Newton symmetric functions and the arithmetic of algebraically closed fields

Abstract

In this paper we analyze some points regarding computations on algebraically closed fields.

We give an explicit formula for the computation of a polynomial whose roots are the sum (resp. the product) of the roots of two given polynomials. This formula is obtained considering a transformation sending a polynomial in the sequence of its Newton symmetric functions, and allows to obtain a better bound for the complexity of the computation of the above polynomial than the bound that can be obtained with the resultant formula used e.g. by Loos.

We show how to represent elements in the algebraic closure of a field of finite transcendency with a choice of algebraically independent complex numbers; we show how a careful choice can give a good estimate for the root separation of the polynomials involved, and this in turn may be used to bound the complexity of the operations in the corresponding representation of the algebraically closed field.

We give an algorithm for absolute factorization which generalizes, with a much simpler proof, an absolute irreducibility test of Kaltofen.

Research with the contribution of Ministero della Pubblica Istruzione