On maximal immediate extensions of valued fields

We study the maximal immediate extensions of valued fields whose residue fields are perfect and whose value groups are divisible by the residue characteristic if it is positive. In the case where there is such an extension which has finite transcendence degree we derive strong properties of the field and the extension and show that the maximal immediate extension is unique up to isomorphism, although these fields need not be Kaplansky fields. If the maximal immediate extension is an algebraic extension, we show that it is equal to the perfect hull and the completion of the field.     

Isometric embeddings of finite fields

By regarding a finite field as a vector space over the prime field with a basis consisting of powers of an element, a Hamming distance is defined on the finite field with respect to the power basis. We consider the existence of isometric homomorphisms between such finite fields, and characterize the isometric embeddings for even characteristic by arithmetical conditions. Moreover, a canonical Hamming metric is defined in a certain infinite dimensional algebraic extension of a finite field.     

A ruled residue theorem for function fields of conics

The ruled residue theorem characterises residue field extensions for valuations on a rational function field. Under the assumption that the characteristic of the residue field is different from 2 this theorem is extended here to function fields of conics. The main result is that there is at most one extension of a valuation on the base field to the function field of a conic for which the residue field extension is transcendental but not ruled. Furthermore the situation when this valuation is present is characterised.     

Energy of Radial Vector Fields on Compact Rank One Symmetric Spaces

We consider the energy (or the total bending) of unit vector fields oncompact Riemannian manifolds for which the set of its singularitiesconsists of a finite number of isolated points and a finite number ofpairwise disjoint closed submanifolds. We determine lower bounds for theenergy of such vector fields on general compact Riemannian manifolds andin particular on compact rank one symmetric spaces. For this last classof spaces, we compute explicit expressions for the total bending whenthe unit vector field is the gradient field of the distance function toa point or to special totally geodesic submanifolds (i.e., for radialunit vector fields around this point or these submanifolds).     

正域为无限区间的初等关联函数构造

定义在有限区间上的初等关联函数在可拓论中发挥了重要作用.然而,对于一些量值表现为越大越好或越小越优的特征,如何描述事物的可拓域未有探讨.构造了正域为无限区间的初等关联函数并得到若干性质.模拟案例说明该函数的实用性.该关联函数符合数据处理的无量纲化和一致化要求,能消除数量量级和量纲的影响;使基元特征量值表现的四种类型都可以用初等关联函数表示,拓宽了关联函数描述事物符合要求程度的范围;正域为有限区间且在端点取最大值的初等关联函数是该函数的特例.     

Note on a Variation of the Schröder-Bernstein Problem for Fields

In this note we study fields F with the property that the simple transcendental extension F(u) of F is isomorphic to some subfield of F but not isomorphic to F. Such a field provides one type of solution of the Schröder-Bernstein problem for fields.     

Vanishing Ideals of Projective Spaces over Finite Fields and a Projective Footprint Bound

We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr¨obner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre's inequality for the number of points of projective hypersurfaces over finite fields is included.     

有限域上遍历矩阵的特性研究

对有限域上遍历矩阵的性质进行了分析,给出了有限域上遍历矩阵的计数定理,并对遍历矩阵序对(A,B)关于矩阵M的双侧幂乘集〈A〉M〈B〉的秩及基数进行了全面分析.给出了R_k(A,B)集的构成及其基数的有关定理,所得到的结论对利用遍历矩阵实现有关的公钥密码具有理论上的指导意义.     

Analysis and Prbability over Infinite Extensions of a Local Field

We consider an infinite extension K of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. K is equipped with an inductive limit topology; its conjugate K; is a completion of K with respect to a topology given by certain explicitly written semi-norms. We construct and study a Gaussian measure, a Fourier transform, a fractional differentiation operator and a cadlag Markov process on K. If we deal with Galois extensions then all these objects are Galois-invariant.     

Cosine transforms over fields of characteristic 2

In this paper, we introduce cosine transforms over fields of characteristic 2. Our approach complements previous definitions of finite field trigonometric transforms, which only hold for fields whose characteristic is an odd prime. Besides introducing some new concepts related to trigonometry in finite fields, we discuss the eigenstructure and other important properties of the proposed transforms.     

Efficient Software-Implementation of Finite Fields with Applications to Cryptography

In this work, we present a survey of efficient techniques for software implementation of finite field arithmetic especially suitable for cryptographic applications. We discuss different algorithms for three types of finite fields and their special versions popularly used in cryptography: Binary fields, prime fields and extension fields. Implementation details of the algorithms for field addition/subtraction, field multiplication, field reduction and field inversion for each of these fields are discussed in detail. The efficiency of these different algorithms depends largely on the underlying micro-processor architecture. Therefore, a careful choice of the appropriate set of algorithms has to be made for a software implementation depending on the performance requirements and available resources.     

A Note on Class Field Theory for Two-Dimensional Local Rings

The class field theory for the fraction field of a two-dimensional complete normal local ring with finite residue field is established by S. Saito. In this paper, we investigate the index of the norm group in the K ₂-idele class group for a finite Abelian extension of such fields and deduce that the existence theorem does not hold for almost fields in this case.     

Geodesic vector fields and Eikonal equation on a Riemannian manifold

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.     

Generalized Gabidulin codes over fields of any characteristic

We generalize Gabidulin codes to a large family of fields, non necessarily finite, possibly with characteristic zero. We consider a general field extension and any automorphism in the Galois group of the extension. This setting enables one to give several definitions of metrics related to the rank-metric, yet potentially different. We provide sufficient conditions on the given automorphism to ensure that the associated rank metrics are indeed all equal and proper, in coherence with the usual definition from linearized polynomials over finite fields. Under these conditions, we generalize the notion of Gabidulin codes. We also present an algorithm for decoding errors and erasures, whose complexity is given in terms of arithmetic operations. Over infinite fields the notion of code alphabet is essential, and more issues appear that in the finite field case. We first focus on codes over integer rings and study their associated decoding problem. But even if the code alphabet is small, we have to deal with the growth of intermediate values. A classical solution to this problem is to perform the computations modulo a prime ideal. For this, we need study the reduction of generalized Gabidulin codes modulo an ideal. We show that the codes obtained by reduction are the classical Gabidulin codes over finite fields. As a consequence, under some conditions, decoding generalized Gabidulin codes over integer rings can be reduced to decoding Gabidulin codes over a finite field.     

On a problem of Arnold on uniform distribution

In this note we consider some quantitative versions of conjectures made by Arnold related to Galois dynamics in finite fields. We refine some results by Shparlinski using exponential sum results.     

M-P invertible matrices and unitary groups over Fq2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field Fq2, which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for an m×n matrix A over Fq2 having an M-P inverse are obtained, which make clear the set of m×n matrices over Fq2 having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

代数扩张域上的多项式分解在密码学中的应用

用有理数域或特征p的素域上的有n个独立变量的有理函数域的有限代数扩张域上的多项式的不可约分解,建议了一类密码系统.     

Sur le nombre de points rationnels des variétés abéliennes et des Jacobiennes sur les corps finis

We give upper and lower bounds for the number of points on abelian and Jacobian varieties over finite fields. We also determine the values for the maximum and minimum number of points on Jacobian surfaces on a given finite field.     

Generalized trigonometry and Chebyshev functions in finite fields

Trigonometry in finite fields was introduced by de Souza et al. and further developed by Lima and Panario and others, giving functions with many properties similar to trigonometric functions over the reals. Those explorations used a degree-2 extension of a base field. While this corresponds most closely to trigonometry over the reals, in finite fields we can have extensions of other degrees. In this paper we generalize the definitions of trigonometric functions and their related Chebyshev polynomials to arbitrary degrees and explore their properties. Many familiar results carry over into the generalized setting.     

Self-dual normal integral bases for infinite unramified extensions

We prove a generalization to infinite Galois extensions of local fields, of a classical result by Noether on the existence of normal integral bases for finite tamely ramified Galois extensions. We also prove a self-dual normal integral basis theorem for infinite unramified Galois field extensions of local fields with finite residue fields of characteristic different from 2. This generalizes a result by Fainsilber for the finite case. To do this, we obtain an injectivity result concerning pointed cohomology sets, defined by inverse limits of norm-one groups of free finite-dimensional algebras with involution over complete discrete valuation rings.