Let $\iota:K\hookrightarrow L\cong K(x)$ be a simple transcendental extension of valued fields, where $K$ is equipped with valuation $\nu$ rank 1. That is, we assume given 1 and its $\nu'$ to $L$. $(R_\nu,M_\nu,k_\nu)$ denote the ring $\nu$. The purpose this paper present refined version MacLane's theory key polynomials, similar those considered by M. Vaqui\'e, reminiscent related objects studi...

We prove a tight bound on the number of realized 0=1 patterns (or equivalently Vapnik–Chervonenkis codensity) definable families in models theory algebraically closed valued fields with non-archimedean valuation. Our result improves best known this direction proved by Aschenbrenner, Dolich, Haskell, Macpherson and Starchenko, who weaker restricted case where characteristics field $K$ its residu...