Of limit key polynomials

Abstract

Let ν be a valuation of arbitrary rank on the polynomial ring $K[x]$ with coefficients in a field K. We prove comparison theorems between MacLane–Vaquié key polynomials for valuations $\mathrm{\mu }\le \mathrm{\nu }$ and abstract key polynomials for ν. Also, some results on invariants associated to limit key polynomials are obtained. In particular, if $char(K)=0$, we show that all the limit key polynomials of unbounded continuous families of augmentations have the numerical character equal to one.