313,961 research outputs found

### Quasi-Valuations Extending a Valuation

Suppose $F$ is a field with valuation $v$ and valuation ring $O_{v}$, $E$ is
a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We
study quasi-valuations on $E$ that extend $v$; in particular, their
corresponding rings and their prime spectrums. We prove that these ring
extensions satisfy INC (incomparability), LO (lying over), and GD (going down)
over $O_{v}$; in particular, they have the same Krull Dimension. We also prove
that every such quasi-valuation is dominated by some valuation extending $v$.
Under the assumption that the value monoid of the quasi-valuation is a group
we prove that these ring extensions satisfy GU (going up) over $O_{v}$, and a
bound on the size of the prime spectrum is given. In addition, a 1:1
correspondence is obtained between exponential quasi-valuations and integrally
closed quasi-valuation rings.
Given $R$, an algebra over $O_{v}$, we construct a quasi-valuation on $R$; we
also construct a quasi-valuation on $R \otimes_{O_{v}} F$ which helps us prove
our main Theorem. The main Theorem states that if $R \subseteq E$ satisfies $R
\cap F=O_{v}$ and $E$ is the field of fractions of $R$, then $R$ and $v$ induce
a quasi-valuation $w$ on $E$ such that $R=O_{w}$ and $w$ extends $v$; thus $R$
satisfies the properties of a quasi-valuation ring.Comment: 51 page

### The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences

Let $V$ be a valuation domain with quotient field $K$. Given a
pseudo-convergent sequence $E$ in $K$, we study two constructions associating
to $E$ a valuation domain of $K(X)$ lying over $V$, especially when $V$ has
rank one. The first one has been introduced by Ostrowski, the second one more
recently by Loper and Werner. We describe the main properties of these
valuation domains, and we give a notion of equivalence on the set of
pseudo-convergent sequences of $K$ characterizing when the associated valuation
domains are equal. Then, we analyze the topological properties of the
Zariski-Riemann spaces formed by these valuation domains.Comment: any comment is welcome! Trans. Amer. Math. Soc. 373 (2020), no. 11,
7959-799

### Valuation equilibrium

We introduce a new solution concept for games in extensive form with perfect information, valuation equilibrium, which is based on a partition of each player's moves into similarity classes. A valuation of a player'is a real-valued function on the set of her similarity classes. In this equilibrium each player's strategy is optimal in the sense that at each of her nodes, a player chooses a move that belongs to a class with maximum valuation. The valuation of each player is consistent with the strategy profile in the sense that the valuation of a similarity class is the player's expected payoff, given that the path (induced by the strategy profile) intersects the similarity class. The solution concept is applied to decision problems and multi-player extensive form games. It is contrasted with existing solution concepts. The valuation approach is next applied to stopping games, in which non-terminal moves form a single similarity class, and we note that the behaviors obtained echo some biases observed experimentally. Finally, we tentatively suggest a way of endogenizing the similarity partitions in which moves are categorized according to how well they perform relative to the expected equilibrium value, interpreted as the aspiration level

### Algebraic matroids and Frobenius flocks

We show that each algebraic representation of a matroid $M$ in positive
characteristic determines a matroid valuation of $M$, which we have named the
{\em Lindstr\"om valuation}. If this valuation is trivial, then a linear
representation of $M$ in characteristic $p$ can be derived from the algebraic
representation. Thus, so-called rigid matroids, which only admit trivial
valuations, are algebraic in positive characteristic $p$ if and only if they
are linear in characteristic $p$.
To construct the Lindstr\"om valuation, we introduce new matroid
representations called flocks, and show that each algebraic representation of a
matroid induces flock representations.Comment: 21 pages, 1 figur

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