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Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... 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 ..."
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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.
Stewart Platforms Without Computer?
, 1992
"... . 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 ..."
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Cited by 12 (0 self)
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. 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 ...
Fundamental Solutions of Real Homogeneous Cubic Operators of Principal Type in Three Dimensions
, 1998
"... . { 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 princ ..."
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Cited by 7 (4 self)
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. { 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 nonconvex 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 consideredto my knowledgefor 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 realvalued symbol (cf. [20, II, pp. 14{ 22], [6, Ch. 11, pp. 146148]). An explicit formula for a fundamental solution was given in [19]. The o...
Formulae For The Singularities At Infinity Of Plane Algebraic Curves
, 2000
"... 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 ..."
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Cited by 6 (0 self)
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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.
ARCS AND RESOLUTION OF SINGULARITIES
, 2004
"... 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 f ..."
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Cited by 4 (1 self)
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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.
A note on singularity and nonproper value set of polynomial maps
 of C
"... Abstract. Some properties of the relation between the singular point set and the nonproper value curve of polynomial maps of C 2 are expressed in terms of NewtonPuiseux expansions. 1. ..."
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Cited by 3 (3 self)
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Abstract. Some properties of the relation between the singular point set and the nonproper value curve of polynomial maps of C 2 are expressed in terms of NewtonPuiseux expansions. 1.
Local properties of Jcomplex curves in Lipschitz structures
, 2009
"... We prove the existence of primitive curves and positivity of intersections of Jcomplex curves for Lipschitzcontinuous almost complex structures. These results are deduced from the Strong Comparison Theorem for Jholomorphic maps in Lipschitz structures previously known for J of class C 2. We also ..."
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We prove the existence of primitive curves and positivity of intersections of Jcomplex curves for Lipschitzcontinuous almost complex structures. These results are deduced from the Strong Comparison Theorem for Jholomorphic maps in Lipschitz structures previously known for J of class C 2. We also give the optimal regularity of curves in Lipschitz structures. It occurs to be C 1,LnLip, i.e., a Jcomplex curve for Lipschitz J has its first derivatives LogLipschitz continuous. A simple example that nothig better could be achieved is given.
Plane Jacobian conjecture for rational polynomials
, 2008
"... By an approach of the NewtonPuiseux data and the geometry of rational ruled surfaces we present a geometric proof of the plane Jacobian conjecture for the rational case: 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 ..."
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By an approach of the NewtonPuiseux data and the geometry of rational ruled surfaces we present a geometric proof of the plane Jacobian conjecture for the rational case: 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 2dimensional topological sphere with a finite number of punctures.
CHARACTERIZATION OF NONDEGENERATE PLANE CURVE SINGULARITIES
, 2007
"... Abstract. We characterize plane curve germs (nondegenerate in Kouchnirenko’s sense) in terms of characteristics and intersection multiplicities of branches. 1. Introduction. In this paper we consider (reduced) plane curve germs C, D,... centered at a fixed point O of a complex nonsingular surface. ..."
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Abstract. We characterize plane curve germs (nondegenerate in Kouchnirenko’s sense) in terms of characteristics and intersection multiplicities of branches. 1. Introduction. In this paper we consider (reduced) plane curve germs C, D,... centered at a fixed point O of a complex nonsingular surface. Two germs C and D are equisingular if there exists a bijection between their branches which preserves characteristic pairs and intersection numbers. Let (x, y) be a chart centered at O. Then a plane curve germ has a local equation of