Itoh-Tsujii Inversion in Standard Basis and Its Application in Cryptography and Codes

This contribution is concerned with a generalization of Itoh and Tsujii's algorithm for inversion in extension fields . Unlike the original algorithm, the method introduced here uses a standard (or polynomial) basis representation. The inversion method is generalized for standard basis representation and relevant complexity expressions are established, consisting of the number of extension field multiplications and exponentiations. As the main contribution, for three important classes of fields we show that the Frobenius map can be explored to perform the exponentiations required for the inversion algorithm efficiently. As an important consequence, Itoh and Tsujii's inversion method shows almost the same practical complexity for standard basis as for normal basis representation for the field classes considered.     

On four-sheeted polynomial mappings of ℂ2. I. The case of an irreducible ramification curve

The paper is devoted to the Jacobian Conjecture: a polynomial mappingf²² with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a nonzero constant such that after the resolution of the indeterminacy points at infinity there is only one added curve whose image is not a point and does not belong to infinity.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 847–862, December, 1998.The authors are grateful to A. G. Vitushkin and P. Cassou-Nogues for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01218. The work of the second author was done under the financial support of DGICYT (Spain), grant No. SAB95-0502.     

Characteristic and Ehrhart Polynomials

Let A be a subspace arrangement and let (A,t) be the characteristic polynomial of its intersection lattice L( A). We show that if the subspaces in A are taken from , where is the type B Weyl arrangement, then (A,t) counts a certain set of lattice points. One can use this result to study the partial factorization of (A,t) over the integers and the coefficients of its expansion in various bases for the polynomial ring R[t]. Next we prove that the characteristic polynomial of any Weyl hyperplane arrangement can be expressed in terms of an Ehrhart quasi-polynomial for its affine Weyl chamber. Note that our first result deals with all subspace arrangements embedded in while the second deals with all finite Weyl groups but only their hyperplane arrangements.     

Extendibility of homogeneous polynomials on Banach spaces

We study the -homogeneous polynomials on a Banach space that can be extended to any space containing . We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible -homogeneous polynomials on and we characterize the extendible 2-homogeneous polynomials on when is a Hilbert space, an -space or an -space.

     

Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus

We consider dynamical systems with two degrees of freedom whose configuration space is a torus and which admit first integrals polynomial in velocity. We obtain constructive criteria for the existence of conditional linear and quadratic integrals on the two-dimensional torus. Moreover, we show that under some additional conditions the degree of an irreducible integral of the geodesic flow on the torus does not exceed 2.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 37–44, July, 1998.The author wishes to express his thanks to V. V. Kozlov for his interest and his help in this work.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00747.     

Use of complex analysis for deriving lower bounds for trigonometric polynomials

It is shown that for any distinct natural numbersk ₁,...,k _n and arbitrary real numbersa ₁,...,a _n the following inequality holds:

A Fully Polynomial Approximation Scheme for Minimizing Makespan of Deteriorating Jobs

A fully polynomial approximation scheme for the problem of scheduling n deteriorating jobs on a single machine to minimize makespan is presented. Each algorithm of the scheme runs in O(n ⁵ L ⁴³) time, where L is the number of bits in the binary encoding of the largest numerical parameter in the input, and is required relative error. The idea behind the scheme is rather general and it can be used to develop fully polynomial approximation schemes for other combinatorial optimization problems. Main feature of the scheme is that it does not require any prior knowledge of lower and/or upper bounds on the value of optimal solutions.     

On the calculation of the acyclic polynomial

Graphical methods are developed for recursive evaluation of the acyclic polynomial. Analytical formulas of the acyclic polynomials for several specific series of graphs are given. Mathematical properties of the derivatives of the acyclic polynomial are given.     

Minimax estimation of a multivariate normal mean under polynomial loss

Let X be an observation from a p-variate (p ≥ 3) normal random vector with unknown mean vector θ and known covariance matrix

. The problem of improving upon the usual estimator of θ, δ⁰(X) = X, is considered. An approach is developed which can lead to improved estimators, δ, for loss functions which are polynomials in the coordinates of (δ ? θ). As an example of this approach, the loss L(δ, θ) = |δ ? θ|⁴ is considered, and estimators are developed which are significantly better than δ⁰. When

is the identity matrix, these estimators are of the form $\text{δ(X) = (1 ? (}\text{b}\text{(d + |X|}{}^2\text{)}\text{))X}$.     

The Clar covering polynomial of hexagonal systems Ⅱ.An application to resonance energy of condensed aromatic hydrocarbons

The Clar covering polynomial of hexagonal systems is a recently proposed1,2 concept which contains much more topological properties of condensed aromatic hydrocarbons, such as Kekule structure count, Clar number, first Herndon number, etc. It is shown that this polynomial can be used for calculating the resonance energy of condensed aromatic hydrocarbons with better accuracy.