Related papers: Exploring tropical differential equations
We prove that a semiring isomorphism between the rational function semifields of two tropical curves induces an expansive map between those tropical curves. This semiring isomorphism and the expansive map respect zeros and poles of rational…
This survey is meant to provide an introduction to the fundamental theorem of linear algebra and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to linear algebra. Specifically, we…
The purpose of this survey is to summarize known results about tropical hypersurfaces and the Cayley Trick from polyhedral geometry. This allows for a systematic study of arrangements of tropical hypersurfaces and, in particular,…
Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by…
This paper is devoted to the bounding and computation of the dimension of deformation spaces of tropical curves and hypersurfaces. This characteristic is interesting in light of the fact that it often coincides with the dimension of…
Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables.…
Finding a common factor of two multivariate polynomials with approximate coefficients is a problem in symbolic-numeric computing. Taking a tropical view on this problem leads to efficient preprocessing techniques, applying polyhedral…
In order to give a formal treatment of differential equations in positive characteristic p, it is necessary to use divided powers. One runs into an analog problem in the theory of q-difference equations when q is a pth root of unity. We…
We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are enriched with an information of multiplicity, sign, or argument. We obtain existence or uniqueness…
We develop a framework to apply tropical and nonarchimedean analytic techniques to multiplication maps on linear series and study degenerations of these multiplications maps when the special fiber is not of compact type. As an application,…
Given an algebraic variety defined over a discrete valuation field and a skeleton of its Berkovich analytification, the tropicalization process transforms function field of the variety to a semifield of tropical functions on the skeleton.…
This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's…
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but…
Ducros, Hrushovski, and Loeser gave maps from families of archimedean diffrential forms to non-archiemedean (or tropical) ones, which are compatible with integrals on algebraic varieties. In this paper, we introduce slight modifications of…
We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our…
This paper lays out a foundation for a theory of supertropical algebraic geometry, relying on commutative $\nu$-algebra. To this end, the paper introduces $\mathfrak{q}$-congruences, carried over $\nu$-semirings, whose distinguished ghost…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
We introduce algebraic structures on the polyvector fields of an algebraic torus that serve to compute multiplicities in tropical and log Gromov-Witten theory while also connecting to the mirror symmetry dual deformation theory of complex…
The purpose of this paper is to propose the implementation of some methods from algebraic geometry in the theory of gravitation, and more especially in the variational formalism. It has been assumed that the metric tensor depends on two…
New hyperfields, that is fields in which addition is multivalued, are introduced and studied. In a separate paper these hyperfields are shown to provide a base for the tropical geometry. The main hyperfields considered here are classical…