’s conjecture over a global function field, and prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over a finite field. 1.

We present a sequential deterministic polynomial-time algorithm for testing dense multivariate polynomials over a large finite field for irreducibility. All previously known algorithms were of a probabilistic nature. Our deterministic solution is based on our algorithm for absolute irreducibility testing combined with Berlekamp's algorithm.

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The Exact Geometric Computation (EGC) mode of computation has been developed over the last decade in response to the widespread problem of numerical non-robustness in geometric algorithms. Its technology has been encoded in libraries such as LEDA, CGAL and Core Library. The key feature of EGC is the necessity to decide zero in its computation. This paper addresses the problem of providing a foundation for the EGC mode of computation. This requires a theory of real computation that properly addresses the Zero Problem. The two current approaches to real computation are represented by the analytic school and algebraic school. We propose a variant of the analytic approach based on real approximation. • To capture the issues of representation, we begin with a reworking of van der Waerden’s idea of explicit rings and fields. We introduce explicit sets and explicit algebraic structures. • Explicit rings serve as the foundation for real approximation: our starting point here is not R, but F ⊆ R, an explicit ordered ring extension of Z that is dense in R. We develop the approximability of real functions within standard Turing machine computability, and show its connection to the analytic approach. • Current discussions of real computation fail to address issues at the intersection of continuous and discrete computation. An appropriate computational model for this purpose is obtained by extending Schönhage’s pointer machines to support both algebraic and numerical computation. • Finally, we propose a synthesis wherein both the algebraic and the analytic models coexist to play complementary roles. Many fundamental questions can now be posed in this setting, including transfer theorems connecting algebraic computability with approximability. 1

Here we consider some simple decision problems to do with fields and abelian groups and show that they are algorithmically insoluble and, indeed, that they can occur with arbitrary complexity. THEOREM. For any recursively enumerable set A, there exists a computable field F

on topological spaces via domain representations