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.

. Let S 0 ; : : : ; S 5 be 6 spheres in R 3 and H a hexahedron with vertices P 0 ; : : : ; P 5 . How many ways are there to move H in such a way that P i belongs to S 1 , i = 0; : : : ; 5 ? We show in this paper that generically there are at most 40 solutions. This problem is the geometric version of a control problem for Stewart robots. 1991 Mathematics Subject Classification: 70B15, 14C17, 51M20. A (generalized) Stewart platform is a solid with 6 points P 0 ; : : : ; P 5 on it attached through 6 legs to 6 fixed points Q 0 ; : : : ; Q 5 in the space R 3 . Assuming that the lengths of the legs can be varied arbitrarily (within the physical limits), the problem, first considered by D. Stewart [8], is to control the position of the body. y In mathematical terms, we can identify the space of positions of the solid with the space SO(3) \Theta R 3 of rotations and translations of R 3 . We have a map: \Phi = \Phi P i ;Q i : SO(3) \Theta R 3 ! R 6 ; \Phi(R; T ) = i kT +R(P i ...

. { Certain fundamental solutions E a of the partial dierential operators @ 3 1 + @ 3 2 + @ 3 3 + 3a@ 1 @ 2 @ 3 ; a 2 R n f1g; are represented by elliptic integrals of the rst kind. These operators are (apart from a linear change of variables) the most general homogeneous operators of real principal type in three variables and irreducible of degree three. The fundamental solutions E a show interesting non-convex lacunas, and their remaining level sets are algebraic surfaces of degree 6, explicitly calculated. 1. INTRODUCTION 1.1. The operator @ 3 1 +@ 3 2 +@ 3 3 was considered|to my knowledge|for the rst time in 1913 in N. Zeilon's article [20], wherein he generalizes I. Fredholm's method of construction of fundamental solutions (see [5]) from homogeneous elliptic equations to arbitrary homogeneous equations in three variables with a real-valued symbol (cf. [20, II, pp. 14{ 22], [6, Ch. 11, pp. 146-148]). An explicit formula for a fundamental solution was given in [19]. The o...

This paper collects together formulae concerning singularities at infinity of plane algebraic curves. For every polynomial f : C 2 ! C with isolated critical points we consider the well known topological invariants (f) (the global Milnor number) and (f) (the sum of all the "jumps" in the Milnor number at infinity). We prove new estimations for (f) +(f) and show that the number of critical values at infinity of f is less than or equal to ((d \Gamma 1) 2 \Gamma (f) \Gamma (f))=d where d is the degree of f . We give also some estimations for the / Lojasiewicz exponent at infinity.

Abstract. For a certain class of varieties X, we derive a formula for the valuation dX on the arc space L(Y) of a smooth ambient space Y, in terms of an embedded resolution of singularities. A simple transformation rule yields a formula for the geometric Poincaré series. Furthermore, we prove that for this class of varieties, the arithmetic and the geometric Poincaré series coincide. We also study the geometric valuation for plane curves. 1.

Abstract. Some properties of the relation between the singular point set and the non-proper value curve of polynomial maps of C 2 are expressed in terms of Newton-Puiseux expansions. 1.

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Singular mobiles were introduced by Encinas and Hauser in order to conceptualize the information which is necessary to prove strong resolution of singularities in characteristic zero. It turns out that after Hironaka’s Annals paper from 1964 essentially all proofs rely – either implicitly or explicitly – on the data collected in a mobile, often with only small technical variations. The present text explains why mobiles are the appropriate resolution datum and how they are used to build up the induction argument of the proof.

This paper is to present a geometrical proof of the plane Jacobian conjecture for rational polynomials by an approach of Newton-Puiseux data and geometry of rational surfaces. The obtained result shows that a polynomial map F = (P, Q) : C 2 − → C 2 with PxQy − PyQx ≡ const. ̸ = 0 has a polynomial inverse if the component P is a rational polynomial, i.e. if the generic fiber of P is the 2-dimensional topological sphere with a finite number of punctures.

Let Γ be a plane curve of degree d with δ ordinary nodes and no other singularities. Let