A Basic Result on the Theory of Subresultants

Given the polynomials f, g ∈ Z[x] the main result of our paper, Theorem 1, establishes a direct one-to-one correspondence between the modified Euclidean and Euclidean polynomial remainder sequences (prs’s) of f, g computed in Q[x], on one hand, and the subresultant prs of f, g computed by determinant evaluations in Z[x], on the other. An important consequence of our theorem is that the signs of Euclidean and modified Euclidean prs’s - computed either in Q[x] or in Z[x] - are uniquely determined by the corresponding signs of the subresultant prs’s. In this respect, all prs’s are uniquely "signed". Our result fills a gap in the theory of subresultant prs’s. In order to place Theorem 1 into its correct historical perspective we present a brief historical review of the subject and hint at certain aspects that need - according to our opinion - to be revised. ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2

Over-constrained Weierstrass iteration and the nearest consistent system

We propose a generalization of the Weierstrass iteration for over-constrained systems of equations and we prove that the proposed method is the Gauss-Newton iteration to find the nearest system which has at least $k$ common roots and which is obtained via a perturbation of prescribed structure. In the univariate case we show the connection of our method to the optimization problem formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate case we generalize the expressions of Karmarkar and Lakshman, and give explicitly several iteration functions to compute the optimum. The arithmetic complexity of the iterations is detailed

Sturm and Sylvester algorithms revisited via tridiagonal determinantal representations

International audienceFirst, we show that Sturm algorithm and Sylvester algorithm, which compute the number of real roots of a given univariate polynomial, lead to two dual tridiagonal determinantal representations of the polynomial. Next, we show that the number of real roots of a polynomial given by a tridiagonal determinantal representation is greater than the signature of this representation

On interpolation-based decoding of a class of maximum rank distance codes

In this paper we present an interpolation-based decoding algorithm to decode a family of maximum rank distance codes proposed recently by Trombetti and Zhou. We employ the properties of the Dickson matrix associated with a linearized polynomial with a given rank and the modified Berlekamp-Massey algorithm in decoding. When the rank of the error vector attains the unique decoding radius, the problem is converted to solving a quadratic polynomial, which ensures that the proposed decoding algorithm has polynomial-time complexity.acceptedVersio