Polynomials with complex coefficients: size of the factors, repartition of the zeros

Abstract

We relate the size of the factors of a polynomial with the repartition of its zeroz. First, we show that a polynomial with zeros on the unit circle always has a factor which is exponentially large. Then we give a symbolic formula, valid in the distribution sense, which allows one to reconstruct a polynomial from the repartition function of its zeros. From this formula we deduce a reciprocal to a well-known result of Erdos and Turan. We deal here with polynomials with complex coefficients, normalized with leading coefficient 1. We write such a polynomial under the form

A factor of P is a polynomial Q=II_j(z-z_j), where ,J is any subset of {1,…,n}. We are interested in relating the size of coefficients in the factors of P and the repartition of the zeors of P. Upper bounds for the size of the factors were given by the author in [1] and [2]; we deal here with lower bounds. In the first part of the present paper, using a result of Erdos-Turan [3], we show that any polynomial with zeros on the unit circle has a factor whic is exponentially large. In the second part, we give a symbolic formula, valid in the distribution sense, which allows to reconstruct the polynomial from the repartition function of its zeros.

Keywords:

• Zeros of polynomials

^*Supported in part by Contract 89/1377, Ministry of Defecse, D.G.A./D.R.E.T.-France

^*Supported in part by Contract 89/1377, Ministry of Defecse, D.G.A./D.R.E.T.-France

Notes

^*Supported in part by Contract 89/1377, Ministry of Defecse, D.G.A./D.R.E.T.-France