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We carefully study the number of arithmetic operations required to compute rational Puiseux expansions of a bivariate polynomial F over a finite field. Our approach is based on the rational Newton-Puiseux algorithm introduced by D. Duval. In particular, we prove that coefficients of F may be significantly truncated and that certain complexity upper bounds may be expressed in terms of the output size. These preliminary results lead to a more efficient version of the algorithm with a complexity upper bound that improves previously published results. We also deduce consequences for the complexity of the computation of the genus of an algebraic curve defined over a finite field or an algebraic number field. Our results are practical since they are based on well established subalgorithms, such as fast multiplication of univariate polynomials with coefficients in a finite field.

The research work presented in this thesis concern to the stability analysis of linear time-delay systems with low-order controllers. This thesis is divided into three parts.The first part of the thesis focus on the study of linear SISO (single-input/single-output) systems with input/output delays, where the feedback loop is closed with a controller of PID-type. Inspired by the geometrical approach developed by Gu et al. we propose an analytical method to find the stability regions of all stabilizing controllers of PID-type for the time-delay system. Based on this same approach, we propose an algorithm to calculate the degree of fragility of a given controller of PID- type (PI, PD and PID).The second part of the thesis focuses on the stability analysis of linear systems under an NCS (Networked System Control) based approach. More precisely, we first focus in the stabilization problem by taking into account the induced network delays and the effects induced by the sampling period. To carry out such an analysis we have adopted an eigenvalue perturbation-based approach.Finally, in the third part of the thesis we tackle certain problems concerning to the behavior of the zeros of a certain class of sampled-data SISO systems. More precisely, given a continuous-time system, we obtain the sampling intervals guaranteeing the invariance of the number of unstable zeros in each interval. To perform such an analysis, we adopt an eigenvalue perturbation-based approach., Les travaux de recherche présentés dans cette thèse concernent des contributions à l’étude de stabilité des systèmes linéaires à retards avec contrôleurs d’ordre réduit. Cette mémoire est partagée en trois parties.La première partie est axée sur l’étude des systèmes linéaires à retard mono-entré /mono-sortie, bouclées avec un contrôleur de type PID. Inspiré par l’approche géométrique développée par Gu et al. Nous avons proposé une méthode analytique pour trouver la région (ou les régions) de tous les contrôleurs de type PID stabilisant pour le système à retard. Basée sur cette même approche, on a développé un algorithme pour calculer le dégrée de fragilité d’un contrôleur donné de type PID (PI, PD et PID).La deuxième partie de la thèse est axée sur l’étude de stabilité sous une approche NCS (pour son acronyme en anglais : Networked Control System). Plus précisément, nous avons d’abord étudié le problème de la stabilisation en tenant compte des retards induit par le réseau et les effets induits par la période d’échantillonnages. Pour mener une telle analyse nous avons adopté une approche basée sur la théorie des perturbations. Finalement, dans la troisième partie de la thèse nous abordons certains problèmes concernant le comportement des zéros d’une certaine classe de systèmes échantillonnés mono-entré /mono-sortie. Plus précisément, étant donné un système à temps continu, on obtient les intervalles d’échantillonnage garantissant l’invariance du nombre de zéros instables dans chaque intervalle. Pour développer cette analyse, nous adoptons une approche basée sur la perturbation aux valeurs propres.

19 pages; International audience; The beta-conjugates of a base of numeration $\beta > 1$, $\beta$ being a Parry number, were introduced by Boyd, in the context of the Rényi-Parry dynamics of numeration system and the beta-transformation. These beta-conjugates are canonically associated with $\beta$. Let $\beta > 1$ be a real algebraic number. A more general definition of the beta-conjugates of $\beta$ is introduced in terms of the Parry Upper function $f_{\beta}(z)$ of the beta-transformation. We introduce the concept of a germ of curve at $(0,1/\beta) \in \mathbb{C}^{2}$ associated with $f_{\beta}(z)$ and the reciprocal of the minimal polynomial of $\beta$. This germ is decomposed into irreducible elements according to the theory of Puiseux, gathered into conjugacy classes. The beta-conjugates of $\beta$, in terms of the Puiseux expansions, are given a new equivalent definition in this new context. If $\beta$ is a Parry number the (Artin-Mazur) dynamical zeta function $\zeta_{\beta}(z)$ of the beta-transformation, simply related to $f_{\beta}(z)$, is expressed as a product formula, under some assumptions, a sort of analog to the Euler product of the Riemann zeta function, and the factorization of the Parry polynomial of $\beta$ is deduced from the germ.

Given an algebraically closed field K of characteristic zero, we present the Abhyankar–Jung theorem for any excellent henselian ring whose completion is a formal power series ring K[[z]]. In particular, examples include the local rings which form a Weierstrass system over the field K. The Abhyankar–Jung theorem may be regarded as a higher dimensional counterpart of the Newton–Puiseux theorem. It asserts that the roots of a Weierstrass (formal or convergent) polynomial over an algebraically closed field of characteristic zero with discriminant being a normal crossing are fractional (formal or convergent) series. The first proof for the case of two complex variables was due to H.W. Jung [6]. The general result for the algebroid case (for several variables and an arbitrary ground field) was established by S.S. Abhyankar [1] by means of purely algebraic methods, namely, some properties of the Galois group of the polynomial under study. The methods of Jung and Abhyankar are described e.g. in [8]. The classical proofs of the Newton–Puiseux theorem applied Newton’s algorithm to compute, term by term, the fractional series (called Puiseux series) arising as t-roots of an algebraic equation f(x, t) = 0. This algorithm had been invented in [11], ”Methodus fluxionum et serierum infinitorum” (see also [19]). It consists in computation using the so-called Newton polygon, determined by the exponents of a given polynomial. 2010 MSC: 13F25, 13F40.

The parameter space $\mathcal{S}_p$ for monic centered cubic polynomial maps with a marked critical point of period $p$ is a smooth affine algebraic curve whose genus increases rapidly with $p$. Each $\mathcal{S}_p$ consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by an essentially unique Puiseux series. This note will describe the topology of $\mathcal{S}_p$, and of its smooth compactification, in terms of these escape regions. It concludes with a discussion of the real sub-locus of $\mathcal{S}_p$., 51 pages, 16 figures

Let A(z) be an analytic square matrix and $\lambda_{0}$ an eigenvalue of A(0) of multiplicity m. Then under the generic condition, the characteristic polynomial of A(z) evaluated at $\lambda_{0}$ has a simple zero at z=0, we prove that the Jordan normal form of A(0) corresponding to the eigenvalue $\lambda_{0}$ consists of a single m-by-m Jordan block, the perturbed eigenvalues near $\lambda_{0}$ and their eigenvectors can be represented by a single convergent Puiseux series containing only powers of z^{1/m}, and there are explicit recursive formulas to compute all the Puiseux series coefficients from just the derivatives of A(z) at the origin. Using these recursive formulas we calculate the series coefficients up to the second order and list them for quick reference. This paper gives, under a generic condition, explicit recursive formulas to compute the perturbed eigenvalues and eigenvectors for non-selfadjoint analytic perturbations of matrices with degenerate eigenvalues., Comment: 21 pages, no figures. Revision including title change and typos fixed. To appear in SIAM J. Matrix Anal. Appl

We introduce a family of fields of series with support in strongly convex rational cones. All these fields contain the polynomials in several variables. We prove that they are algebraically closed with a construction that is analogous to the Newton polygon for algebraic curves. As a corollary we show the existence of fractional power solutions with support in cones for systems of equations

We construct bases of the Riemann-Roch space of a divisor D of the function field of a plane curve given by an equation F ( x , y ) = 0. When F has coefficients in an algebraic number field and D is defined over this field, we give bounds (depending on the height and the degree of F ) for the archimedean as well as the non-archimedean absolute values of the coefficients of the Puiseux series of the constructed bases. These bounds supersede estimates given by Coates in 1970.

We study the problem of subharmonic bifurcations for analytic systems in the plane with perturbations depending periodically on time, in the case in which we only assume that the subharmonic Melnikov function has at least one zero. If the order of zero is odd, then there is always at least one subharmonic solution, whereas if the order is even in general other conditions have to be assumed to guarantee the existence of subharmonic solutions. Even when such solutions exist, in general they are not analytic in the perturbation parameter. We show that they are analytic in a fractional power of the perturbation parameter. To obtain a fully constructive algorithm which allows us not only to prove existence but also to obtain bounds on the radius of analyticity and to approximate the solutions within any fixed accuracy, we need further assumptions. The method we use to construct the solution -- when this is possible -- is based on a combination of the Newton-Puiseux algorithm and the tree formalism. This leads to a graphical representation of the solution in terms of diagrams. Finally, if the subharmonic Melnikov function is identically zero, we show that it is possible to introduce higher order generalisations, for which the same kind of analysis can be carried out., 30 pages, 6 figures

In [12], we sketched a numeric-symbolic method to compute Puiseux series with floating point coefficients. In this paper, we address the symbolic part of our algorithm. We study the reduction of Puiseux series coefficients modulo a prime ideal and prove a good reduction criterion sufficient to preserve the required information, namely Newton polygon trees. We introduce a convenient modification of Newton polygons that greatly simplifies proofs and statements of our results. Finally, we improve complexity bounds for Puiseux series calculations over finite fields, and estimate the bit-complexity of polygon tree computation.

We give an algorithm to compute term by term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities., Comment: 19 pages. To appear in International Journal of Mathematics. Major changes: Several sections added explaining the geometrical meaning of the series solutions and a corollary about quasi-ordinary singularities

Let $A(t,x) = (A_{ij} (t,x))$ be a square matrix with $A_{ij}$ being a polynomial in t and x. This paper proposes an algorithm for computing the Puiseux (= fractional power) series solutions $x = x(t)$ to the equation ${\operatorname{det}}A(t, x) = 0$. The algorithm is based on an observation which links the Newton diagram (polygon) for det ${\operatorname{det}}A(t, x)$ with the perfect matchings of a bipartite graph associated with A. The algorithm is efficient, making full use of available fast network-type algorithms.

We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algorithm is based on a differential version of the Newton-Puiseux procedure for algebraic equations.

The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseux-valued ``lift'' of this point in the algebraic variety. This theorem is so fundamental because it justifies why a tropical variety (defined combinatorially using initial ideals) carries information about algebraic varieties: it is the image of an algebraic variety over the Puiseux series under the valuation map. We have implemented the ``lifting algorithm'' using Singular and Gfan if the base field are the rational numbers. As a byproduct we get an algorithm to compute the Puiseux expansion of a space curve singularity in (K^{n+1},0)., Comment: 33 pages

The notion of an adapted coordinate system, introduced by V.I.Arnol'd, plays an important role in the study of asymptotic expansions of oscillatory integrals. In two dimensions, A.N.Varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for a class of analytic functions without multiple components. Varchenko's proof is based on Hironaka's theorem on the resolution of singularities. In this article, we present a new, elementary and concrete approach to these results, which is based on the Puiseux series expansion of roots of the given function. Our method applies to arbitrary real analytic functions, and even extends to arbitrary smooth functions of finite type. Moreover, by avoiding Hironaka's theorem, we can give necessary and sufficient conditions for the adaptedness of a given coordinate system in the smooth, finite type setting., Comment: 31 pages, 6 figures

We study the final/initial behavior of a dust Universe with spatial spherical symmetry. This study is done in proximity of the collapse/generation times by an expansion in fractional Puiseux series. Even if the evolution of the universe has different behaviours depending on the initial data (in particular on the initial spatial curvature), we show that, in proximity of generation or collapse time, the Universe expands or collapses with the same behavior.

The purpose of this paper is to investigate the limits of multivariate rational functions with the aid of the theory of real valuations. The following is one of our main results. For two nonzero polynomials f, g ∈ ℝ[x1,…,xn] and (a1,…,an) ∈ ℝn, the (finite) limit of the rational function [Formula: see text] at (a1,…,an) does not exist if and only if (1) there exists a sequence u1(x),…,un(x) of polynomials over ℝ in one variable x such that ui(0) = ai for i = 1,…,n, g(u1(x),…,un(x)) ≠ 0, but [Formula: see text]; or (2) there exist two sequences u1(x),…,un(x) and w1(x),…,wn(x) of polynomials over ℝ in one variable x such that ui(0) = wi(0) = ai for i = 1,…,n, g(u1(x),…,un(x)) ⋅ g(w1(x),…,wn(x)) ≠ 0, but [Formula: see text].

Tropical polytopes are images of polytopes in an affine space over the Puiseux series field under the degree map. This viewpoint gives rise to a family of cellular resolutions of monomial ideals which generalize the hull complex of Bayer and Sturmfels, instances of which improve upon the hull resolution in the sense of being smaller. We also suggest a new definition of a face of a tropical polytope, which has nicer properties than previous definitions; we give examples and provide many conjectures and directions for further research in this area., 19 pages, 13 figures

In this note we consider a perturbed mathematical programming problem where both the objective and the constraint functions are polynomial in all underlying decision variables and in the perturbation parameter e. Recently, the theory of Grobner bases was used to show that solutions of the system of first order optimality conditions can be represented as Puiseux series in e in a neighbourhood of e = 0. In this paper we show that the determination of the branching order and the order of the pole (if any) of these Puiseux series can be achieved by invoking a classical technique known as the "Newton's polygon" and using it in conjunction with the Grobner bases techniques.

We consider a perturbed mathematical programming problem where both the objective and the constraint functions are polynomial in all underlying decision variables and in the perturbation parameter \(\varepsilon.\) We study the behaviour of the solutions of such a perturbed problem as \(\varepsilon \rightarrow 0.\) Though the solutions of the programming problems are real, we consider the Karush–Kuhn–Tucker optimality system as a one-dimensional complex algebraic variety in a multi-dimensional complex space. We use the Buchberger’s elimination algorithm of the Grobner bases theory to replace the defining equations of the variety by its Grobner basis, that has the property that one of its elements is bivariate, that is, a polynomial in \(\varepsilon\) and one of the decision variables only. Changing the elimination order in the Buchberger’s algorithm, we obtain such a bivariate polynomial for each of the decision variables. Thus, the solutions of the original system reduces to a number of algebraic functions in \(\varepsilon\) that can be represented as a Puiseux series in \(\varepsilon\) a neighbourhood of \(\varepsilon = 0\). A detailed analysis of the branching order and the order of the pole is also provided. The latter is estimated via characteristics of these bivariate polynomials of Grobner bases.

This paper investigates the finite-dimensional asymptotic regulator design for nonlinear flexible structures with arbitrary constant disturbances. The problem is to design a feedback controller such that resulting closed-loop nonlinear system will be stable and the controlled output will be regulated. With some assumptions, this paper presents explicit sufficient conditions for the existence of the finite dimensional asymptotic regulator which stabilizes and regulates nonlinear flexible structures.

The Painleve test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonometric ones. The standard methods for constructing the elliptic solutions consist of two independent steps: transformation of a nonlinear polynomial differential equation into a nonlinear algebraic system and a search for solutions of the obtained system. It has been demonstrated by the example of the generalized Henon-Heiles system that the use of the Laurent series solutions of the initial differential equation assists to solve the obtained algebraic system. This procedure has been automatized and generalized on some type of multivalued solutions. To find solutions of the initial differential equation in the form of the Laurent or Puiseux series we use the Painleve test. This test can also assist to solve the inverse problem: to find the form of a polynomial potential, which corresponds to the required type of solutions. We consider the five-dimensional gravitational model with a scalar field to demonstrate this., LaTeX, 14 pages, the paper has been published in the Journal of Nonlinear Mathematical Physics (http://www.sm.luth.se/math/JNMP/)

We study polynomials with coefficients in a field L as dynamical systems where L is any algebraically closed and complete ultrametric field with dense valuation group and characteristic zero residual field. We give a complete description of the dynamical and parameter space of cubic polynomials. In particular we characterize cubic polynomials with compact Julia sets. Also, we prove that any infraconnected connected component of a filled Julia set (of a cubic polynomial) is either a point or eventually periodic. A smallest field S with the above properties is, up to isomorphism, the completion of the field of formal Puiseux series with coefficients in an algebraic closure of Q. We show that some elements of S naturally correspond to the Fourier series of analytic almost periodic functions (in the sense of Bohr) which parametrize (near infinity) the quasiconformal classes of non-renormalizable complex cubic polynomials. Our techniques are based on the ideas introduced by Branner and Hubbard to study complex cubic polynomials., 46 pages

4.1. Puiseux Series and Valuations 4.2. Tree Structure 4.2.1. Nonmetric Tree Structure 4.2.2. Puiseux Parameterization 4.2.3. Multiplicities 4.3. Galois Action 4.3.1. The Galois Group 4.3.2. Action on \(\widehat{\mathcal{V}}_x\) 4.3.3. The Orbit Tree 4.4. A Tale of Two Trees 4.4.1. Minimal Polynomials 4.4.2. The Morphism 4.4.3. Proof 4.5. The Berkovich Projective Line 4.6. The Bruhat-Tits Metric 4.7. Dictionary

In the first section, we define real and real closed fields and state some of their basic properties. In the third section, we define semi-algebraic sets and prove that the projection of a semi-algebraic set is semi-algebraic. This is done using a parametric version of real root counting techniques described in the second section. The fourth section is devoted to several important applications of the projection theorem, of logical and geometric nature. In the last section, an important example of a non-archimedean real closed field is described: the field of Puiseux series.

This thesis deals with rational normal forms of matrix perturbations and the eigenvalue perturbation problem : the asymptotic behavior of perturbed eigenvalues only depends on a few monomials of the characteristic polynomial and the goal is thus to be able to "recover" these monomials directly from the initial matrix (quasi-generic perturbations) or at least from a similar matrix perturbation (reduced form). Following Moser and Lidskii, we define two reduced forms that are associated with two different families of quasi-generic matrix perturbations. We also extend Lidskii's perturbation theorem in order to give a third reduced form and to provide a complete solution to the problem. Additionally, a Newton diagram interpretation is detailed for each of these results and some Maple functionalities for handling matrix perturbations are described., Cette thèse étudie les formes normales rationnelles de perturbations de matrices en vue de la résolution du problème de perturbations pour les valeurs propres : le comportement asymptotique des valeurs propres d'une perturbation de matrice pouvant être entièrement décrit à partir de seulement quelques monômes du polynôme caractéristique, il s'agit essentiellement d'arriver à "lire" ces invariants matriciels directement sur la matrice de départ (perturbations quasi-génériques) ou, à défaut, sur une perturbation qui lui soit semblable (forme réduite). Partant des travaux de Moser et de Lidskii, on propose deux premières formes réduites, chacune étant associée à une famille de perturbations quasi-génériques. Des algorithmes de réduction par similitude polynomiale ainsi que les formes normales correspondantes sont également présentés. Enfin, une généralisation d'un théorème de Lidskii indique une troisième forme réduite, pour laquelle le problème de départ est complètement résolu. L'ensemble de ces résultats trouve une interprétation simple avec le polygone de Newton et l'implantation en Maple des algorithmes proposés a permis de développer une première "boîte à outils" pour les perturbations de matrices.

Suppose that f is irreducible in a power series ring in two variables over an algebraically closed field k of characteristic 0. The characteristic pairs of f can be defined from a fractional power series expansion of a solution of f . The singularity of f can be resolved by a finite number of blow ups of points. This subject, which can be traced back to Newton, has been studied extensively. A few references are Abhyankar [1], Brieskorn and Knorrer [2], Campillo [3], Enriques and Chisini [4] and Zariski [7]. In Sections 1 and 2 we give an exposition of the basic results in the theory of Puiseux series. In Section 3 we give a formula for the characteristic pairs of the transform of f along the sequence of blow ups of points resolving the singularity. As a corollary, we obtain the classical theorem of Enriques and Chisini relating the multiplicity sequence of a resolution and the characteristic pairs of f , and we recover the classical result that the characteristic pairs are an invariant of f . We use an inversion formula of Abhyankar to obtain the results of this paper. 1. The Puiseux series. Let R be a power series ring in two variables over an algebraically closed field k. Then we have the following well-known theorem (see [2, pp. 405 406], [7, p. 7]). Theorem 1.1. Suppose that f ∈ R is irreducible and (x, y) are regular parameters for R such that the multiplicity ν(f) = ν(f(0, y)). Then a fractional power series exists (called a Puiseux series) of y in terms of x. The expansion has the form

In semiconductor theory, applying the kp -method to the monodimensional Schrodinger equation leads to a symmetric perturbed eigenvalue problem (Kane E O 1967 The kp method Semiconductors and Semimetals (New York: Academic) p 75), i.e. to the diagonalization of a matrix A ( ) depending on a small parameter , symmetric . The eigenelements of A ( ) admit expansions in fractional powers of (Puiseux series). Usually, physicists solve this problem by using Schrodinger perturbation formulae under some restrictive conditions, which make perturbed eigenvector symbolic approximation impossible. This is illustrated by the modified Kane matrix (Fishman G 1997 Quasi-cube et Wurtzite: Application au GaN (Montpellier: Ecole thematique du CNRS)). To solve this problem completely from a symbolic computing point of view, we consider the symmetric perturbed eigenvalue problem in the case of analytic perturbations (Baumgartel H 1985 Analytic Perturbation Theory for Matrices and Operators (Basle: Birkhauser), Kato T 1980 Perturbation Theory For Linear Operators (Berlin: Springer)). We first review the classical characteristic polynomial approach, showing why it may be not optimal. We also present a direct matricial algorithm (Jeannerod C P and Pflugel E 1999 Int. Symp. on Symbolic and Algebraic Computation (Vancouver, Canada, July 1999) (New York: ACM) pp 121-8): transforming the analytic matrix A ( ) into its so-called q -reduced form allows to recover the information we need for the eigenvalues. This alternative method, as well as the classical one, can be described in terms of the Newton polygon. However, our approach uses only a finite number of terms of A ( ) and is more suitable for large matrices and a low approximation order. Besides, we show that the q -reduction process can simultaneously provide symbolic approximations of both the perturbed eigenvalues and eigenvectors. The implementation of this algorithm in MAPLE is used to diagonalize the modified Kane matrix up to a given order.

We consider a perturbed mathematical programming problem where both the objective and the constraint functions are analytical in both the underlying decision variables and in the perturbation variable/parameter that is denoted by ϵ. The following question arises: what is the description of the solutions of such a perturbed problem when ϵ → 0? We demonstrate that, under weak conditions, the solutions of the perturbed problems are obtained as Puiseux series expansions in ϵ. The results are obtained by application of the Remmert–Stein representation theorem for complex analytic varieties.

Let be given a homogeneous linear ordinary differential equation with coefficients which are polynomial over a field K. The well known Frobenius’ algorithm [7] computes the fundamental solutions system in the neighborhood of regular singularities. These solutions are from \(\mathop K\limits^ \sim \) [[x]][ln(x)] where \(\mathop K\limits^ \sim \) is an algebraic extension of K. We present a new method based on the Frobenius’ algorithm. This method allows to build solutions with hypergeometric coefficients of Laurent or Puiseux series.

In 1976 Bewley and Kohlberg proved that solutions of the limit discount equation of finite stochastic games are given by Puiseux series (i.e., fractional power series) in the discount factor, when β is sufficiently near 1. Their proof relied on a theorem from formal logic due to Tarski. In a recent paper, Szczechla et al. have given an alternative proof, and an extension, of this result with the help of powerful methods from a branch of the theory of functions of several complex variables known as complex analytic varieties.

Algebraic functions over Q(X) may indeed be expressed as Puiseux series. We examine an algorithm for the development of the series expansion for algebraic functions. From the algorithm we show why the series developed over fields of finite characteristic will not necessarily be Puiseux.

In many applications, one encounters asymptotic expansions of a form more complicated than mere Puiseux series. Existing computer algebra systems lack good algorithms for handling such asymptotic expansions. We present tools that permit the representation and automatic handling of general exp-log asymptotic expansions.

We study special solutions of the sixth Painleve equation with generic values of parameters whose linear monodromy can be calculated explicitly. We give two kinds of special solutions. One is the meromorphic solutions at a fixed singularity and the other is the solutions expressed in the Puiseux series at t=0. We give the respective character of these special solutions.

In 1887 Runge [13] proved that a binary Diophantine equation F ( x , y ) = 0 F(x,y) = 0 , with F F irreducible, in a class including those in which the leading form of F F is not a constant multiple of a power of an irreducible polynomial, has only a finite number of solutions. It follows from Runge’s method of proof that there exists a computable upper bound for the absolute value of each of the integer solutions x x and y y . Runge did not give such a computation. Here we first deduce Runge’s Theorem from a more general theorem on Puiseux series that may be of interest in its own right. Second, we extend the Puiseux series theorem and deduce from the generalized version a generalized form of Runge’s Theorem in which the solutions x x and y y of the polynomial equation F ( x , y ) = 0 F(x,y) = 0 are integers, satisfying certain conditions, of an arbitrary algebraic number field. Third, we compute bounds for the solutions ( x , y ) ∈ Z 2 (x,y) \in {{\mathbf {Z}}^2} in terms of the height of F F and the degrees in x x and y y of F F .

In this paper we study the relation between the tangent structure of an analytic set V at a point p and the local representation of V as a branched covering. A prototype for our type of result is the fact that one obtains a covering of minimal degree by projecting transverse to the Zariski tangent cone C 3 ( V , p ) {C_3}(V,p) . We show, for instance, that one obtains the smallest possible branch locus for a branched covering if one projects transverse to the cone C 4 ( V , p ) {C_4}(V,p) . This and similar results show that points where the various tangent cones C i ( V , p ) , i = 4 , 5 , 6 {C_i}(V,p),i = 4,5,6 , have minimal dimension give rise to the simplest branched coverings. This observation leads to the idea of “Puiseux series normalization", generalizing the situation in one dimension. These Puiseux series allow us to strengthen some results of Hironaka and Whitney on the local structure of certain types of singularities.

We give an exposition of some of the basic results on singularities of plane algebraic curves, in terms of polynomials and formal power series.

In 1957, M. S. Livsic introduced into scattering theory a certain dissipative operator-valued function B(x) [22-251. His theorem, which relates the characteristic function of B(z) to a certain S-matrix, is based on a simple formula for the compression of a resolvent to a subspace. This formula, proved below as Proposition 1, and the matrix B(z) apparently have been rediscovered several times, and occur in the physics literature under a variety of names. The earliest reference known to the author‘is [22], and we shall therefore refer to B(z) as the Livsic matrix. The purpose of the present article is to indicate an application of this formula to the perturbation theory of embedded eigenvalues. A perturbation of an isolated eigenvalue may be called regular if the perturbed resolvent is analytic in the perturbation parameter K. The perturbed eigenvalues then have convergent Puiseux expansions in K. The Livsic matrix leads to a natural extension of the idea of a regular perturbation to eigenvalues which are not isolated, in which the Puiseux series describe resonances rather than eigenvalues [14]. Moreover, calculation of the coefficients of these series is reduced to the known procedures of the theory of isolated eigenvalues. A somewhat similar reduction, relying on dilatation analyticity, was obtained by Simon [28] for Balslev-Combes Hamiltonians. The general properties of B(z) are given in Section 1, and Livsic’s Theorem is proved in an abstract framework. A general theorem on resonnaces and spectral concentration is proved in Section 2. Regular perturbations of embedded eigenvalues occupy the next two sections, which include a remark on the breakdown of the ordinary perturbation method. The final section indicates how eigenvalues which are not regular can often be discussed, and gives an example in which spectral concentration occurs, but there is no analytic family of resonances. The method used here may be considered as a combination of the two approaches to the problem referred to by Simon in his introduction [28]. This article by no means exhausts the applications of the Livsic matrix, and the author recommends its use to other writers on perturbation theory.

Let (g,f) be an analytic map germ from ( C 2 ,0) into ( C 2 ,0) and denote by (u,v) the canonical coordinates in (g,f)( C 2 ) ; it is (g(x,y),f(x,y))=(u,v). In [J. London Math. Soc. (2) 59 (1999) 207–226], we showed that the set constituted of the first (not necessarily characteristic) Puiseux exponent (in the (u,v)-coordinates) of each branch δ of the discriminant curve of (g,f) is a topological invariant of (g,f). Here we prove that for each branch δ there exists an integer k(δ) such that the set constituted of the first (not necessarily characteristic) k(δ) exponents of the Puiseux series in the (u,v)-coordinates of each δ is a topological invariant of (g,f). We give different ways to compute these invariants.

The algorithm DELIRE (Linear Differential Equation Irregular and REgular singularities) presented in this paper gives the formal solutions (normal and sub-normal) at an irregular singular point of a linear homogeneous differential operator, whose coefficients are formal Puiseux series [11].