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We present in this note a correction to Theorem 17 in Ran and Wojtylak (Compl. Anal. Oper. Theory 15:44, 2021) and sharpen the estimates for eigenvalues of parametric rank one perturbations given in that theorem. more...

Given an autonomous first order algebraic ordinary differential equation F ( y , y ′ ) = 0 , we prove that every formal Puiseux series solution of F ( y , y ′ ) = 0 , expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide an algorithm to describe all such Puiseux series solutions. Moreover, we show that for any point in the complex plane there exists a solution of the differential equation which defines an analytic curve passing through this point. more...

A new efficient method that combines the Puiseux series asymptotic technique with an augmented compact finite volume method is proposed to develop a numerical approximate solution for the Thomas–Fermi equation on semi-infinity domain. By using the asymptotic series of solution at infinity and the Puiseux series expansion at origin to characterize the singularities, the natural and precise boundary conditions are obtained. The expansions contain undetermined parameters which associate with the singularity as the augmented variables. A regular boundary value problem is derived, for which an augmented compact finite volume method is used. The computational results show that the method not only obtains the high precise numerical solution, but also obtains the high precise initial slope. In particular, we find that the initial slope is exactly equal to the augmented variable related to the singularities in the Puiseux series. The initial slope not only has an important physical significance, but also its calculation accuracy has become an important criteria to measure the quality of the algorithm. more...

Spectral abscissa (SA) is defined as the real part of the rightmost characteristic root(s) of a dynamical system, and it can be regarded as the decaying rate of the system, the smaller the better from the viewpoint of fast stabilization. Based on the Puiseux series expansion of complex-valued functions, this paper shows that the SA can be minimized within a given delay interval at values where the characteristic equation has repeated roots with multiplicity 2 or 3. Four sufficient conditions in terms of the partial derivatives of the characteristic function are established for testing whether the SA is minimized or not, and they can be tested directly and easily. more...

We solve completely the problem of Liouvillian integrability for cubic Li\'{e}nard differential equations with quadratic damping. %Our results are applicable for a wide family of dynamical systems. Our main tool is the method of Puiseux series. We find necessary and sufficient conditions for equations under consideration to have Jacobi last multipliers of a special form. It turns out that some particular sub--families being Liouvillian non--integrable possess Jacobi last multipliers. The Jacobi last multipliers give rise to non--standard Lagrangians and it is an interesting property of these dynamical systems. In addition, we prove that cubic Li\'{e}nard differential equations with quadratic damping do not have algebraic limit cycles. more...

In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x0,y0)∈C2, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x0)=y0., Agencia Estatal de Investigación, Ministerio de Economía y Competitividad, Austrian Science Fund more...

The paper discusses several methods of analytic continuation of a multivalued function of one variable given on a part of its Riemann surface in the form of a Puiseux series generated by the power function z = w1/ρ, where ρ 1/2 and ρ ≠ 1. We present a many-sheeted variant of G.Polya’s theorem describing the relation between the indicator and conjugate diagrams for entire functions of exponential type. The description is based on V. Bernstein’s construction for the many-sheeted indicator diagram of an entire function of order ρ ≠ 1 and normal type. The summation domain of the “proper” Puiseux series (the many-sheeted “Borel polygon”) is found with the use of a generalization of the Borel method. This result seems to be new even in the case of a power series. The theory is applied to describe the domains of analytic continuation of Puiseux series representing the inverses of rational functions. As a consequence, a new approach to the solution of algebraic equations is found. more...

This study focuses on sound attenuation in a two-dimensional waveguide with arbitrary admittance boundary conditions on both sides of the guide. The emphasis is on understanding the formation and potential applications of the exceptional points (EPs) which arise when two (EP2) or three (EP3) modes degenerate into a single mode. A perturbation approach is used to obtain asymptotic expressions for the trajectories of the axial wavenumbers in the complex plane as they coalesce to form an EP. The numerical results presented herein suggest that the first triple root (EP3) assures maximum modal attenuation along the waveguide. Further, it is demonstrated that the classical Green’s function is degenerate at an EP. Modified Green’s functions which are valid at EP2 and EP3 are presented. more...

The beta-type functions play an important role in many applied sciences. The partial derivatives of the beta function and the incomplete beta function are integrals involving algebraic and logarithmic endpoint singularities. In this paper, some series expansions for these beta-type functions are found, which are easily used to evaluate these functions with prescribed precision. On the other hand, an accurate Gauss-Legendre quadrature formula is designed to compute these beta-type functions and their partial derivatives based on the Puiseux series for the integrands at their singularities. Some numerical examples confirm the high accuracy and high efficiency of the two algorithms, and also show that the algorithms can be used to effectively evaluate the generalized beta-type functions. more...

In this paper, we present an algorithm for determining the limits of real rational functions in two variables, based on Sturm's familiar theorem and the general Sturm–Tarski theorem for counting certain roots of univariate polynomials in a real closed field. Let R [ x , y ] be the ring of polynomials with real coefficients in two variables x, y, and let u ( x , y ) , v ( x , y ) ∈ R [ x , y ] be two non-zero polynomials such that u ( a , b ) = v ( a , b ) = 0 for a, b ∈ R . The purpose of this paper is to decide the existence of lim ( x , y ) → ( a , b ) ⁡ u ( x , y ) v ( x , y ) and compute the limit if it exists. Our algorithm needs no assumption on the denominators and does not involve the computation of Puiseux series. more...

Purpose This paper aims to analyze soil electrical properties based on fractional calculus theory due to the fact that the frequency dependence of soil electrical parameters at high frequencies exhibits a fractional effect. In addition, for the fractional-order formulation, this paper aims to provide a more accurate numerical algorithm for solving the fractional differential equations. Design/methodology/approach This paper analyzes the frequency-dependence of soil electrical properties based on fractional calculus theory. A collocation method based on the Puiseux series is proposed to solve fractional differential equations. Findings The algorithm proposed in this paper can be used to solve fractional differential equations of arbitrary order, especially for 0.5th-order equations, obtaining accurate numerical solutions. Calculating the impact response of the grounding electrode based on the fractional calculus theory can obtain a more accurate result. Originality/value This paper proposes an algorithm for solving fractional differential equations of arbitrary order, especially for 0.5th-order equations. Using fractional calculus theory to analyze the frequency-dependent effect of soil electrical properties, provides a new idea for ground-related transient calculation. more...

In the theory of the N-point gravita- tional lens equation, two groups of problems can be dis- tinguished. These are the so-called primal and inverse problems. Primal problems include problems of image definition in a specified lens for a specified source. In- verse problems include problems of determining a lens, source, or multiple images from one or more specified images. Inverse problem have an important applica- tions. We studied the equation of the N-point gravitational lens in a complex form. These studies became the basis for the solution of the inverse problem in the following formulation. N-point gravitational lens has specified. It is necessary to determine all other images from one of the images of a point source in N-point gravitational lens. Determine the necessary and sufficient conditions under which this problem has solutions. The algebraic formulation of the problem has the following form. The equation (of N-point gravitational lens) has specified. It is necessary to solve the problem of solutions unification (to express unequivocally all of the equation solutions through one parameter). To solve the inverse problem, we used methods of al- gebraic geometry and function theory. Branches equa- tions of any algebraic function admit unequivocal pa- rameterization by Puiseux series. The solutions of the N-point gravitational lens equation are algebraic functions defined by a certain irreducible polynomial. That polynomial has unequivocally defined by the N- point gravitational lens equation. Thus, the polyno- mial roots also admits parameterization by Puiseux se- ries. In simple cases, for lenses with a small number of point masses, the solution can be obtained in a sim- pler form. In particular, for the Schwarzschild lens and binary lens, the inverse problem has a solution in radicals. more...

For regular expressions in (strong) star normal form a large set of efficient algorithms is known, from conversions into finite automata to characterisations of unambiguity. In this paper we study the average complexity of this class of expressions using analytic combinatorics. As it is not always feasible to obtain explicit expressions for the generating functions involved, here we show how to get the required information for the asymptotic estimates with an indirect use of the existence of Puiseux expansions at singularities. We study, asymptotically and on average, the alphabetic size, the size of the [Formula: see text]-follow automaton and of the position automaton, as well as the ratio and the size of these expressions to standard regular expressions. more...

In 1669 and 1671, Isaac Newton resolved an algebraic equation f(x, y)=0 by expressing y as an infinite series of x . In this paper, we formulate Newton's resolution as a contemporary algorithm along the line of his original text and prove that the series converges asymptotically to the implicit function or one of the branches under certain conditions., "The study of the history of mathematics 2018". September 18-21, 2018. edited by Shigeru Jochi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed. more...

Let $F\in \mathbb{K}[X, Y ]$ be a polynomial of total degree $D$ defined over a perfect field $\mathbb{K}$ of characteristic zero or greater than $D$. Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes the singular parts of all Puiseux series of $F$ above $X = 0$ in less than $\tilde{\mathcal{O}}(D\delta)$ operations in $\mathbb{K}$, where $\delta$ is the valuation of the resultant of $F$ and its partial derivative with respect to $Y$. To this aim, we use a divide and conquer strategy and replace univariate factorization by dynamic evaluation. As a first main corollary, we compute the irreducible factors of $F$ in $\mathbb{K}[[X]][Y ]$ up to an arbitrary precision $X^N$ with $\tilde{\mathcal{O}}(D(\delta + N ))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by $F$ with $\tilde{\mathcal{O}}(D^3)$ arithmetic operations and, if $\mathbb{K} = \mathbb{Q}$, with $\tilde{\mathcal{O}}((h+1)D^3)$ bit operations using a probabilistic algorithm, where $h$ is the logarithmic heigth of $F$., Comment: 27 pages, 2 figures more...

International audience; There exist several methods for computing exact solutions of algebraic differential equations. Most of the methods, however, do not ensure existence and uniqueness of the solutions and might fail after several steps, or are restricted to linear equations. The authors have presented in previous works a method to overcome this problem for autonomous first order algebraic ordinary differential equations and formal Puiseux series solutions and algebraic solutions. In the first case, all solutions can uniquely be represented by a sufficiently large truncation and in the latter case by its minimal polynomial. The main contribution of this paper is the implementation, in a MAPLE-package named FirstOrderSolve, of the algorithmic ideas presented therein. More precisely, all formal Puiseux series and algebraic solutions, including the generic and singular solutions, are computed and described uniquely. The computation strategy is to reduce the given differential equation to a simpler one by using local parametrizations and the already known degree bounds. more...

Let $\mathcal{C}$ be a plane curve given by an equation $f(x,y)=0$ with $f\in K[x][y]$ a monic squarefree polynomial. We study the problem of computing an integral basis of the algebraic function field $K(\mathcal{C})$ and give new complexity bounds for three known algorithms dealing with this problem. For each algorithm, we study its subroutines and, when it is possible, we modify or replace them so as to take advantage of faster primitives. Then, we combine complexity results to derive an overall complexity estimate for each algorithm. In particular, we modify an algorithm due to B\"ohm et al. and achieve a quasi-optimal runtime., Comment: Preliminary version more...

International audience; A numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods. The radius of convergence of eigenvalue perturbation methods, based on Taylor series, is known to decrease when eigenvalues are getting closer to each other. This phenomenon, knwon as veering in structural dynamics, is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective. When this degeneracy, referred to as Exceptional Point (EP), is close to the real axis, the veering becomes stronger. The main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity. To do so, two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter. This yields a new robust eigenvalue reconstruction scheme which is compared to Taylor and Puiseux series. In all cases, theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model. This system illustrate the ability of the method to handle the vibrations of a structure with a randomly varying parameter. Computationally efficient, the proposed algorithm could also be relevant for actual numerical models of large size, arising from other applications involving parametric eigenvalue problems, e.g., waveguides, rotating machinery or instability problems such as squeal or flutter. more...

In this work, we investigate the asymptotic stability of LTI differential commensurate time-delay systems whose dynamics are defined in the frequency domain by quasi-polynomials of the form f(S, T) = Σmj=0 pj(s) e-jTS. We propose a new approach for studying the stability with respect to the delay: we determine the asymptotic behavior of the roots of the quasi-polynomial near to the critical pairs of f(s, T) by analysing the intersection of an associated algebraic space curve with sufficiently refined 3D real boxes. Compared to the existing methods, our method does not require any Puiseux series computation. more...

Picard–Vessiot theorem (1910) provides a necessary and sufficient condition for solvability of linear differential equations of order n by quadratures in terms of its Galois group. It is based on the differential Galois theory and is rather involved. Liouville in 1839 found an elementary criterium for such solvability for $$n=2$$ . Ritt simplified Liouville’s theorem (1948). In 1973 Rosenlicht proved a similar criterium for arbitrary n. Rosenlicht work relies on the valuation theory and is not elementary. In these notes we show that the elementary Liouville–Ritt method based on developing solutions in Puiseux series as functions of a parameter works smoothly for arbitrary n and proves the same criterium. more...

We study the support function in the neighborhood of inflections of oriented planar curves. Even for a regular curve, the support function is not regular at the inflection and is multivalued on its neighborhood. We describe this function using an implicit algebraic equation and the rational Puiseux series of its branches. Based on these results we are able to approximate the curve at its inflection to any desired degree by curves with a simple support function, which consequently possess rational offsets. We also study the G 1 Hermite interpolation at two points of a planar curve. It is reduced to the functional C 1 interpolation of the support function. For the sake of comparison and better understanding, we show (using standard methods) that its approximation order is 4 for inflection-free curves. In the presence of inflection points this approximation is known to be less efficient. We analyze this phenomenon in detail and prove that by applying a nonuniform subdivision scheme it is possible to receive the best possible approximation order 4, even in the inflection case. more...

We consider the extension of the method of Gauss–Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the coefficient matrix is a series with matrix coefficients, and provide a characterization for when the matrix series is regular based on the algebraic variety of an augmented system. The structure of the linear system leads to a block triangular system. In the regular case, solving the linear system is equivalent to solving a Hermite interpolation problem. We show that this solution has cost cubic in the problem size. In general, at singular points, we rely on methods of tropical algebraic geometry to compute Puiseux series. With a few illustrative examples, we demonstrate the application to polynomial homotopy continuation. more...

The Puiseux series generated by the power function z = w1/ρ, where ρ > 0,ρ ≠ 1, is considered. A version of the Polya–Bernstein theorem for an entire function of order ρ ≠ 1 and normal type is proposed and applied to describe the domain of analytic continuation of this series. The domain of summability of a “regular” Puiseux series is found (this is a many-sheeted “Borel polygon”); in the case ρ = 1, the “one-sheeted” result of Borel is substantially extended. These results make it possible to describe domains of analytic continuation of the Puiseux expansions of popular many-sheeted functions (such as inverses of rational functions). more...

We show for $n,k\geq1$, and an $n$-dimensional complex vector space $V$ that if an element $A\in\text{End}(V)[[z]]$ has constant term similar to a Jordan block, then there exists a polynomial gauge transformation $g$ such that the first $k$ coefficients of $gAg^{-1}$ have a controlled normal form. Furthermore, we show that this normal form is unique by demonstrating explicit relationships between the first $nk$ coefficients of the Puiseux series expansion of the eigenvalues of $A$ and the entries of the first $k$ coefficients of $gAg^{-1}$., Comment: 13 pages, to appear in Involve: A Journal of Mathematics more...

An accurate semi-decoupling numerical method has been proposed for nonlinear singular differential equation with interface, which combines Puiseux series asymptotic technique with augmented compact finite volume method. The main motivation is to decouple the singular interface problem and get high order accurate numerical solution. Key to our proposed new method is introducing three augmented variables involving the interface and the singularity, and reconstructing the representative of the solution as the Puiseux series expansions on the interface to decouple original problem as two standard nonlinear singular problems with the second jump condition. In this way, the augmented variables related with semi-analytic solutions near the interface and numerical solutions can be simultaneously solved in the remaining interval on both sides of the interface. It demonstrates that our method does not take more heavier works for handling jump conditions like other methods, and is independent of the interface and jump ratio. A rigorous error estimate for the solution of nonlinear singular differential equation with interface and augmented variables is obtained. Numerical experiments for those singular differential equations with interface confirm the theoretical analysis and accuracy of the new approach. In particular, an interesting example with blow-up coefficient at singular point shows that our approach can be extended to numerically solve strongly singular interface problem. more...

Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue measure and integration theory such that the main results from the classical setting hold. The construction involves methods from model theory, o-minimal geometry and valuation theory. We set up the construction in such a way that it is determined by a section of the valuation. If the value group is isomorphic to the group of rational numbers the construction is uniquely determined up to isomorphism. The range of the measure and integration is obtained in a controlled and tame way from the real closed field we start with. The main example is given by the case of the field of Puiseux series where the range is the polynomial ring in one variable over this field. more...

This paper develops an efficient algorithm to solve nonlinear Volterra integral equation of the second kind with weakly singular convolution kernel. First, we show that the general Puiseux series for the solution about zero exists under smooth assumptions for the nonlinear function, and then design an algorithm to recover the finite-term truncation of the asymptotic expansion by Picard iteration. This asymptotic expansion can easily yield a more accurate Pade approximation. Second, we use trapezoidal rule to discretize the singular integral and derive the Euler–Maclaurin asymptotic expansion using the known Puiseux expansion of the solution. By accumulating some lower order error terms to the quadrature formula, we obtain high precision evaluations to the nonlinear Volterra integral equation. Third, we prove that the scheme is convergent by extending the Gronwall inequality to be held for the scheme. Fourth, an example is provided to illustrate that the combination of the Puiseux expansion and the numerical integration can effectively increase the computational accuracy of the equation. Finally, we apply the method to solve the Lighthill integral equation, and obtain the asymptotic expansions of the solution near zero and infinity, respectively. The computation shows that the trapezoidal rule is only necessary in a small finite range of the semi-infinite interval. more...

The quantum problem of four particles in R d ( d ≥ 3 ), with arbitrary masses m 1 , m 2 , m 3 and m 4 , interacting through a harmonic oscillator potential is considered. This model allows exact solvability and a critical analysis of the Born–Oppenheimer approximation. The study is restricted to the ground state level. We pay special attention to the case of two equally heavy masses m 1 = m 2 = M and two light particles m 3 = m 4 = m . It is shown that the sum of the first two terms of the Puiseux series, in powers of the dimensionless parameter σ = m M , of the exact phase Φ of the wave function ψ 0 = e − Φ and the corresponding ground state energy E 0 , coincide exactly with the values obtained in the Born–Oppenheimer approximation. A physically relevant rough model of the H 2 molecule and of the chemical compound H 2 O 2 (Hydrogen peroxide) is described in detail. The generalization to an arbitrary number of particles n , with d degrees of freedom ( d ≥ n − 1 ), interacting through a harmonic oscillator potential is briefly discussed as well. more...

International audience; For applications dealing with dissipative acoustic waveguides strong modal attenuation are often achieved close to exceptional points (EP). These EP correspond to a particular tuning of certain design parameters which render the associated eigenvalue problem degenerate where both eigenvalues, i.e. the wavenumber, and eigenvectors of a pair of guided modes coalesce. These non-Hermitian degeneracies have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems. Here, a new algorithm is proposed to quickly explore the parametric space and to locate EPs. The method requires the computation of successive derivatives of two selected eigenpairs with respect to the parameter so that, after recombination, regular functions can be constructed. This algebraic manipulation permits the EP localization, using standard root-finding algorithms and the computation of the associated Puiseux series up to an arbitrary order, useful to follow modal branches. Examples related to the acousic propagation in straight ducts with absorbing walls and in periodic guiding structures are given to illustrate the versatility of the proposed method and its ability to handle large size matrices arising from finite element discretization techniques. more...

We present the Painleve test on various examples of nonlinear ODEs, $$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle E(x,u^{(N)},\ldots,u^{\prime},u)=0,\, ^\prime = \frac{\mathrm{d}}{\mathrm{d} x}, \, \ u^{(N)}=\frac{\mathrm{d}^N u}{\mathrm{d} x^N}, {} {} \end{array} \end{aligned} $$ (2.1) This is a local analysis which can be implemented as an algorithm to provide necessary conditions for the Painleve property. more...

Mathieu functions of period $\pi$ or $2\pi$, also called elliptic cylinder functions, were introduced in 1868 by \'Emile Mathieu together with so-called modified Mathieu functions, in order to help understand the vibrations of an elastic membrane set in a fixed elliptical hoop. These functions still occur frequently in applications today: our interest, for instance, was stimulated by a problem of pulsatile blood flow in a blood vessel compressed into an elliptical cross-section. This paper surveys and recapitulates the historical development of the theory and methods of computation for Mathieu functions and modified Mathieu functions and identifies some gaps in current software capability, particularly to do with double eigenvalues of the Mathieu equation. We demonstrate how to compute Puiseux expansions of the Mathieu eigenvalues about such double eigenvalues, and give methods to compute the generalized eigenfunctions that arise there. In examining Mathieu's original contribution, we bring out that his use of anti-secularity predates that of Lindstedt. For interest, we also provide short biographies of some of the major mathematical researchers involved in the history of the Mathieu functions: \'Emile Mathieu, Sir Edmund Whittaker, Edward Ince, and Gertrude Blanch., Comment: 65 pages, 15 figures more...

In this paper, we present an adaptive step-size homotopy tracking method for computing bifurcation points of nonlinear systems. There are four components in this new method: (1) an adaptive tracking technique is developed near bifurcation points; (2) an inflation technique is backed up when the adaptive tracking fails; (3) Puiseux series interpolation is used to compute bifurcation points; and (4) the tangent cone structure of the bifurcation point is approximated numerically to compute solutions on different branches. Various numerical examples of nonlinear systems are given to illustrate the efficiency of this new approach. This new adaptive homotopy tracking method is also applied to a system of nonlinear PDEs and shows robustness and efficiency for large-scale nonlinear discretized systems. more...

We study analytic properties of the dispersion relations in classical hydrodynamics by treating them as Puiseux series in complex momentum. The radii of convergence of the series are determined by the critical points of the associated complex spectral curves. For theories that admit a dual gravitational description through holography, the critical points correspond to level-crossings in the quasinormal spectrum of the dual black hole. We illustrate these methods in ${\cal N}=4$ supersymmetric Yang-Mills theory in 3+1 dimensions, in a holographic model with broken translation symmetry in 2+1 dimensions, and in conformal field theory in 1+1 dimensions. We comment on the pole-skipping phenomenon in thermal correlation functions, and show that it is not specific to energy density correlations., V3: 54 pages, 18 figures. Appendix added. Version to appear in JHEP more...

International audience; This paper aims at studying the asymptotic stability of retarded type linear differential systems with commensurate delays. Within the frequency-domain approach, it is well-known that the asymptotic stability of such a system is ensured by the condition that all the roots of the corresponding quasipolynomial have negative real parts. A classical approach for checking this condition consists in computing the set of critical zeros of the quasipolynomial, i.e., the roots (and the corresponding delays) of the quasipolynomial that lie on the imaginary axis, and then analyzing the variation of these roots with respect to the variation of the delay. Following this approach, based on solving algebraic systems techniques, we propose a certified and efficient symbolic-numeric algorithm for computing the set of critical roots of a quasipolynomial. Moreover, using recent algorithmic results developed by the computer algebra community, we present an efficient algorithm for the computation of Puiseux series at a critical zero which allows us to finely analyze the stability of the system with respect to the variation of the delay. Explicit examples are given to illustrate our algorithms.; Ce papier a pour but l’étude de la stabilité asymptotique des systèmes différentiels linéaires à retards commensurables de type retardé. Dans l’approche fréquentielle, il est bien connu que la stabilité asymptotique d’un tel système est assurée par la condition que les racines du quasipolynôme correspondant ont des parties réelles négatives. Une approche classique pour tester cette condition consiste à calculer l’ensemble des zéros critiques du quasipolynôme, c’est-à-dire les racines (et les retards correspondants) du quasipolynôme qui sont sur l’axe imaginaire, et d’analyser la variation de ces racines par rapport à la variation du retard [16]. Suivant cette approche, et en nous basant sur des techniques de résolution de systèmes algébriques, nous proposons un algorithme symbolique-numérique efficace et certifié pour le calcul de l’ensemble des racines critiques d’un quasipolynôme. De plus, en utilisant des résultats algorithmiques récents développés par la communauté du calcul formel, nous présentons un algorithme efficace pour le calcul des séries de Puiseux en une racine critique nous permettant d’analyser finement la stabilité du système par rapport à la variation du retard [15]. Nous donnons des exemples explicites qui illustrent nos algorithmes. more...

We present a new algorithm for computing integral bases in algebraic function fields of one variable, or equivalently for constructing the normalization of a plane curve. Our basic strategy makes use of the concepts of localization and completion, together with the Chinese remainder theorem, to reduce the problem to the task of finding integral bases for the branches of each singularity of the curve. To solve the latter task, in turn, we work with suitably truncated Puiseux expansions. In contrast to van Hoeij's algorithm, which also relies on Puiseux expansions (but pursues a different strategy), we use Hensel's lemma as a key ingredient. This allows us at some steps of the algorithm to compute factors corresponding to conjugacy classes of Puiseux expansions, without actually computing the individual expansions. In this way, we make substantially less use of the Newton-Puiseux algorithm. In addition, our algorithm is inherently parallel. As a result, it outperforms in most cases any other algorithm known to us by far. Typical applications are the computation of adjoint ideals and, based on this, the computation of Riemann-Roch spaces and the parametrization of rational curves., 47 pages; revised structure more...

© 2017 Society for Industrial and Applied Mathematics. We contribute to the perturbation theory of nonlinear eigenvalue problems in three ways. First, we extend the formula for the sensitivity of a simple eigenvalue with respect to a variation of a parameter to the case of multiple nonsemisimple eigenvalues, thereby providing an explicit expression for the leading coefficients of the Puiseux series of the emanating branches of eigenvalues. Second, for a broad class of delay eigenvalue problems, the connection between the finitedimensional nonlinear eigenvalue problem and an associated infinite-dimensional linear eigenvalue problem is emphasized in the developed perturbation theory. Finally, in contrast to existing work on analyzing multiple eigenvalues of delay systems, we develop all theory in a matrix framework, i.e., without reduction of a problem to the analysis of a scalar characteristic quasi-polynomial. ispartof: SIAM Journal on Matrix Analysis and Applications vol:38 issue:2 pages:599-620 status: published more...

Let $f(x,y)$ be a complex irreducible formal power series without constant term. One may solve the equation $f(x,y)=0$ by choosing either $x$ or $y$ as independent variable, getting two finite sets of Newton-Puiseux series. In 1967 and 1968, Abhyankar and Zariski published proofs of an \emph{inversion theorem}, expressing the \emph{characteristic exponents} of one set of series in terms of those of the other ones. In fact, a more general theorem, stated by Halphen in 1876 and proved by Stolz in 1879, relates also the \emph{coefficients} of the characteristic terms of both sets of series. This theorem seems to have been completely forgotten. We give two new proofs of it and we generalize it to a theorem concerning equations with an arbitrary number of variables., 27 pages. This is the final published version. The introduction and several proofs were modified according to the recommendations of the referee. The bibliography was augmented, Mathematische Annalen. Online first on 03.12.2016 more...

A general framework is constructed for efficiently and stably evaluating the Hadamard finite-part integrals by composite quadrature rules. Firstly, the integrands are assumed to have the Puiseux expansions at the endpoints with arbitrary algebraic and logarithmic singularities. Secondly, the Euler-Maclaurin expansion of a general composite quadrature rule is obtained directly by using the asymptotic expansions of the partial sums of the Hurwitz zeta function and the generalized Stieltjes constant, which shows that the standard numerical integration formula is not convergent for computing the Hadamard finite-part integrals. Thirdly, the standard quadrature formula is recast in two steps. In step one, the singular part of the integrand is integrated analytically and in step two, the regular integral of the remaining part is evaluated using the standard composite quadrature rule. In this stage, a threshold is introduced such that the function evaluations in the vicinity of the singularity are intentionally excluded, where the threshold is determined by analyzing the roundoff errors caused by the singular nature of the integrand. Fourthly, two practical algorithms are designed for evaluating the Hadamard finite-part integrals by applying the Gauss-Legendre and Gauss-Kronrod rules to the proposed framework. Practical error indicator and implementation involved in the Gauss-Legendre rule are addressed. Finally, some typical examples are provided to show that the algorithms can be used to effectively evaluate the Hadamard finite-part integrals over finite or infinite intervals. more...

An accurate and efficient numerical method is proposed for nonlinear singular two point boundary value problems. The scheme combines Puiseux series asymptotic technique with augmented compact finite volume method. The main motivation is to get high order accurate solution for nonlinear singular problems. The key of the new method is to express the solution as the Puiseux series expansion on a small subinterval involving the singular point. The expansion contains an undetermined parameter which is an augmented variable in the numerical method. In this way, a nonlinear system is constructed in other subinterval and the parameter related with the semi-analytic solution near the singular point and the numerical solution can be simultaneously obtained. A rigorous error estimate for the solution of the singular differential equation is conducted and fourth order accuracy is obtained. Numerical examples confirm the theoretical analysis and efficiency of the new method. more...

This paper derives an explicit formula for a type of fractional power series, known as a Puiseux series, arising in a wide class of applied problems in the physical sciences and engineering. Detailed consideration is given to the gaps which occur in these series (lacunae); they are shown to be determined by a number-theoretic argument involving the greatest common divisor of a set of exponents appearing in the Newton polytope of the problem, and by two number-theoretic objects, called here Sylvester sets, which are complements of Frobenius sets. A key tool is Faà di Bruno’s formula for high derivatives, as implemented by Bell polynomials. Full account is taken of repeated roots, of arbitrary multiplicity, in the leading-order polynomial which determines a fractional-power expansion, namely the facet polynomial. For high multiplicity, the fractional powers are shown to have large denominators and contain irregularly spaced gaps. The orientation and methods of the paper are those of applications, but in a concluding section we draw attention to a more abstract approach, which is beyond the scope of the paper. more...

The algebraic and geometric properties for the local stability crossing curves (SCCs) of systems with two free parameters have attracted considerable interest, but some fundamental issues still remain unsolved. In this paper, we will develop a systematic approach for addressing such SCCs. First, we will parametrize the local SCCs through proposing an algorithm, with which the parametrization in the general case can be obtained. It will be seen that the parametrization in the general case is subject to some systems of Puiseux series, a new notion introduced in this paper. With the systems of Puiseux series, we can further explore the algebraic as well as the geometric properties of local SCCs. Next, all possible cases regarding the systems of Puiseux series and the topological structure will be appropriately classified. The resultant topological classifications, which take into account the distribution of the characteristic roots (from the stability standpoint), are insightful for the stability and stabilization studies. Finally, the asymptotic behavior issue when the parameters move along any given curve is addressed. We will show that the asymptotic behavior, in this case, corresponds to a group of Puiseux series, from which the detailed information on the local root loci is available. Moreover, this approach helps to verify the approaches proposed previously, such that all the approaches presented in our paper can be cross-validated. more...

Helton and Nie conjectured that every convex semialgebraic set over the field of real numbers can be written as the projection of a spectrahedron. Recently, Scheiderer disproved this conjecture. We show, however, that the following result, which may be thought of as a tropical analogue of this conjecture, is true: over a real closed nonarchimedean field of Puiseux series, the convex semialgebraic sets and the projections of spectrahedra have precisely the same images by the nonarchimedean valuation. The proof relies on game theory methods., Comment: 20 pages, 7 figures, this is an extended version of the preprint presented at MEGA 2017 more...

We study a Chemostat system of the form x = − q x + R K + y x y , y = ( c − y ) q − R a ( K + y ) x y , where q > 0 , R > 0 , K > 0 , c > 0 and a ≠ 0 . This system appears in competition modelling in biology. We describe its global dynamics on the Poincare disc and study its Liouvillian integrability. For the first topic we use the well-known Poincare compactification theory and for the second one we make use of the Puiseux series to derive the structure of all the irreducible invariant algebraic curves. more...

This paper aims at studying the asymptotic stability of retarded type linear differential systems with commensurate delays. Within the frequency-domain approach, it is well-known that the asymptotic stability of such a system is ensured by the condition that all the roots of the corresponding quasipolynomial have negative real parts. A classical approach for checking this condition consists in computing the set of critical zeros of the quasipolynomial, i.e., the roots (and the corresponding delays) of the quasipolynomial that lie on the imaginary axis, and then analyzing the variation of these roots with respect to the variation of the delay. Following this approach, based on solving algebraic systems techniques, we propose a certified and efficient symbolic-numeric algorithm for computing the set of critical roots of a quasipolynomial. Moreover, using recent algorithmic results developed by the computer algebra community, we present an efficient algorithm for the computation of Puiseux series at a critical zero which allows us to finely analyze the stability of the system with respect to the variation of the delay. Explicit examples are given to illustrate our algorithms. more...

A numerical algorithm is proposed to explore in a systematic way the trajectories of the eigenvalues of non-Hermitian matrices in the parametric space and exploit this in order to find the locations of defective eigenvalues in the complex plane. These non-Hermitian degeneracies also called exceptional points (EP) have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems. The method requires the computation of successive derivatives of two selected eigenvalues with respect to the parameter so that, after recombination, regular functions can be constructed. This algebraic manipulation permits the localization of exceptional points (EP), using standard root-finding algorithms and the computation of the associated Puiseux series up to an arbitrary order. This representation, which is associated with the topological structure of Riemann surfaces allows to efficiently approximate the selected pair in a certain neighbourhood of the EP. Practical applications dealing with guided acoustic waves propagating in straight ducts with absorbing walls and in periodic guiding structures are given to illustrate the versatility of the proposed method and its ability to handle large size matrices arising from finite element discretization techniques. The fact that EPs are associated with optimal dissipative treatments in the sense that they should provide best modal attenuation is also discussed., Comment: Fix typo in Eq. (17), add EasterEig library url, few other enhancements more...

International audience; This paper proposes an analytical method to characterize the behavior of multiple critical roots of a retarded system with two delays. Expressing locally the related characteristic function, as a Weierstrass polynomial, we derive several results to analyze the stability behavior of such characteristic roots with respect to small variations on the delay parameters. The proposed results are illustrated by considering several numerical examples. more...

Mixed-signal system-on-chip (SoC) devices offer single-chip solutions, but face challenges of hardware-software co-design optimization, device signal range constraints, and limited precision. These issues are addressed by developing a multi-level evolutionary approach to realize complex computational circuits called Embedded-Cascaded Hierarchically Evolved Logic Output Networks (ECHELON). The ECHELON technique utilizes analog evolved building blocks and refines their output using digital fabric to compose power series expansions of transcendental functions which are all routed under intrinsic control on a field-programmable SoC (PSoC). The result for the evolution of seven different powers of the independent variable is a reduction of 31.24% in the overall error as compared to the analog circuits that produce the raw inputs to a differential digital correction phase. Computation blocks developed on a Cypress PSoC-5LP mixed-signal SoC reduced error in the final mathematical approximation to the range of 40–150 mV. In doing so, speedups of roughly 1.4-fold to 6.6-fold with an average of 2.72-fold reduction in function execution times were attained. In particular, this approach achieved a 41.7-fold reduction in error with respect to the largest power of the independent variable used as an input to compute an erf(x) function. more...

In 1994 Lazard proposed an improved projection operation for cylindrical algebraic decomposition (CAD). For the proof he introduced a certain notion of valuation of a multivariate Puiseux series at a point. However a gap in one of the key supporting results for the improved projection was subsequently noticed. In this paper we show that Lazard's projection is valid for CAD construction for so-called well-oriented polynomial sets. Our proof does not make use of Lazard's notion of valuation, however. more...

International audience; This paper is devoted to the study of the stability of linear differential systems with commensurate delays. Within the frequency-domain approach, it is well-known that the asymptotic stability of such systems is ensured by the condition that all the roots of the corresponding quasipolynomial have negative real parts. A classical approach for checking this condition consists in computing the set of critical zeros of the quasipolynomial, i.e., the roots (and the corresponding delays) of the quasipolynomial that lie on the imaginary axis, and then analyzing the variation of these roots with respect to the variation of the delay. Following this approach, based on solving algebraic systems techniques, we propose a certified and efficient symbolic-numeric algorithm for computing the set of critical roots of a quasipolynomial. Moreover, using recent algorithmic results developed by the computer algebra community, we present an efficient algorithm for the computation of Puiseux series at a critical zero which allows us to finely analyze the stability of the system with respect to the variation of the delay. Explicit examples are given to illustrate our algorithms. more...