From an Elementary Result on Real Closed Fields to Positive Operators on Ordered Fields

Abstract

Let K be a (commutative totally) ordered field, let K[X ₁,…, X _n ] be the K-vector space of the polynomials with n variables.

An operator T (i.e., an endomorphism of K[X ₁,…,X _n ] into itself) is said to be “positive” if the image of every positive polynomial is a positive polynomial, where a positive polynomial is a polynomial which takes only non-negative values. First we prove that in ℝ[X], the sum of the derivatives of a positive polynomial is a positive polynomial too. Then we give what we believe to be a good framework to prove that this result remains true for every ordered field and we propose generalizations.

Notes

Communicated by A. Prestel.