155 research outputs found
Segre Class Computation and Practical Applications
Let be closed (possibly singular) subschemes of a smooth
projective toric variety . We show how to compute the Segre class
as a class in the Chow group of . Building on this, we give effective
methods to compute intersection products in projective varieties, to determine
algebraic multiplicity without working in local rings, and to test pairwise
containment of subvarieties of . Our methods may be implemented without
using Groebner bases; in particular any algorithm to compute the number of
solutions of a zero-dimensional polynomial system may be used
Computational Aspects of Retrieving a Representation of an Algebraic Geometry Code
ProducciĂłn CientĂficaCode-based cryptography is an interesting alternative to classic number-theoretic public key cryptosystem since it is conjectured to be secure against quantum computer attacks. Many families of codes have been proposed for these cryptosystems such as algebraic geometry codes. In [Designs, Codes and Cryptography, pages 1-16, 2012] -for so called very strong algebraic geometry codes , where is an algebraic curve over , is an -tuple of mutually distinct -rational points of and is a divisor of with disjoint support from --- it was shown that an equivalent representation can be found. The -tuple of points is obtained directly from a generator matrix of , where the columns are viewed as homogeneous coordinates of these points. The curve is given by , the homogeneous elements of degree of the vanishing ideal . Furthermore, it was shown that can be computed efficiently as the kernel of certain linear map. What was not shown was how to get the divisor and how to obtain efficiently an adequate decoding algorithm for the new representation. The main result of this paper is an efficient computational approach to the first problem, that is getting . The security status of the McEliece public key cryptosystem using algebraic geometry codes is still not completely settled and is left as an open problemThis research was partly supported by the Danish National Research Foundation and the National Science Foundation of China (Grant No.\ 11061130539) for the Danish-Chinese Center for Applications of Algebraic Geometry in Coding Theory and Cryptography and by Spanish grants MTM2007-64704, MTM2010-21580-C02-02 and MTM2012-36917-C03-03. Part of the research of the second author is also funded by the Vernon Wilson Endowed Chair at Eastern Kentucky University during his sabbatical leave
Gauge Backgrounds and Zero-Mode Counting in F-Theory
Computing the exact spectrum of charged massless matter is a crucial step
towards understanding the effective field theory describing F-theory vacua in
four dimensions. In this work we further develop a coherent framework to
determine the charged massless matter in F-theory compactified on elliptic
fourfolds, and demonstrate its application in a concrete example. The gauge
background is represented, via duality with M-theory, by algebraic cycles
modulo rational equivalence. Intersection theory within the Chow ring allows us
to extract coherent sheaves on the base of the elliptic fibration whose
cohomology groups encode the charged zero-mode spectrum. The dimensions of
these cohomology groups are computed with the help of modern techniques from
algebraic geometry, which we implement in the software gap. We exemplify this
approach in models with an Abelian and non-Abelian gauge group and observe
jumps in the exact massless spectrum as the complex structure moduli are
varied. An extended mathematical appendix gives a self-contained introduction
to the algebro-geometric concepts underlying our framework.Comment: 41 pages + extended appendice
An F4-Style Involutive Basis Algorithm
How to solve a linear equation system? The echelon form of this system will be obtained by Gaussian elimination then give us the solution. Similarly, Gröbner Basis is the “nice form” of nonlinear equation systems that can span all the polynomials in the given ideal [4]. So we can use Gröbner Basis to analyze the solution of a nonlinear equation system.
But how to compute a Gröbner Basis? There exist several ways to do it. Buchberger’s algorithm is the original method [2]. Gebauer-Möller algorithm [6] is a refined Buchberger’s algorithm. The F4 algorithm [5] uses matrix reduction to compute efficiently. Involutive Basis algorithm [8, 1, 12] is an effective method avoiding much ambiguity in the other algorithms.
In Chapters 1 and 2 we describe two well-known methods of computing Gröbner Basis called Buchberger’s and F4 algorithm. In Chapter 3 after presenting the definition of involutive division we give a detailed formulation of basic and improved Involutive Basis algorithm. We will see that there exists ambiguity both in Buchberger’s and F4 algorithm. But in the method of Involutive Basis Algorithm, the ambiguity for the choice of prolongation has been avoided. So in Chapter 4 we combine the F4 algorithm and Involutive Basis algorithm in order to obtain a new approach that can reduce polynomials faster as well as avoid ambiguity. The combined algorithm called F4-involutive is a partial result due to its efficiency. More work such as implementing Buchberger’s criteria would be done in the future
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