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Let K be the (real closed) field of Puiseux series in t over R endowed with the natural linear order. Then the elements of the formal power series rings R[[ξ1, . . . , ξn]] converge t-adically on [−t, t]n, and hence define functions [−t, t]n → K. Let L be the language of ordered fields, enriched with symbols for these functions. By Corollary 3.17, K is o-minimal in L. This result is obtained from a quantifier elimination theorem. The proofs use methods from non-Archimedean analysis.

A package facilitating search for analytic solutions of nonintegrable systems of autonomous ordinary differential equations is suggested. The algorithm discussed, which is a generalization of the Conte-Musette method, allows one to obtain single-valued and multivalued analytical solutions that were known to exist only in the form of formal Laurent and Puiseux series (obtained, for example, by means of the Painleve analysis), respectively. Solutions in the form of the formal Laurent series and procedures from the package discussed are used not only for constructing elliptic solutions but also for proving impossibility of such construction. The suggested procedures have been implemented as packages in Maple and REDUCE.

A detailed algebraic-geometric background is presented for the tropical approach to enumeration of singular curves on toric surfaces, which consists of reduc- ing the enumeration of algebraic curves to that of non-Archimedean amoebas, the images of algebraic curves by a real-valued non-Archimedean valuation. This idea was proposed by Kontsevich and recently realized by Mikhalkin, who enumerated the nodal curves on toric surfaces. The main technical tools are a refined tropicalization of one-parametric equisingular families of curves and the patchworking construction of singular algebraic curves. The case of curves with a cusp and the case of real nodal curves are also treated. §1. Introduction The rapid development of tropical algebraic geometry over recent years has led to interesting applications to enumerative geometry of singular algebraic curves, proposed by Kontsevich (16). The first result in this direction was obtained by Mikhalkin (18), who counted the curves with a given number of nodes on toric surfaces via lattice paths in con- vex lattice polygons. Our main goal in the present paper is to explain this breakthrough result, notably the link between algebraic curves and non-Archimedean amoebas, which is the core of the tropical approach to enumerative geometry. Our point of view is purely algebraic-geometric and differs from Mikhalkin's method, which is based on symplectic geometry techniques. Briefly speaking, we count equisingular families of curves over a punctured disk. The tropicalization procedure extends such families to the central point, and these tropical limits are basically encoded by non-Archimedean amoebas. In its turn, the patchworking construction restores an equisingular family out of the central fiber. Tropicalization. Let ∆ ⊂ R 2 be a convex lattice polygon, and let TorK(∆) be the toric surface associated with the polygon ∆ and defined over an algebraically closed field K of characteristic zero. We denote by ΛK(∆) the tautological linear system on TorK(∆) generated by the monomials x i y j ,( i, j) ∈ ∆ ∩ Z 2 . We would like to count the n-nodal curves belonging to ΛK(∆) and passing through r =d im Λ K(∆) − n = |∆ ∩ Z 2 |− 1 − n generic points in TorK(∆), i.e., we want to find the degree of the so-called Severi variety Σ∆(nA1). Let K be the field of convergent Puiseux series over C, i.e., power series of the form b(t )= � τ ∈R cτ t τ ,w hereR ⊂ Q is contained in an arithmetic progression bounded from below, and � τ ∈R |cτ |t τ < ∞ for sufficiently small positive t. The field K is equipped with a non-Archimedean valuation Val(b )= − min{τ ∈ R : cτ � } ,w hich

In this paper we will present the family of Newton algorithms. From the computer algebra point of view, the most basic of them is well known for the local analysis of plane algebraic curves f(x,y)=0 and consists in expanding y as Puiseux series in the variable x. A similar algorithm has been developped for multi-variate algebraic equations and for linear differential equations, using the same basic tools: a “regular” case, associated with a “simple” class of solutions, and a “simple” method of calculus of these solutions; a Newton polygon; changes of variable of type ramification; changes of unknown function of two types y=ctμ+ϕ or y=exp (c/tμ)ϕ.

Let F(X,Y) be an absolutely irreducible polynomial in \(\mathbb{Z}\left[ {X,Y} \right]\) such that the algebraic curve C: F(X,Y) = 0 has infinitely many integer points. In this paper we obtain an explicit estimate on the distribution of integer points of C.

In this chapter, we confirm the invariance property for the case where the partial derivative of the characteristic function with respect to delay is nonzero for all the critical pairs. It is proved that the asymptotic behavior of a critical imaginary root with any multiplicity can be determined from the corresponding frequency-sweeping curve. As the frequency-sweeping curves are independent of delay, the invariance property concerning the critical imaginary root’s asymptotic behavior is confirmed. Moreover, from the development in this chapter we come to a useful idea: The asymptotic behavior of the frequency-sweeping curves may be treated as a reference object in studying the invariance property in the forthcoming chapters.

Universitat Politècnica de Catalunya. Facultat de Matemàtiques i Estadística. Departament de Matemàtica Aplicada I. Treball Final de Màster: Master of Science in Advanced Mathematics and Mathematical Engineering., This works focus on computational aspects of the theory of singularities of plane algebraic curves. We show how to use the Puiseux factorization of a curve, computed through the Newton-Puiseux algorithm, to study the equisingularity type of a curve. We present a novel version of the Newton-Puiseux algorithm that can compute all the Puiseux factorization of any arbitrary polynomial, removing the restriction of reduced inputs. Next, we introduce the theory of infinitely near points and the concept of base points of an ideal. Finally, we develop a novel algorithm that, using our novel version of the Newton-Puiseux algorithm, computes the weighted cluster of base points of any two dimensional ideal from any set of generators.

International audience; MSC: 93D20 93C15 93B25 34D05 34D20 Keywords: Exponential stability Differential behavior Time-varying Duality Serre category We develop a new approach to exponential stability of linear time-varying (LTV) differential behaviors that is analogous to that in our paper on exponential stability of discrete LTV behaviors (H. Bourlès et al., 2015 [5]). Stability theory for differential state space systems with smooth coefficients is an important subject in the literature. For differential LTV behaviors with arbitrary smooth coefficients there is no reasonable stability theory. Therefore we restrict the smooth varying coefficients to functions that are defined by means of locally convergent Puiseux series. All rational functions are of this type. We introduce a new kind of behaviors and prove a module-behavior duality for these. We define a new notion of weak exponential stability (w.e.s.) of a behavior B and its associated finitely generated (f.g.) module M and show that the w.e.s. modules and behaviors are closed under isomor-phisms, subobjects, factor objects and extensions. The standard uniform exponential stability of state space equations is not preserved under behavior isomorphisms and unsuitable for a behavioral theory. In the main result we assume a nonzero f.g. torsion module M and its associated autonomous behavior B. Such a module may be regular or irregular singular according to the Galois theory of differential equations. If it is nonzero and regular singular it is never w.e.s. For irregular singular M we characterize w.e.s. of most B algebraically and constructively.

We investigate the bifurcation of limit cycles in one-parameter unfoldings of quadractic differential systems in the plane having a degenerate critical point at infinity. It is shown that there are three types of quadratic systems possessing an elliptic critical point which bifurcates from infinity together with eventual limit cycles around it. We establish that these limit cycles can be studied by performing a degenerate transformation which brings the system to a small perturbation of certain well-known reversible systems having a center. The corresponding displacement function is then expanded in a Puiseux series with respect to the small parameter and its coefficients are expressed in terms of Abelian integrals. Finally, we investigate in more detail four of the cases, among them the elliptic case (Bogdanov-Takens system) and the isochronous center S3. We show that in each of these cases the corresponding vector space of bifurcation functions has the Chebishev property: the number of the zeros of each function is less than the dimension of the vector space. To prove this we construct the bifurcation diagram of zeros of certain Abelian integrals in a complex domain.

Given a mapping F from Rn to Rm with $n\geq m,$ a basic computational problem that has applications in robotics is to solve for paths x(t) in the domain of F given a target path z(t) in the range of F so that F(x(t))=z(t). Results are given for the existence and characterization of solution paths through singularities of F, with emphasis on folds and cusps. Reparametrizations using Puiseux series are derived in cases where x(t) is not differentiable.

We reconstruct [Formula: see text] gravity theory with an exponential scale factor to realize the bouncing behavior in the early universe and examine the asymptotic behavior of late-time solutions in this model. We propose an approach for the construction of asymptotic expansions of solutions of the Friedmann equations on the basis of Puiseux series.

We show that an effective version of Siegel's Theorem on finiteness of integer solutions and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the k >= 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. This dichotomy holds even when restricted to planar graphs. A special case of this result is that counting edge k-colorings is #P-hard over planar 3-regular graphs for k >= 3. In fact, we prove that counting edge k-colorings is #P-hard over planar r-regular graphs for all k >= r >= 3. The problem is polynomial-time computable in all other parameter settings. The proof of the dichotomy theorem for Holant(f) depends on the fact that a specific polynomial p(x,y) has an explicitly listed finite set of integer solutions, and the determination of the Galois groups of some specific polynomials. In the process, we also encounter the Tutte polynomial, medial graphs, Eulerian partitions, Puiseux series, and a certain lattice condition on the (logarithm of) the roots of polynomials., 75 pages, 29 figures, 4 tables

It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisfied for any lift. This solves a conjecture of Develin and Yu., 18 pages, 6 figures; v2: major revision

We present a new algorithm for the computation of the irreducible factors of degree at most $d$, with multiplicity, of multivariate lacunary polynomials over fields of characteristic zero. The algorithm reduces this computation to the computation of irreducible factors of degree at most $d$ of univariate lacunary polynomials and to the factorization of low-degree multivariate polynomials. The reduction runs in time polynomial in the size of the input polynomial and in $d$. As a result, we obtain a new polynomial-time algorithm for the computation of low-degree factors, with multiplicity, of multivariate lacunary polynomials over number fields, but our method also gives partial results for other fields, such as the fields of $p$-adic numbers or for absolute or approximate factorization for instance. The core of our reduction uses the Newton polygon of the input polynomial, and its validity is based on the Newton-Puiseux expansion of roots of bivariate polynomials. In particular, we bound the valuation of $f(X,\phi)$ where $f$ is a lacunary polynomial and $\phi$ a Puiseux series whose vanishing polynomial has low degree., Comment: 22 pages

We show that the weak limit of the maximal measures for any degenerating sequence of rational maps on the Riemann sphere ${\hat{{\mathbb{C}}}} $ must be a countable sum of atoms. For a one-parameter family $f_t$ of rational maps, we refine this result by showing that the measures of maximal entropy have a unique limit on $\hat{{\mathbb{C}}}$ as the family degenerates. The family $f_t$ may be viewed as a single rational function on the Berkovich projective line $\mathbf{P}^1_{\mathbb{L}}$ over the completion of the field of formal Puiseux series in $t$, and the limiting measure on $\hat{{\mathbb{C}}}$ is the ‘residual measure’ associated with the equilibrium measure on $\mathbf{P}^1_{\mathbb{L}}$. For the proof, we introduce a new technique for quantizing measures on the Berkovich projective line and demonstrate the uniqueness of solutions to a quantized version of the pullback formula for the equilibrium measure on $\mathbf{P}^1_{\mathbb{L}}$.

When a time-delay system involves multiple imaginary roots (MIRs), the stability analysis will become much more complicated than that in the case with only simple imaginary roots (SIRs). An MIR may exhibit different splitting behaviors and, to the best of the authors' knowledge, their properties have not been fully investigated. In this paper, we focus on characterizing the invariance properties for MIRs with any multiplicity. Furthermore, the proposed methodology makes it possible to also cover some degenerate cases already encountered and discussed in the literature. In addition, we propose an easily implemented frequency-sweeping method, making it possible to derive the asymptotic behavior without invoking the Puiseux series.

We investigate the existence of distinct polynomialsF, G having roots of prescribed multiplicities and deg(F −G) as small as predicted by Mason’sabc theorem. The case of characteristic zero has been treated completely in a previous paper, but those methods do not apply in positive characteristic. Here we study this problem through reduction; it turns out that what we require amounts to proving good reduction for certain covers of the projective line, unramified except above 0, 1, ∞. We shall give sufficient conditions for good reduction of those covers, which sometimes go beyond known criteria due to Grothendieck, Fulton and Beckmann. The methods are completely different from those used by such authors and rely on results by Dwork and Robba onp-adic analytic continuation of Puiseux series.

A Maple procedure is described by means of which an algebraic function given by an equation f (x, y) e 0 can be expanded into a fractional power series (Puiseux series) y(x)=\sum_{n={n_0}}^\infty \alpha_n(x-x_0)^{n/\varepsilon}, where \varepsilon \in \mathbb{N}, n_0 \in \mathbb{Z}, of special (“nice”) type. It may be a series with polynomial, rational, hypergeometric coefficients, or m-sparse or m-sparse m-hypergeometric series. First, a linear ordinary differential equation with polynomial coefficients Ly(x) e 0 is constructed which is satisfied by the given algebraic function. The e, n0 , and a required number of initial coefficients a0, …, an are computed by using Maple algcurves package. By means of Maple Slode package, a solution to the equation Ly(x) e 0 is constructed in the form of a series with “nice” coefficients, the initial coefficients of which correspond to the calculated a0 , …, an. The procedure suggested can construct an expansion at a user-given point x0 , as well as determine points where an expansion of such a special type is possible.

Kinks and rotations are studied in long Josephson junctions for small and large surface losses. Geometric singular perturbation theory is used to prove existence for small surface losses, while numerical continuation is necessary to handle large surface losses. A survey of the system behaviour in terms of dissipation parameters and bias current is given. Linear orbital stability for kinks is proved for small surface losses by calculating the spectrum of the linearized problem. The spectrum is split into essential spectrum and discrete spectrum. For the determination of the discrete spectrum, robustness of exponential dichotomies is used. Puiseux series together with perturbation theory for linear operators are an essential tool. In a final step, a smooth Evans function together with geometric singular perturbation theory is used to count eigenvalues. For kinks, non-linear orbital stability is shown. For this purpose, the asymptotic behaviour of a semigroup is given and the theory of centre and stable manifolds is applied. Copyright © 2001 John Wiley & Sons, Ltd.

Given a multivariate polynomial F(x, y, ...,z), this paper deals with calculating the roots ofF w.r.t.x in terms of formal power series or fractional-power series iny, ...,z. If the problem is regular, i.e. the expansion point is not a singular point of a root, then the calculation is easy, and the irregular case is considered in this paper. We extend the generalized Hensel construction slightly so that it can be applied to the irregular case. This extension allows us to calculate the roots of bivariate polynomial F(x, y) in terms of Puiseux series iny. For multivariate polynomial F(x, y, ...,z), we consider expanding the roots into fractional-power series w.r.t. the total-degree ofy, ...,z, and the roots are expressed in terms of the roots of much simpler polynomials.

This thesis deals with hypergeometric functions in several complex variables and systems of partial differential equations of hypergeometric type. One of the main objects of study in the thesis is the so-called Horn system of equations: xiPi(θ)y(x) = Qi(θ)y(x), i = 1, ..., n.Here x ∈ℂn, θ = (θ1, ..., θn),θi = xi ∂/∂xi , Pi and Qi are nonzero polynomials. By definition hypergeometric functions are (multi-valued) analytic solutions to this system of equations. The main purpose of the thesis is to systematically investigate the Horn system of equations and properties of its solutions.To construct solutions to the Horn system we use one of the variants of the Laplace transform which leads to a system of linear difference equations with polynomial coefficients. Solving this system we represent a solution to the Horn system in the form of an iterated Puiseux series.We give an explicit formula for the dimension of the space of analytic solutions to the Horn system at a generic point under some assumptions on its parameters. The proof is based on the study of the module over the Weyl algebra of linear differential operators with polynomial coefficients associated with the Horn system. Combining this formula with the theorem which allows one to represent a solution to the Horn system in the form of an iterated Puiseux series, we obtain a basis in the space of analytic solutions to this system of equations.Another object of study in the thesis is the singular set of a nonconuent hypergeometric function in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We give a description of such hypersurfaces in terms of the Newton polytopes of their defining polynomials. In particular we obtain a geometric description of the zero set of the discriminant of a general algebraic equation.In the case of two variables one can say much more about singularities of nonconuent hypergeometric functions. We give a complete description of the Newton polytope of the polynomial whose zero set naturally contains the singular locus of a nonconuent double hypergeometric series. We show in particular that the Hadamard multiplication of such series corresponds to the Minkowski sum of the Newton polytopes of polynomials whose zero loci contain the singularities of the factors.

For certain classes of lattice models of nanosystems the eigenvalues of the row-to-row transfer matrix and the components of the corner transfer matrix truncations are algebraic functions of the fugacity and of Boltzmann weights. Such functions can be expanded in Puiseux series using techniques from algebraic geometry. Each successive term in the expansions in powers a Boltzmann weight is obtained exactly without modifying previous terms. We are able to obtain useful analytical expressions for any thermodynamic function for these systems from the series in circumstances in which no exact solutions can be found.

In this paper we give a new proof of the existence of Puiseux series expansion of the limit discount equation of finite state stochastic games. Unlike the original proof, due to Bewley and Kohlberg [ Math. Oper. Res., 3 (1976), pp. 197--208], our proof is not algebraic and does not invoke Tarski's principle. Instead we use only the theory of functions of complex variables and complex analytic varieties.

We study the tropicalization of the cone of positive semidefinite matrices over the ordered field of real Puiseux series. The tropical PSD matrices form the normal cone of the Newton polytope of the symmetric determinant at the vertex corresponding to the product of diagonal entries. We find generators and defining inequalities of the cone. The PSD tropical quadratic forms are those that induce the trivial subdivision on the standard simplex dilated by two. We also show that the tropical PSD cone is the tropical convex hull of the set of symmetric matrices of tropical rank one and that every tropical PSD matrix can be factored as a tropical product of a matrix and its transpose., 5 pages

Obtaining the Puiseux series of multiple imaginary (characteristic) roots (MIRs) is a fundamental issue in the stability analysis of time-delay systems. However, to the best of the authors' knowledge, this issue has not been fully investigated up to date. This note focuses on the Puiseux series expansion of MIRs of linear time-invariant systems including commensurate delays. For an MIR of any multiplicity, we propose an algorithm for defining the structure of the Puiseux series, as well as the explicit computation of the corresponding coefficients. By using the proposed method, we can find all the Puiseux series corresponding to all the root loci.

We consider a weak extension of the recently introduced perturbative Painleve test, whereby as in the weak extension of the standard Painleve test finite branching is allowed. This extension allows us to capture even more information about the singularity structure of the solutions to nonlinear ordinary (and partial) differential equations, providing a means of uncovering logarithmic branching which might otherwise be missed. Such logarithmic branching might be thought of as providing a greater obstacle to integrability than does finite branching alone. When our new test is passed the resulting expansion is a Puiseux series in ξ = (x+x0), i.e. doubly infinite in rational powers of ξ. The above-mentioned perturbative Painleve test is included as a special case. However, we are also interested in finding the most efficient means of uncovering such logarithmic branching; we find that for some examples the way to do this is by constructing a descending series solution. For some equations this means that logarithmic branching is exhibited in a particular solution rather than in the general solution. The construction of descending series solutions can be undertaken as a special case of the perturbative Painleve algorithm. The techniques developed here then allow us to reconsider the analysis of the ordinary differential equation Uxx + UUx + kU3 = 0.

We propose two notions of multiplicity for the solutions (Puiseux series) of differential polynomials. The first one is based on the Newton-Puiseux Fine polygon and it is of a combinatorial nature. The second one is of a differential nature and extends the standard notion of algebraic multiplicity. We show that both concepts, defined independently, are equivalent. We also establish the limit of the multiplicity for a given differential polynomial, and we give a precise procedure to reduce the multiplicity. Finally, we shall present the problem of computability of these entities, which still remains open.

The integrability of systems of ordinary differential equations with polynomial vector fields is investigated by using the singularity analysis methods. Three types of results are obtained. First, a general relationship between the degrees of first integrals and the so‐called Kowalevskaya exponents is derived. Second, it is shown that all solutions of algebraically integrable systems can be expanded in Puiseux series. Third, a new method to study partially integrable systems is studied. These different aspects allow us to study algorithmically the integrability, partial integrability, and nonintegrability of differential systems.

Lety=f(x) be an algebraic function which is defined as a root of bivariate polynomialP(x, y). GivenP(x, y), we present a practical computation method for analytic continuation and Riemann surface determination off(x). First, we determine the branch points off(x) by calculating the roots of a univariate polynomial numerically. Next, we expandf(x) into the Puiseux series at each branch point by the extended Hensel construction method proposed by Sasaki and Kako recently. Then, using well-known Smith’s theorem, we determine which series are connected the each other for each pair of branch points, by numerically estimating the values of the series at some middle point between the branch points. This allows us to determine the Riemann surface off(x) completely. Analytic continuation is performed similarly. We employ floating-point number arithmetic to perform these calculations.

There are several well-known algorithms to calculate the Puiseux series developments of the branches of an algebraic function. None of them, however, generates the series in closed form, even in those cases where such a formal result is available. They produce, instead, truncated series, and give information that can be used to handle the series as streams. Here we give a solution to the given problem. We combine an algorithm of D. V. and G. V. Chudnovsky that transforms the given algebraic equation into a differential equation for the function, and further into a recurrence equation for the Puiseux coefficients, with an algorithm of Koepf which in the case of hypergemetric type results in the formal series. A finite linear recurrence equation is optimal for a representation by streams. D. V. and G. V. Chudnovsky point out that their algorithm requires only0(M) field operations ifM is the order of the number of series terms considered. However, from a practical point of view, it is of importance that the complexity of the resulting recurrence equation — as well as of the differential equation — can be extremely high, a fact, which we illustrate by an example. It turns out, that many algebraic functions of low order with a sparse representation are of hypergeometric type, and so closed form representations for the corresponding series can be given.

This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous invariants involved in this classification are the left equivalence classes of period or monodromy functions, and the cohomology classes of period cocycles, which can be expressed in terms of Puiseux series. We also study the problem of Hamiltonianization of these integrable vector fields by a compatible symplectic or Poisson structure., 31 pages, 12 figures, submitted to a special issue of Acta Mathematica Vietnamica

This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real tropical hypersurfaces. If V is a tropical hypersurface defined over the field of real Puiseux series, it has a real part RV which is a polyhedral complex. We define the total curvature of V (resp. RV) by using the total curvature of Amoebas and passing to the limit. We also define the "polyhedral total curvature" of the real part RV of a generic tropical hypersurface. The main results we prove about these notions are the following: - The fact that the total curvature and the polyhedral total curvature coincide for real non-singular tropical hypersurfaces. - A universal inequality between the total curvatures of V and RV and another between the logarithmic curvatures of the real and complex parts of a real algebraic hypersurface. -The fact that this inequality is sharp in the non-singular case., 23 pages, 11 figures

We explore the concept of real tropical basis of an ideal in the field of real Puiseux series. We show explicit tropical bases of zero-dimensional real radical ideals, linear ideals and hypersurfaces coming from combinatorial patchworking. But we also show that there exist real radical ideals that do not admit a tropical basis. As an application, we show how to compute the set of singular points of a real tropical hypersurface. i.e. we compute the real tropical discriminant., Comment: 17 pages, figures

For a regular chain R in dimension one, we propose an algorithm which computes the (non-trivial) limit points of the quasi-component of R, that is, the set $\overline{W(R)} \setminus W(R)$ . Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of R. We provide experimental results illustrating the benefits of our algorithms.

The classical Chevalley-Weil theorem asserts that for an \'etale covering of projective varieties over a number field K, the discriminant of the field of definition of the fiber over a K-rational point is uniformly bounded. We obtain a fully explicit version of this theorem in dimension 1., Comment: version 4: minor inaccuracies in Lemma 3.4 and Proposition 5.2 corrected

In this paper we outline an algorithmic approach to compute Puiseux series expansions for algebraic surfaces. The series expansions originate at the intersection of the surface with as many coordinate planes as the dimension of the surface. Our approach starts with a polyhedral method to compute cones of normal vectors to the Newton polytopes of the given polynomial system that defines the surface. If as many vectors in the cone as the dimension of the surface define an initial form system that has isolated solutions, then those vectors are potential tropisms for the initial term of the Puiseux series expansion. Our preliminary methods produce exact representations for solution sets of the cyclic $n$-roots problem, for $n = m^2$, corresponding to a result of Backelin., Comment: accepted for presentation at ISSAC 2012

Obtaining the Puiseux Series expansion of multiple imaginary (characteristic) roots (MIRs) is a fundamental issue in the stability analysis of time-delay systems. However, this issue has not been fully investigated up to date. This work focuses on the Puiseux Series expansion of MIRs for a class of linear time-delay systems including commensurate delays. For an MIR of any multiplicity, we propose an algorithm for defining the structure of the Puiseux Series, as well as the explicit computation of the corresponding coefficients. By using the proposed method, we can find all the Puiseux Series corresponding to all the root trajectories. Some illustrative examples complete the work.

This paper addresses the stability analysis problem of a class of linear system with commensurate delays in a frequency-domain setting. In Part I of this paper, only the simple imaginary roots (SIRs) case is considered. In Part II of the paper, the case of multiple imaginary roots (MIRs) will be studied and general results on the issues of Puiseux series expansion, invariance, and instability persistence will be presented. By the approach proposed in this two-part paper, we can study the complete stability for the time-delay systems under consideration.

This paper addresses the stability analysis problem of a class of linear system with commensurate delays in a frequency-domain setting. In Part I of this paper, only the simple imaginary roots (SIRs) case is considered. First, the characteristic quasipolynominal is reformulated into a factorization form. Then, it is found that (i) the crossing direction of imaginary roots (CDIRs) can be reflected by an appropriate frequency-sweeping test, (ii) the CDIRs are determined by a certain order of the imaginary roots by magnitude and finally, (iii) a time-delay system is ultimately unstable if there is at least one crossing imaginary root. To apply the properties (i)-(iii), an easily implemented frequency-sweeping method is proposed, by which the CDIRs can be obtained without any further computation. The method can be used for the complete stability of the system. In Part II of the paper, the MIR case will be discussed.

Puiseux series are power series in which the exponents can be fractional and/or negative rational numbers. Several computer algebra systems have one or more built-in or loadable functions for computing truncated Puiseux series. Some are generalized to allow coefficients containing functions of the series variable that are dominated by any power of that variable, such as logarithms and nested logarithms of the series variable. Some computer algebra systems also have built-in or loadable functions that compute infinite Puiseux series. Unfortunately, there are some little-known pitfalls in computing Puiseux series. The most serious of these is expansions within branch cuts or at branch points that are incorrect for some directions in the complex plane. For example with each series implementation accessible to you: Compare the value of (z^2 + z^3)^(3/2) with that of its truncated series expansion about z = 0, approximated at z = -0.01. Does the series converge to a value that is the negative of the correct value? Compare the value of ln(z^2 + z^3) with its truncated series expansion about z = 0, approximated at z = -0.01 + 0.1i. Does the series converge to a value that is incorrect by 2pi i? Compare arctanh(-2 + ln(z)z) with its truncated series expansion about z = 0, approximated at z = -0.01. Does the series converge to a value that is incorrect by about pi i? At the time of this writing, most implementations that accommodate such series exhibit such errors. This article describes how to avoid these errors both for manual derivation of series and when implementing series packages., 36 pages, 5 tables, to appear in ACM Communications in Computer Algebra

Puiseux series are power series in which the exponents can be fractional and/or negative rational numbers. Several computer algebra systems have one or more built-in or loadable functions for computing truncated Puiseux series -- perhaps generalized to allow coefficients containing functions of the series variable that are dominated by any power of that variable, such as logarithms and nested logarithms of the series variable. Some computer-algebra systems also offer functions that can compute more-general truncated recursive hierarchical series. However, for all of these kinds of truncated series there are important implementation details that haven't been addressed before in the published literature and in current implementations. For implementers this article contains ideas for designing more convenient, correct, and efficient implementations or improving existing ones. For users, this article is a warning about some of these limitations. Many of the ideas in this article have been implemented in the computer-algebra within the TI-Nspire calculator, Windows and Macintosh products., 21 pages; 1 Algorithm; 2 Tables; To appear in the ACM Communications in Computer Algebra

Let K be a field, and δ a derivation of K. The right zeros of skew polynomials in the formal differential operator ring K[T,δ], with commutation law Tα - αT = δ(α) for any α ∊ K, are logarithmic derivatives of solutions of linear differential equations in some differential extension of K. For K the Puiseux series field over an algebraically closed field of characteristic zero, and δ the classical derivation, we prove that any polynomial of degree > 1 in K[T,δ] splits, without unicity, into factors of degree one.

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.

International audience; We present in this paper a detailed note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation. This paper is a continuation of [1] which was on the complexity of solving arbitrary ordinary polynomial differential equations in terms of Puiseux series.

We give a complete description for the dynamics of quadratic rational maps with coefficients in the completion of the field of formal Puiseux series.