Of limit key polynomials

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 μ ≤ ν and abstract key polynomials for ν. Also, some results on invariants associated to limit key polynomials are obtained. In particular, if char.K/ D 0, we show that all the limit key polynomials of unbounded continuous families of augmentations have the numerical character equal to one.