Karatsuba and Toom-Cook are well-known methods used to efficiently multiply univariate polynomials and long integers. For multivariate polynomials, asymptotically good approaches like Kronecker’s trick combined with FFT become truly effective only when the degree is above some threshold. In this paper we analyze Karatsuba and some of Toom-Cook methods for multivariate polynomials, considering d...

QUAD is a provably secure stream cipher, whose security is based on the hardness assumption of solving multivariate quadratic polynomial systems over a finite field, which is known to be NP-complete. However, such provable security comes at a price, and QUAD is slower than most other stream ciphers that do not have security proofs. In this paper, we discuss two efficient parallelization techniq...

We find zero-free regions in the complex plane at large |q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) ZG(q,w) of a graph G with general complex edge weights w = {we}. This generalizes a result of Sokal [22] that applied only within the complex antiferromagnetic regime |1+we| ≤ 1. Our proof uses the polymer-gas representatio...

We find zero-free regions in the complex plane at large |q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) ZG(q,w) of a graph G with general complex edge weights w = {we}. This generalizes a result of Sokal [22] that applied only within the complex antiferromagnetic regime |1+we| ≤ 1. Our proof uses the polymer-gas representatio...

A novel semi-blind defocused image deconvolution technique is proposed, which is based on multivariate local polynomial regression and iterative Wiener filtering. In this technique, firstly a multivariate local polynomial regression model is trained in wavelet domain to estimate defocus parameter. After obtaining the point spread function (PSF ) parameter, iterative wiener filter is adopted to ...

The lecture addresses topics in multivariate approximation which have caught the author’s interest in the last ten years. These include: the approximation by functions with fewer variables, correct points for polynomial interpolation, the B(ernstein,-ézier, -arycentric)-form for polynomials and its use in understanding smooth piecewise polynomial (pp) functions, approximation order from spaces ...

A new algorithm is introduced which computes the multivariate leading coefficients of polynomial factors from their univariate images. This algorithm is incorporated into a sparse Hensel lifting scheme and only requires the factorization of a single univariate image. The algorithm also provides the content of the input polynomial in the main variable as a by-product. We show how we can take adv...

The asymmetric cipher protocol, based on decomposition problem in matrix semiring M over semiring of natural numbers N is presented. The security of presented cipher protocol is based on matrix decomposition problem (MDP), which is linked to the problem of solution of multivariate polynomial system of equations. Compromitation of proposed scheme relies on the solution of system of multivariate ...

In one-dimensional case, various important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz–Zygmund, Nikolskii, Schur, Remez, etc., have been proved under the doubling condition or the slightly stronger A∞ condition on the weights by Mastroianni and Totik in a recent paper [G. Mastroianni, V. Totik, Weighted polynomial inequalities with doubling and A∞ weights, Constr. Approx...

In this paper, we consider the problem of computing estimates of the domain-ofattraction for non-polynomial systems. A polynomial approximation technique, based on multivariate polynomial interpolation and error analysis for remaining functions, is applied to compute an uncertain polynomial system, whose set of trajectories contains that of the original non-polynomial system. Experiments on the...