We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.

We consider three monads on $\mathsf{Top}$ , the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second one is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads $V \to H$ . In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. We show that V can be restricted to a submonad of $\tau$ -smooth probability measures on $\mathsf{Top}$ . By composing these morphisms of monads, we obtain that taking the supports of $\tau$ -smooth probability measures is also a morphism of monads.

We show that over any field $F$ of characteristic 2 and 2-rank $n$, there exist $2^{n}$ bilinear $n$-fold Pfister forms that have no slot in common. This answers a question of Becher [‘Triple linkage’, Ann.$K$-Theory, to appear] in the negative. We provide an analogous result also for quadratic Pfister forms.

Li introduced the normalized volume of a valuation due to its relation to K-semistability. He conjectured that over a Kawamata log terminal (klt) singularity there exists a valuation with smallest normalized volume. We prove this conjecture and give an explicit example to show that such a valuation need not be divisorial.

We generalize the $\mathbb{Z}/p$metabelian birational $p$-adic section conjecture for curves, as introduced and proved in Pop [On the birational$p$-adic section conjecture, Compos. Math. 146 (2010), 621–637], to all complete smooth varieties, provided $p>2$. The condition $p>2$ seems to be of technical nature only, and might be removable.

We study the question of which Henselian fields admit definable Henselian valuations (with or without parameters). We show that every field that admits a Henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) Henselian valuation. In equicharacteristic 0, we give a complete characterization of Henselian fields admitting a parameter-definable (non-trivial) Henselian valuation. We also obtain partial characterization results of fields admitting -definable (non-trivial) Henselian valuations. We then draw some Galois-theoretic conclusions from our results.

In this note we investigate the question when a henselian valued field carries a nontrivial ∅-definable henselian valuation (in the language of rings). This is clearly not possible when the field is either separably or real closed, and, by the work of Prestel and Ziegler, there are further examples of henselian valued fields which do not admit a ∅-definable nontrivial henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definition. In particular, we show that a henselian valued field admits a nontrivial henselian ∅-definable valuation when the residue field is separably closed or sufficiently nonhenselian, or when the absolute Galois group of the (residue) field is nonuniversal.

We show that the valuation ring $F_q [[t]]$ in the local field $F_q \left( t \right)$ is existentially definable in the language of rings with no parameters. The method is to use the definition of the henselian topology following the work of Prestel-Ziegler to give an ∃-$F_q $-definable bounded neighbouhood of 0. Then we “tweak” this set by subtracting, taking roots, and applying Hensel’s Lemma in order to find an ∃-$F_q $-definable subset of $F_q [[t]]$ which contains $tF_q [[t]]$. Finally, we use the fact that $F_q $ is defined by the formula $x^q - x = 0$ to extend the definition to the whole of $F_q [[t]]$ and to rid the definition of parameters.

Several extensions of the theorem are obtained, notably an ∃-∅-definition of the valuation ring of a nontrivial valuation with divisible value group.

Nous considérons une prise de contrôle dans laquelle les motivations des repreneurs potentiels sont multiples et étudions I’impact du degré d’affiliation de leurs évaluations sur la stratégie optimale. Nous montrons que cet impact dépend du rapport des dotations initiales. Nous trouvons que lorsque les dotations initiales sont symétriques, la stratégie optimale des enchérisseurs est indépendante du degré d’affiliation de leurs évaluations. En revanche, lorsque les dotations initiales sont asymétriques, cet impact dépend du niveau du signal privé de l’enchérisseur. Si les acqueréurs potentiels surenchérissent en présence de dotations initiales, l’étendue de la surenchére décroît avec l’augmentation du degré d’affiliation lorsque le signal privé est inférieur un certain seuil. L’agressivité des enchérisseurs profite à la firme cible dont le revenu espéré augmente avec le degré d’affiliation des évaluations.

The spaces of Sp(n)-, Sp(n) · U(1)- and Sp(n) · Sp(1)-invariant, translation-invariant, continuous convex valuations on the quaternionic vector space ℍn are studied. Combinatorial dimension formulae involving Young diagrams and Schur polynomials are proved.

Klep and Velušček generalized the Krull–Baer theorem for higher level preorderings to the non-commutative setting. A $n$-real valuation $v$ on a skew field $D$ induces a group homomorphism $\overline{v}$. A section of $\overline{v}$ is a crucial ingredient of the construction of a complete preordering on the base field $D$ such that its projection on the residue skew field ${{k}_{v}}$ equals the given level 1 ordering on ${{k}_{v}}$. In the article we give a proof of the existence of the section of $\overline{v}$, which was left as an open problem by Klep and Velušček, and thus complete the generalization of the Krull–Baer theorem for preorderings.

In this article we introduce and prove a ℤ/p meta-abelian form of the birationalp-adic section conjecture for curves. This is a much stronger result than the usual p-adic birational section conjecture for curves, and makes an effective p-adic section conjecture for curves quite plausible.

We introduce a valuation-theoretic approach to the problem of semistable reduction (i.e. existence of logarithmic extensions on suitable covers) of overconvergent isocrystals with Frobenius structure. The key tool is the quasicompactness of the Riemann–Zariski space associated to the function field of a variety. We also make some initial reductions, which allow attention to be focused on valuations of height 1 and transcendence degree 0.

We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (i.e., the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimprič regarding $*$-orderability of quantum groups.