Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.

In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic number field. In practice, this problem is often considered to be wellsolved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for factoring integers. Applying a variant of a standard algorithm for finding rings of integers, one finds a subring of the number field that one may view as the "best guess" one has for the ring of integers. This best guess is probably often correct. Our main concern is what can be proved about this subring. We show that it has a particularly transparent local structure, which is reminiscent of the structure of tamely ramified extensions of local fields. A major portion of the paper is devoted to the study of rings that are "tame" in our more general sense. As a byproduct, we prove complexity results that elaborate upon a ...

. Let S be a nite set of positive integers. A \coprime base for S" means a set P of positive integers such that (1) each element of P is coprime to every other element of P and (2) each element of S is a product of powers of elements of P . There is a natural coprime base for S. This paper introduces an algorithm that computes the natural coprime base for S in essentially linear time. The best previous result was a quadratic-time algorithm of Bach, Driscoll, and Shallit. This paper also shows how to factor S into elements of P in essentially linear time. The algorithms apply to any free commutative monoid with fast algorithms for multiplication, division, and greatest common divisors; e.g., monic polynomials over a eld. They can be used as a substitute for prime factorization in many applications. 1.

Given a squarefree polynomial P 2 k 0 [x; y], k 0 a number eld, we construct a linear dierential operator that allows one to calculate the genus of the complex curve dened by P = 0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k 0 , and calculate information concerning the Galois group of P over k 0 (x) as well as over k 0 (x).

. We give a new algorithm for resolving singularities of plane curves. The algorithm is polynomial time in the bit complexity model, does not require factorization, and works over Q or finite fields. 1 Introduction Resolving singularities is a central problem in computational algebraic geometry. In this paper we describe a new algorithm for resolving singularities of irreducible plane curves. The algorithm runs in polynomialtime in the bit complexity model, does not require polynomial factorization, and works over Q or any finite field. Classical algorithms for resolving singularities [2, 15, 7] use a combination of methods involving -- the Newton polygon, a polygon in Z 2 whose vertices are the exponents of terms in f ; -- Puiseux series, power series with fractional exponents. These algorithms take polynomial time if we assume efficient factorization over algebraic extensions of the base field and unit-time arithmetic these extensions. Teitelbaum [13] establishes bounds on the d...

Contents 1 Summary 2 2 Goal of the Project 4 2.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 What is an m-adjoint? . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Eciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Importance of the Goal 7 3.1 Rational Parameterization and Problems in CAD/CAM . . . . . 7 3.2 Solving Geometric Problems on Irrational Surfaces . . . . . . . . 9 4 State of the Art 10 4.1 The curve case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 The theory of adjoints . . . . . . . . . . . . . . . . . . . . . . . . 10 4.3 The only algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3.1 A severe technical problem . . . . . . . . . . . . . . . . . 12 4.3.2 A severe theoretical problem . . . . . . . . . . . . . . . . 13 4.4 The related problem of the resolution of singularities . . . . . . . 14 4.4.1 Abhyankar's method