Genetics of polynomials over local fields

Abstract

Let $(K,v)$ be a discrete valued field with valuation ring $\oo$, and let $\oo_v$ be the completion of $\oo$ with respect to the $v$-adic topology. In this paper we discuss the advantages of manipulating polynomials in $\oo_v[x]$ in a computer by means of OM representations of prime (monic and irreducible) polynomials. An OM representation supports discrete data characterizing the Okutsu equivalence class of the prime polynomial. These discrete parameters are a kind of DNA sequence common to all individuals in the same Okutsu class, and they contain relevant arithmetic information about the polynomial and the extension of $K_v$ that it determines.