On the complexity of solving ordinary differential equations in terms of Puiseux series

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

Exact linear modeling using Ore algebras

Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas the operators with varying coefficients were addressed only recently. This question can be tackled either using Gr\"obner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods to compute "most powerful unfalsified models" (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples

Extremal Growth of Multiple Toeplitz Operators and Implications for Numerical Stability of Approximation Schemes

We relate the power bound and a resolvent condition of Kreiss-Ritt type and characterize the extremal growth of two families of products of three Toeplitz operators on the Hardy space that contain infinitely many points in their spectra. Since these operators do not fall into a well-understood class, we analyze them through explicit techniques based on properties of Toeplitz operators and the structure of the Hardy space. Our methods apply mutatis mutandis to operators of the form $T_{g(z)}^{-1}T_{f(z)}T_{g(z)}$ where $f(z)$ is a polynomial in $z$ and $\bar{z}$ and $g(z)$ is a polynomial in $z$. This collection of operators arises in the numerical solution of the Cauchy problem for linear ordinary, partial, and delay differential equations that are frequently used as models for processes in the sciences and engineering. Our results provide a framework for the stability analysis of existing numerical methods for new classes of linear differential equations as well as the development of novel approximation schemes

Computational methods in algebra and analysis

This paper describes some applications of Computer Algebra to Algebraic Analysis also known as D-module theory, i.e. the algebraic study of the systems of linear partial differential equations. Gröbner bases for rings of linear differential operators are the main tools in the field. We start by giving a short review of the problem of solving systems of polynomial equations by symbolic methods. These problems motivate some of the later developed subjects.Ministerio de Ciencia y TecnologíaJunta de Andalucí