THE MAIN INVARIANT OF A DEFECTLESS POLYNOMIAL

In an earlier paper the authors calculated the main invariant of a tame polynomial over a valued field in terms of the simple invariants associated with a strict system of polynomial extensions that contained that polynomial. In this note we give upper and lower bounds in terms of such invariants for the main invariant of any defectless polynomial. We also determine precisely the polynomials for which the upper bound is the main invariant; this class strictly contains the set of tame polynomials. A class of examples with the same upper and with the same lower bound for the main invariant is given whose main invariants form a dense subset of the interval between the two bounds. A second class of polynomials is given whose strict systems have arbitrarily long length and whose main invariant is the lower bound. A basic tool is a formula for the main invariant which itself gives an algorithm for computing the main invariants of the polynomials in any strict system; in particular, simple formulas are given for the main invariants of some very special types of defectless polynomials including generalized Schönemann polynomials. The Krasner constants of defectless polynomials are also studied.