313,961 research outputs found

    Quasi-Valuations Extending a Valuation

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    Suppose FF is a field with valuation vv and valuation ring OvO_{v}, EE is a finite field extension and ww is a quasi-valuation on EE extending vv. We study quasi-valuations on EE that extend vv; 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 OvO_{v}; in particular, they have the same Krull Dimension. We also prove that every such quasi-valuation is dominated by some valuation extending vv. 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 OvO_{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 RR, an algebra over OvO_{v}, we construct a quasi-valuation on RR; we also construct a quasi-valuation on R⊗OvFR \otimes_{O_{v}} F which helps us prove our main Theorem. The main Theorem states that if R⊆ER \subseteq E satisfies R∩F=OvR \cap F=O_{v} and EE is the field of fractions of RR, then RR and vv induce a quasi-valuation ww on EE such that R=OwR=O_{w} and ww extends vv; thus RR satisfies the properties of a quasi-valuation ring.Comment: 51 page

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

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    Let VV be a valuation domain with quotient field KK. Given a pseudo-convergent sequence EE in KK, we study two constructions associating to EE a valuation domain of K(X)K(X) lying over VV, especially when VV 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 KK 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

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    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

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    We show that each algebraic representation of a matroid MM in positive characteristic determines a matroid valuation of MM, which we have named the {\em Lindstr\"om valuation}. If this valuation is trivial, then a linear representation of MM in characteristic pp can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic pp if and only if they are linear in characteristic pp. 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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