We give model theoretic criteria for the existence of∃∀ and∀∃formulas in the ring language to define uniformly the valuation rings O of models (K,O) of an elementary theory Σ of henselian valued fields. As one of the applications we obtain the existence of an ∃∀-formula defining uniformly the valuation rings O of valued henselian fields (K,O) whose residue class field k is finite, pseudofinite,...

1. January 12th – Motivation (Bhargav Bhatt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. January 26th – Huber rings (Rankeya Datta). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

Classical Buchberger theory is generalized to a new family of rings. The family includes all subalgebras of the polynomial algebra in one variable. Some subalgebras of polynomial algebras in several variables are included. The new rings are integral domains and have a number of other properties in common with polynomial rings. The rings sit in a field F in appropriate position relative to a val...