In this paper, we consider nonlocal, nonlinear partial differential equations to model anisotropic dynamics of complex root sets of random polynomials under differentiation. These equations aim to generalise the recent PDE obtained by Stefan Steinerberger (2019) in the real case, and the PDE obtained by Sean O'Rourke and Stefan Steinerberger (2020) in the radial case, which amounts to work in 1D. These PDEs approximate dynamics of the complex roots for random polynomials of sufficiently high degree n. The unit of the time t corresponds to n differentiations, and the increment Δt corresponds to 1/n. The general situation in 2D, in particular for complex roots of real polynomials, was not yet addressed. The purpose of this paper is to present a first attempt in that direction. We assume that the roots are distributed according to a regular distribution with a local homogeneity property (defined in the text), and that this property is maintained under differentiation. This allows us to derive a system of two coupled equations to model the motion. Our system could be interesting for other applications. The paper is illustrated with examples computed with the Maple system.

We apply Computer algebra techniques, such as algebraic computations of resultants and discriminants, certified drawing (with a guaranteed topology) of plane curves, to a problem in Fluid dynamics: We investigate ``capillary-gravity'' solitary waves in shallow water, relying on the framework of the Serre-Green-Naghdi equations. So, we deal with 2 dimensional surface waves, propagating in a shallow water of constant depth. By a differential elimination process, the study reduces to describing the solutions of an ordinary non linear first order differential equation, depending on two parameters. The paper is illustrated with examples and pictures computed with the computer algebra system Maple.

We consider a univariate polynomial f with real coefficients having a high degree N but a rather small number d + 1 of monomials, with d ≪ N. Such a sparse polynomial has a number of real root smaller or equal to d. Our target is to find for each real root of f an interval isolating this root from the others. The usual subdivision methods, relying either on Sturm sequences or Moebius transform followed by Descartes's rule of sign, destruct the sparse structure. Our approach relies on the generalized Budan-Fourier theorem of Coste, Lajous, Lombardi, Roy [8] and the techniques developed in Galligo [12]. To such a f is associated a set of d + 1 F-derivatives. The Budan-Fourier function Vf (x) counts the sign changes in the sequence of F-derivatives of the f evaluated at x. The values at which this function jumps are called the F-virtual roots of f, these include the real roots of f. We also consider the augmented F-virtual roots of f and introduce a genericity property which eases our study. We present a real root isolation method and an algorithm which has been implemented in Maple. We rely on an improved generalized Budan-Fourier count applied to both the input polynomial and its reciprocal, together with Newton like approximation steps. The paper is illustrated with examples and pictures.

In isogeometric anlaysis, parameterization of computational domain has great effects as mesh generation in finite element analysis. In this paper, based on the concept of harmonic map from the computational domain to parametric domain, a variational approach is proposed to construct the parameterization of computational domain for 2D isogeometric analysis. Different from the previous elliptic mesh generation method in finite element analysis, the proposed method focus on isogeometric version, and converts the elliptic PDE into a nonlinear optimization problem. A regular term is integrated into the optimization formulation to achieve more uniform grid near convex(concave) parts of the boundary. Several examples are presented to show the efficiency of the proposed method.

After the works of Gonzales-Vega, Lombardi, Mahe [11], and of Coste, Lajous, Lombardi, Roy [6], we consider the virtual roots of a univariate polynomial f with real coefficients. Using fractional derivatives, we associate to f a bivariate polynomial P(x,t) depending on the choice of an origin a, then two type of plan curves we call the FDcurve and stem of f. We show, in the generic case, how to locate the virtual roots of f on the Budan table and on each of these curves. The paper is illustrated with examples and pictures computed with the computer algebra system Maple.

Our probabilistic analysis sheds light to the following questions: Why do random polynomials seem to have few, and well separated real roots, on the average? Why do exact algorithms for real root isolation may perform comparatively well or even better than numerical ones?

Motivated by the computation of intersection loci in Computer Aided Geometric Design (CAGD), we introduce and study the elimination problem for systems of three bivariate polynomial equations with separated variables. Such systems are simple sparse bivariate ones but resemble to univariate systems of two equations both geometrically and algebraically. Interesting structures for generalized Sylvester and bezoutian matrices can be explicited. Then one can take advantage of these structures to represent the objects and speed up the computations. A corresponding notion of subresultant is presented and related to a Gröbner basis of the polynomial system.

We present the computation of selfintersections as a major problem in Computer Aided Geometric Design (CAD) and Geometric Modeling, and particularly for patches of parametrized bicubic surfaces. Then we expose two complementary contributions on that subject with Computer Algebra tools: First, a specific sparse bivariate resultant adapted to the corresponding elimination problem, second a semi-numeric polynomial solver able to deal with large system of equations with floating point coefficients. Examples and timings are provided.

Parametrized surfaces of low degrees are very useful in applications, specially in Computer Aided Geometric Design and Geometric Modeling. The precise description of their geometry is not easy in general. Here we study surfaces of bidegree (1,2). We show that, generically up to linear changes of coordinates, they are classified by two continuous parameters (modulus). We present an elegant combinatorial description where these modulus appear as cross ratios. We provide compact implicit equations for these surfaces and for their singular locus together with a geometric interpretation.

We introduced a general representation of algebraic surfaces, that we called semiimplicit, which encapsulates both usual and less known surfaces. Here we specialize this notion in order to apply it in solid modeling: we view a surface in /spl Ropf//sup 3/ as a one-parameter (algebraic) family of algebraic low-degree curves. The paper mainly addresses the topic of performing the usual CAD operations with semiimplicit representation of surfaces. We derive formulae for computing the normal and the curvatures at a regular point. We provide exact algorithms for computing self-intersections of a surface and more generally its singular locus. We also present some surface/surface intersection algorithms relying on generalized resultant calculations.

In this paper, we propose a new semi-numerical algorithmic method for factoring multivariate polynomials absolutely. It is based on algebraic and geometric properties after reduction to the bivariate case in a generic system of coordinates. The method combines 4 tools: zero-sum relations at triplets of points, partial information on monodromy action, Newton interpolation on a structured grid, and a homotopy method. The algorithm relies on a probabilistic approach and uses numerical computations to propose a candidate factorization (with probability almost one) which is later validated.

In this paper, we propose a semi-numerical algorithm for computing all irreducible branches of a curve in C3 defined by polynomials with rational coefficients. It is based on some properties appearing after a generic change of coordinate. Using numerical computation, Galois group action and rational approximation, it provides an efficient probabilistic algorithm for medium degrees. Our method generalizes our study on absolute factorization of polynomials ([2, 6]).

Article Free Access Share on A numerical absolute primality test for bivariate polynomials Authors: André Galligo Laboratoire de Mathématiques, Université de Nice, France Laboratoire de Mathématiques, Université de Nice, FranceView Profile , Stephen Watt IBM T.J. Watson Research Center and INRIA Sophia Antipolis, France IBM T.J. Watson Research Center and INRIA Sophia Antipolis, FranceView Profile Authors Info & Claims ISSAC '97: Proceedings of the 1997 international symposium on Symbolic and algebraic computationJuly 1997 Pages 217–224https://doi.org/10.1145/258726.258788Published:01 July 1997Publication History 22citation207DownloadsMetricsTotal Citations22Total Downloads207Last 12 Months2Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF