Doru Ştefănescu

Estimates for Polynomial Roots

Abstract. Given a complex polynomial, we obtain lower bounds for the moduli of the roots outside the unit circle. Our main tool is the method of Dandelin–Graeffe, which can be used directly for polynomials with distinct absolute values of the roots. In the general case the arguments of the powers of the roots must be controlled, and we achieve this by two methods: a theorem of Dirichlet, and an argument using linear recurrent sequences.

Inequalities on upper bounds for real polynomial roots

In this paper we propose two methods for the computation of upper bounds of the real roots of univariate polynomials with real coefficients. Our results apply to polynomials having at least one negative coefficient. The upper bounds of the real roots are expressed as functions of the first positive coefficients and of the two largest absolute values of the negative ones.

Construction of Classes of Irreducible Bivariate Polynomials

We describe a method for constructing classes of bivariate polynomials which are irreducible over algebraically closed fields of characteristic zero. The constructions make use of some factorization conditions and apply to classes of polynomials that includes the generalized difference polynomials.

Improvements of Lagrange's bound for polynomial roots

Abstract An upper bound for the roots of X d + a 1 X d − 1 + ⋯ + a d − 1 X + a d is given by the sum of the largest two of the terms | a i | 1 / i . This bound by Lagrange has gained attention from different sides recently, while a succinct proof seems to be missing. We present a short, original proof of Lagranges bound. Our approach leads to some definite improvements. To benefit computationally from these improvements, we construct a modified Lagrange bound which at the same asymptotic computational complexity is at most 11 per cent from optimal for degrees d ≥ 16 .

Bounds for real roots and applications to orthogonal polynomials

We obtain new inequalities on the real roots of a univariate polynomial with real coefficients. Then we derive estimates for the largest positive root, which is a key step for real root isolation. We discuss the case of classic orthogonal polynomials. We also compute upper bounds for the roots of orthogonal polynomials using new inequalities derived from the differential equations satisfied by these polynomials. Our results are compared with those obtained by other methods.

Bounds for Heights of Integer Polynomial Factors

We describe new methods for the estimation of the bounds of the coefficients of proper divisors of integer polynomials in one variable. There exist classes of polynomials for which our estimates are better than those obtained using the polynomial measure or the 2-weighted norm.

Computational Aspects of a Bound of Lagrange

We consider the bound \(R+\rho \) of Lagrange and we obtain some improvements of it. We also discuss the efficiency of this bound of Lagrange and of its refinements.

Applications of the Newton Index to the Construction of Irreducible Polynomials

We use properties of the Newton index associated to a polynomial with coefficients in a discrete valuation domain for generating classes of irreducible polynomials. We obtain factorization properties similar to the case of bivariate polynomials and we give new applications to the construction of families of irreducible polynomials over various discrete valuation domains. The examples are obtained using the package gp-pari.