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Nagoya Mathematical Journal

Formal and Rigid Geometry. Rigid Analytic Curves.

Jacobian Varieties. Raynaud Extensions. Abeloid Varieties. Edelman, E. Kostlan, M.

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Shub and S. Smale on random polynomial system solving, with a particular emphasis on asymptotic studies and complexity theory there is a much influential sequence of papers by Shub and Smale named complexity of Bezout's theorem''. The motivation for a random study came from the necessity, often stimulated by applications, of having a global picture allowing to overcome the noise given by the fluctuations of possible outcomes: since over the real numbers there is in general no generic'' case, the attention was shifted to the random'' case.

Under the influence of the Edelman-Kostlan-Shub-Smale model, this program has first seen a strong progress in the direction of the study of metric properties of random real algebraic sets e. This research direction can be considered as a further step in the original Shub-Smale program: if the complexity of Bezout's theorem'' papers dealt with the basic problem of counting points in the intersection of random hypersurfaces in projective space, here we count solutions to more advanced intersection theory problem. In this talk I will survey the main results and methods of this program and, time permitting, I will introduce some new questions related to random maps and tensor geometry.

Computing an isotopic piecewise-linear approximation of an implicit curve is a problem, which appears in several domains such as geometric modeling, graphics, numerical simulations, etc. The challenge is to compute efficiently and robustly a mesh close to the curve with the same topology, from the algebraic equation defining the curve. Several families of methods have been developed to tackle this problem, such as grid, subdivision or sweeping methods.

We consider the first two types of methods, which are usually applied with finite precision arithmetic and show good performance in practice. Grid methods evaluate the polynomial function at the points of a regular grid and deduce the topology of a level set of the function, from a local analysis. A very popular method is the so-called Marching Square algorithm.

Such approaches can be certified, when the function is a polynomial.


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We provide bounds on their complexity. Subdivision methods are based on an adaptive scheme, which subdivides the geometric domain and use local criteria to deduce the global topology of an algebraic set. An important ingredient of these methods is an adaptive local representation of the equation, such as Bernstein basis representation.

Appendix: Curves and Their Jacobians | SpringerLink

Exploiting interesting properties of this representation, the output size of subdivision methods can be controlled in terms of geometric characteristics of the curve. Another ingredient is a regularity test, which verify when the topology of the curve in a cell can be determined locally. We describe some of these regularity tests for algebraic curves. They require to isolate singular or critical points from regular parts. The topology near singular points is guaranteed through topological degree computation. The topology inside a cell is recovered from information on its boundary. We analyze the cost of these steps in terms of condition numbers of the polynomial systems involved in the problems.

Rigid Geometry of Curves and Their Jacobians

A comparison of subdivision and grid methods will be developed. Some examples will illustrate their behavior. I will then present an application of this technique to the split and non-split Cartan curve of level 13; this is joint work with J. Balakrishnan, N. Dogra, J.


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Tuitman and J. Explicit computations on the Jacobian variety of a smooth algebraic curve are of great interest in number theory and algebraic geometry. Over the complex numbers the analytic Jacobian has the structure of a complex torus which is, up to isomorpishm, determined by a period matrix of the corresponding compact Riemann surface.

Period matrices can be obtained through numerically integrating holomorphic differentials along the first integral homology group. For number-theoretic applications of period matrices it is often necessary to compute these integrals to high precision while using rigorous error bounds. We implemented algorithms in magma that compute period matrices, reliably and fast, using several integration methods in particular, Gauss-Legendre, Clenshaw-Curtis and double-exponential integration to high precision. In joint work with Pascal Molin, we developed a rigorous algorithm that achieves this goal for the special case of superelliptic curves and is much faster than the one for general curves.

Our implementations in magma and arb vastly improve the existing implementation for hyperelliptic curves in terms of speed and numerical stability.

We also compute the Abel-Jacobi map which is closely related to period matrices. Other important applications include the computation of endomorphism rings and theta functions. I will demonstrate a method for tracking precision in p-adic computations that often yields more precise results, at the cost of increased time and memory usage. The method uses differentials applied to p-adic lattices to get the optimal possible precision. I will give examples using an implementation in Sage.

This work is joint with Xavier Caruso and Tristan Vaccon. Slides , Demo. We will discuss asymptotic upper and lower bounds for the expected Betti numbers of those subcomplexes. This is a joint work with Jean-Yves Welschinger. In this talk, I will give an overview on recent progress in computing the topology of a planar algebraic curve C. Here, the curve is implicitly given as the vanishing set of an integer polynomial f in two variables, and the task is to compute a planar straight line graph G with vertices on C that is isotopic to the curve. One of the most crucial subroutines that is used in the corresponding algorithm is a fast method to compute the roots real and complex of a polynomial with algebraic coefficients.

I will give an overview on such methods and show how they can be integrated in algorithms for computing a cylindrical algebraic decomposition. Finally, I will report on an actual implementation of one of our algorithms to compute the topology of a planar algebraic curve.

This is joint work with R. From the Back Cover : The first contribution of this EMS volume on the subject of complex algebraic geometry touches upon many of the central problems in this vast and very active area of current research. Buy New Learn more about this copy. International Edition.

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Workshop - Numerical methods for algebraic curves

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New Paperback Quantity Available: 1. Parshin; Editor-I. Shafarevich; Translator-I.