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Let P : C n → C be an m -homogeneous polynomial given by P ( x ) = ∑ 1 ≤ j 1 ≤ … ≤ j m ≤ n c j 1 … j m x j 1 … x j m . Defant and Schlüters defined a non-symmetric associated m -form L P : ( C n ) m → C by L P ( x ( 1 ) , … , x ( m ) ) = ∑ 1 ≤ j 1 ≤ … ≤ j m ≤ n c j 1 … j m x j 1 ( 1 ) … x j m ( m ) . They estimated the norm of L P on ( C n , ‖ ⋅ ‖ ) m by the norm of P on ( C n , ‖ ⋅ ‖ ) times a ( c log ⁡ n ) m 2 factor for every 1-unconditional norm ‖ ⋅ ‖ on C n . A symmetrization procedure based on a card-shuffling algorithm which (together with Defant and Schlüters' argument) brings the constant term down to ( c m log ⁡ n ) m − 1 is provided. Regarding the lower bound, it is shown that the optimal constant is bigger than ( c log ⁡ n ) m / 2 when n ≫ m . Finally, the case of ℓ p -norms ‖ ⋅ ‖ p with 1 ≤ p < 2 is addressed. [ABSTRACT FROM AUTHOR]

This paper deals with the evaluation of the fractional integrals involving Saigo-Maeda operators of the product of the general class of polynomials and the H-function containing the factor xλ (xk + ck)-p in its argument. Some interesting special cases are derived. The results given by Saigo and Raina [6] and Chaurasia and Gupta [2] follow as special cases of the results proved in this paper. [ABSTRACT FROM AUTHOR]

The analysis of any Hessenberg matrix as a product of a companion matrix and a triangular matrix is presented in this paper. The factors are explicitly given on terms of the entries of the Hessenberg matrix and, further, various factorizations are undertaken. Applications to comrade and confederate matrices, as well as for the corresponding block cases, are included. [ABSTRACT FROM AUTHOR]

A new strategy for updating preconditioners by polynomial interpolation of factors of approximate inverse factorizations is proposed here. The computational cost per iteration is linear in the number of degree of freedom, the same order of most of the strategies for updating an incomplete factorization proposed in the last decade. The effectiveness of the technique is confirmed by some experiments. [ABSTRACT FROM AUTHOR]

In this paper, we present a new method for computing bounded-degree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in lacunary representation and a degree bound d and computes the irreducible factors of degree at most d of f in time polynomial in the lacunary size of f and in d . Our algorithm, which is valid for any field of zero characteristic, is based on a new gap theorem that enables reducing the problem to several instances of (a) the univariate case and (b) low-degree multivariate factorization. The reduction algorithms we propose are elementary in that they only manipulate the exponent vectors of the input polynomial. The proof of correctness and the complexity bounds rely on the Newton polytope of the polynomial, where the underlying valued field consists of Puiseux series in a single variable. [ABSTRACT FROM AUTHOR]

We introduce several techniques which allow to simplify the expression of the cofactor of Darboux polynomials of polynomial differential systems in $\mathbb {R}^{n}$. We apply these techniques to some well-known systems when n=2,3,4. We also propose a general method for computing Darboux polynomials in the plane. As an application we prove that a family of potential systems, that includes the van der Pol one, has no Darboux polynomials, giving in particular a new simple proof that the van der Pol limit cycle is not algebraic. [ABSTRACT FROM AUTHOR]

Given a polynomial f (z)= zd + c ∈ K[z] over a global field K and α0 ∈ K, we study the density of prime ideals of K dividing at least one element of the orbit of α0 under f . We show that for many choices of d and c this density is zero for all α0, assuming K contains a primitive dth root of unity. The proof relies on several new results, including some giving criteria to ensure the number of irreducible factors of the nth iterate of f remains bounded as n grows, and others on the ramification above certain primes in iterated extensions. Together these allow for nearly complete information when K is a global function field or when K =Q(ξd). [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present a straightforward proof of the well-known fact that a positive semi-definite polynomial matrix can be written as the product of a polynomial matrix and its associated (Hermitian) transposed. For polynomial matrices with complex coefficients both factors also have complex coefficients and can be taken of the same size as the original matrix. For polynomial matrices with real coefficients both factors may have real coefficients, but then the left factor may need twice as many columns as the original matrix, and analogously for the right factor. [Copyright &y& Elsevier]

A generalization of the conformal Galilei algebra with Levi subalgebra isomorphic to is introduced and a virtual copy of the latter in the enveloping algebra of the extension is constructed. Explicit expressions for the Casimir operators are obtained from the determinant of polynomial matrices. For the central factor algebra , an exact formula giving the number of invariants is obtained and a procedure to compute invariant functions that do not depend on the variables of the Levi subalgebra is developed. It is further shown that such solutions determine complete sets of invariants provided that the relation is satisfied. [ABSTRACT FROM AUTHOR]