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Louis Rowen

  1. Tensor Products of Division Algebras and Fields.

    Ср, 06/29/2011 - 15:01 — SomNambul
    Authors: Louis Rowen, David J Saltman
    Subjects: Algebraic Geometry
    Abstract

    This paper began as an investigation of the question of whether $D_1
    \otimes_F D_2$ is a domain where the $D_i$ are division algebras and $F$ is an
    algebraically closed field contained in their centers. We present an example
    where the answer is "no", and also study the Picard group and Brauer group
    properties of $F_1 \otimes_F F_2$ where the $F_i$ are fields. Finally, as part
    of our example, we have results about division algebras and Brauer groups over
    curves. Specifically, we give a splitting criterion for certain Brauer group
    elements on the product of two curves over $F$.

  2. Monoid Valuations and Value Ordered Supervaluations.

    Пт, 04/15/2011 - 15:01 — SomNambul
    Authors: Zur Izhakian, Louis Rowen, Manfred Knebusch
    Subjects: Commutative Algebra
    Abstract

    We complement two papers on supertropical valuation theory ([IKR1],[IKR2]) by
    providing natural examples of m-valuations (= monoid valuations), after that of
    supervaluations and transmissions between them. The supervaluations discussed
    have values in totally ordered supertropical semirings, and the transmissions
    discussed respect the orderings. Basics of a theory of such semirings and
    transmissions are developed as far as needed.

  3. Supertropical linear algebra.

    Втр, 08/03/2010 - 15:01 — SomNambul
    Authors: Zur Izhakian, Louis Rowen, Manfred Knebusch
    Subjects: Commutative Algebra
    Abstract

    The objective of this paper is to lay out the algebraic theory of
    supertropical vector spaces and linear algebra, utilizing the key antisymmetric
    relation of ``ghost surpasses.''Special attention is paid to the various
    notions of ``base,'' which include d-base and s-base, and these are compared to
    other treatments in the tropical theory. Whereas the number of elements in a
    d-base may vary according to the d-base, it is shown that when an s-base
    exists, it is unique up to permutation and multiplication by scalars, and can
    be identified with a set of ``critical'' elements.

  4. Supertropical matrix algebra III: Powers of matrices and generalized eigenspaces.

    Втр, 08/03/2010 - 15:01 — SomNambul
    Authors: Zur Izhakian, Louis Rowen
    Subjects: Commutative Algebra
    Abstract

    We investigate powers of supertropical matrices, with special attention to
    the role of the coefficients of the supertropical characteristic polynomial
    (especially the supertropical trace) in controlling the rank of a power of a
    matrix. This leads to a Jordan-type decomposition of supertropical matrices,
    together with a generalized eigenspace decomposition of a power of an arbitrary
    supertropical matrix.

  5. Supertropical semirings and supervaluations.

    Пт, 03/05/2010 - 16:01 — SomNambul
    Authors: Zur Izhakian, Louis Rowen, Manfred Knebusch
    Subjects: Commutative Algebra
    Abstract

    We interpret a valuation $v$ on a ring $R$ as a map $v: R \to M$ into a so
    called bipotent semiring $M$ (the usual max-plus setting), and then define a
    \textbf{supervaluation} $\phi$ as a suitable map into a supertropical semiring
    $U$ with ghost ideal $M$ (cf. [IR1], [IR2]) covering $v$ via the ghost map $U
    \to M$. The set $\Cov(v)$ of all supervaluations covering $v$ has a natural
    ordering which makes it a complete lattice. In the case that $R$ is a field,
    hence for $v$ a Krull valuation, we give a complete explicit description of
    $\Cov(v)$.

  6. Layered supertropical domains.

    Ср, 12/09/2009 - 16:01 — SomNambul
    Authors: Zur Izhakian, Louis Rowen, Manfred Knebusch
    Subjects: Commutative Algebra
    Abstract

    Generalizing supertropical algebras, we consider a "layered" structure which
    permits different ghost layers, and indicate how it is more amenable than the
    unlayered construction to mathematical analysis, in particular with respect to
    calculus. On the other hand, some of the matrix theory developed in [IR2] and
    [IR4] is also developed in this more general setting.

  7. Supertropical Matrix Algebra II: Solving tropical equations.

    Втр, 09/08/2009 - 12:16 — SomNambul
    Authors: Zur Izhakian, Louis Rowen
    Subjects: Commutative Algebra
    Abstract

    We continue the study of matrices over a supertropical algebra, proving the
    existence of a tangible adjoint of $A$, which provides the unique right (resp.
    left) quasi-inverse maximal with respect to the right (resp. left)
    quasi-identity matrix corresponding to $A$; this provides a unique maximal
    (tangible) solution to supertropical vector equations, via a version of
    Cramer's rule.

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