We define and study a notion of ring of formal power series with exponents in a cyclically ordered group. Such a ring is a quotient of various subrings of classical formal power series rings. It carries a two variable valuation function. In the particular case where the cyclically ordered group is actually totally ordered, our notion of formal power series is equivalent to the classical one in ...

We present a set of generators of the full annihilator ideal for the Witt ring of an arbitrary field of characteristic unequal to two satisfying a non-vanishing condition on the powers of the fundamental ideal in the torsion part of the Witt ring. This settles a conjecture of Ongenae and Van Geel. This result could only be proved by first obtaining a new lower bound on the 2-adic valuation of S...

For a field F of characteristic different from 2, containing a square root of -1, endowed with an F-compatible valuation v such that the residue field has at most two square classes, we use a combinatorial analogue of the Witt ring of F to prove that an anisotropic quadratic form over F with even dimension d, trivial discriminant and Hasse-Witt invariant can be written in the Witt ring as the s...

Given a well-ordered semi-group Γ with a minimal system of generators of ordinal type at most ωn 1 and of rational rank r, which satisfies a positivity and increasing condition, we construct a zero-dimensional valuation centered on the ring of polynomials with r variables such that the semigroup of the values of the polynomial ring is equal to Γ. The construction uses a generalization of Favre ...

We compute explicitly the normal zeta functions of the Heisenberg groups H(R), where R is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form H(OK), where OK is the ring of integers of an arbitrary number field K, at the rational primes which are non-split in K. We show that these local z...

It is shown that every dp-minimal integral domain R a local ring and for non-maximal prime ideal ℘ of R, the localization R℘ valuation ℘R℘ = ℘. Furthermore, if only its residue field infinite or finite maximal principal.

In the title compound, C20H22N2O2, the central pyridine ring forms a dihedral angle of 76.32 (8)° with the pseudo-axial benzene ring. The cyclo-octane ring adopts a twisted boat chair conformation. In the crystal, weak inter-molecular C-H⋯π inter-actions between inversion-related mol-ecules result in the formation of linear double chains along the b-axis direction.

Let p be a prime number and let K be a finite extension of Qp. Let R be the valuation ring of K, P the maximal ideal of R, and K̄ = R/P the residue field of K. Let q denote the cardinality of K̄, so K̄ ≃ Fq. For z in K, let ord z denote the valuation of z, and set |z| = q . Let f be a non constant element of K[x1, . . . , xm]. The p-adic Igusa local zeta function Z(s) associated to f (relative to ...

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,...

1.1 The lifting problem The problem we are concerned with in our lectures and which we shall refer to as the lifting problem was originally formulated by Frans Oort in [17]. To state it, we fix an algebraically closed field κ of positive characteristic p. Let W (κ) be the ring of Witt vectors over κ. Throughout our notes, o will denote a finite local ring extension of W (κ) and k = Frac(o) the ...