The paper provides a system theoretic characterisation of the least common multiple (LCM) m(s) of a given set of polynomials P which leads to an efficient numerical procedure for the computation of LCM that avoids root finding and use of greatest common divisor (GCD) procedures. The procedure that is presented also leads to the computation of the associated set of multipliers of P with respect ...

We propose a geometric method to measure the wild ramification of a smooth étale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of characteristic p > 0 with arbitrary residue field. We also define the characteristic cycle of an l-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent b...

Abstract. We give two new expressions of subresultants, nested subresultant and reduced nested subresultant, for the recursive polynomial remainder sequence (PRS) which has been introduced by the author. The reduced nested subresultant reduces the size of the subresultant matrix drastically compared with the recursive subresultant proposed by the authors before, hence it is much more useful for...

Recently, Brenner et al. proposed a symmetric somewhat homomorphic encryption scheme and applied it to solve some practical problems, such as the Millionaires’ problem, which only need to evaluate circuits of limited depth. It is claimed that the security of their scheme is built on the hardness of integer factorization. In this paper, we use the Euclidean Greatest Common Divisor (GCD) algorith...

This lecture will show by example some of the problems that occur when the roots of a polynomial are computed using a standard polynomial root solver. In particular, polynomials of high degree with a large number of multiple roots will be considered, and it will be shown that even roundoff error due to floating point arithmetic, in the absence of data errors, is sufficient to cause totally inco...

We present new algorithms using structured matrix methods for manipulating polynomi-als expressed in generalized form, that is, relative to a polynomial basis fp i (x)g satisfying a three-term recurrence relation. This includes computing the Euclidean remainders as well as the greatest common divisor of two polynomials u(x) and v(x) of degrees less than n. The computational schemes involve solv...

In this paper, we develop a fast structured total least squares (STLS) algorithm for computing an approximate greatest common divisor (GCD) of two univariate polynomials. By exploiting the displacement structure of the Sylvester matrix and applying the generalized Schur algorithm, each single iteration of the proposed algorithm has quadratic computational complexity in the degrees of the given ...

Let E / Q E/\mathbb {Q} </mml:sema...