# Algebraic Geometry - Explore the Science & Experts | ideXlab

## Algebraic Geometry

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### Chaoping Xing – One of the best experts on this subject based on the ideXlab platform.

• ##### Repairing AlgebraicGeometry Codes
IEEE Transactions on Information Theory, 2018
Co-Authors: Lingfei Jin, Yuan Luo, Chaoping Xing

Abstract:

Minimum storage regenerating codes have minimum storage of data in each node and therefore are maximal distance separable (for short) codes. Thus, the number of nodes is upper-bounded by $2^{ {\mathfrak {b}}}$ , where ${\mathfrak {b}}$ is the bits of data stored in each node. From both theoretical and practical points of view (see the details in Section 1 ), it is natural to consider regenerating codes that nearly have minimum storage of data, and meanwhile, the number of nodes is unbounded. One of the candidates for such regenerating codes is an Algebraic Geometry code. In this paper, we generalize the repairing algorithm of Reed–Solomon codes given by Guruswami and Wotters to Algebraic Geometry codes and present a repairing algorithm for arbitrary one-point Algebraic Geometry codes. By applying our repairing algorithm to the one-point Algebraic Geometry codes based on the Garcia–Stichtenoth tower, one can repair a code of rate $1- \varepsilon$ and length $n$ over $\mathbb {F}_{q}$ with bandwidth $(n-1)(1- \tau)\log q$ for any $\varepsilon =2^{(\tau -1/2)\log q}$ with a real $\tau \in (0,1/2)$ . In addition, storage in each node for an Algebraic Geometry code is close to the minimum storage. Due to nice structures of Hermitian curves, repairing of Hermitian codes is also investigated. As a result, we are able to show that Algebraic Geometry codes are regenerating codes with good parameters.

• ##### Efficiently repairing AlgebraicGeometry codes.
arXiv: Information Theory, 2017
Co-Authors: Lingfei Jin, Yuan Luo, Chaoping Xing

Abstract:

Minimum storage regenerating codes have minimum storage of data in each node and therefore are maximal distance separable (MDS for short) codes. Thus, the number of nodes is upper bounded by $2^{\fb}$, where $\fb$ is the bits of data stored in each node. From both theoretical and practical points of view (see the details in Section 1), it is natural to consider regenerating codes that nearly have minimum storage of data, and meanwhile the number of nodes is unbounded. One of the candidates for such regenerating codes is an Algebraic Geometry code. In this paper, we generalize the repairing algorithm of Reed-Solomon codes given in \cite[STOC2016]{GW16} to Algebraic Geometry codes and present an efficient repairing algorithm for arbitrary one-point Algebraic Geometry codes. By applying our repairing algorithm to the one-point Algebraic Geometry codes based on the Garcia-Stichtenoth tower, one can repair a code of rate $1-\Ge$ and length $n$ over $\F_{q}$ with bandwidth $(n-1)(1-\Gt)\log q$ for any $\Ge=2^{(\Gt-1/2)\log q}$ with a real $\tau\in(0,1/2)$. In addition, storage in each node for an Algebraic Geometry code is close to the minimum storage. Due to nice structures of Hermitian curves, repairing of Hermitian codes is also investigated. As a result, we are able to show that Algebraic Geometry codes are regenerating codes with good parameters. An example reveals that Hermitian codes outperform Reed-Solomon codes for certain parameters.

• ##### AlgebraicGeometry in coding theory and cryptography
, 2009
Co-Authors: Harald Niederreiter, Chaoping Xing

Abstract:

This textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying Algebraic Geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and Algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like AlgebraicGeometry codes and elliptic-curve cryptosystems as well as material not treated by other books, including function-field codes, digital nets, code-based public-key cryptosystems, and frameproof codes. Combining a systematic development of theory with a broad selection of real-world applications, this is the most comprehensive yet accessible introduction to the field available.Introduces graduate students and advanced undergraduates to the foundations of Algebraic Geometry for applications to information theory Provides the first detailed discussion of the interplay between projective curves and Algebraic function fields over finite fields Includes applications to coding theory and cryptography Covers the latest advances in AlgebraicGeometry codes Features applications to cryptography not treated in other books

### Andrew J. Sommese – One of the best experts on this subject based on the ideXlab platform.

• ##### What is numerical AlgebraicGeometry
Journal of Symbolic Computation, 2017
Co-Authors: Jonathan D. Hauenstein, Andrew J. Sommese

Abstract:

The foundation of Algebraic Geometry is the solving of systems of polynomial equations. When the equations to be considered are defined over a subfield of the complex numbers, numerical methods can be used to perform Algebraic geometric computations forming the area of numerical Algebraic Geometry. This article provides a short introduction to numerical Algebraic Geometry with the subsequent articles in this special issue considering three current research topics: solving structured systems, certifying the results of numerical computations, and performing Algebraic computations numerically via Macaulay dual spaces.

• ##### Applying Numerical AlgebraicGeometry to Kinematics
21st Century Kinematics, 2013
Co-Authors: Charles W. Wampler, Andrew J. Sommese

Abstract:

Many problems from kinematics are questions about mappings between Algebraic spaces. This chapter presents a mathematical framework for such problems and discusses how numerical Algebraic Geometry, a computational approach based mainly on polynomial continuation, can be applied to solving them. Publicly available software for numerical Algebraic Geometry, such as the Bertini package, facilitates the solution of such problems, allowing kinematicians to solve with ease problems that were previously considered extremely difficult or intractable.

• ##### Numerical AlgebraicGeometry and Algebraic kinematics
Acta Numerica, 2011
Co-Authors: Charles W. Wampler, Andrew J. Sommese

Abstract:

In this article, the basic constructs of Algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this Algebraic framework, they can be attacked by tools from Algebraic Geometry. In particular, we review the techniques of numerical Algebraic Geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical Algebraic Geometry applies broadly to any system of polynomial equations, Algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.

### Charles W. Wampler – One of the best experts on this subject based on the ideXlab platform.

• ##### Applying Numerical AlgebraicGeometry to Kinematics
21st Century Kinematics, 2013
Co-Authors: Charles W. Wampler, Andrew J. Sommese

Abstract:

Many problems from kinematics are questions about mappings between Algebraic spaces. This chapter presents a mathematical framework for such problems and discusses how numerical Algebraic Geometry, a computational approach based mainly on polynomial continuation, can be applied to solving them. Publicly available software for numerical Algebraic Geometry, such as the Bertini package, facilitates the solution of such problems, allowing kinematicians to solve with ease problems that were previously considered extremely difficult or intractable.

• ##### Numerical AlgebraicGeometry and Algebraic kinematics
Acta Numerica, 2011
Co-Authors: Charles W. Wampler, Andrew J. Sommese

Abstract:

In this article, the basic constructs of Algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this Algebraic framework, they can be attacked by tools from Algebraic Geometry. In particular, we review the techniques of numerical Algebraic Geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical Algebraic Geometry applies broadly to any system of polynomial equations, Algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.

• ##### SNC – Numerical AlgebraicGeometry and kinematics
, 2007
Co-Authors: Charles W. Wampler

Abstract:

Numerical Algebraic Geometry uses numerical methods, principally numerical tracking of paths defined by polynomial homotopies, to find and manipulate Algebraic sets defined by systems of polynomial equations. Kinematics is the study of the geometrical aspects of mechanical motion. The kinematical problems arising in the analysis and design of most robots and mechanisms are essentially Algebraic, because these devices are well-modeled as rigid bodies in contact along Algebraic surfaces. In particular, the constraints imposed by the most common types of joints, such as simple hinges or ball-and-socket joints, are equivalent to containments of linear features (points, lines, and planes) that are maintained during rigid body motion of the parts. Kinematical studies have driven the development of numerical Algebraic Geometry and remain one of its most important application areas. Numerical Algebraic Geometry has proven to be particularly apt for the natural parameterizations presented by problems from kinematics. This extended abstract gives brief overviews of basic numerical Algebraic Geometry and kinematics.