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Covariance is a useful property for handling supergravity theories. In this paper, we prove a covariance property of supergravity field equations: under reasonable conditions, field equations of supergravity are covariant modulo other field…
In this paper, the author introduces hyperfields and give some facts about the roots and multiplicities of polynomials over hyperfield based on \cite{2} and \cite{3}. Then he tests the sharpness of an inequality in Baker's former work…
For every natural number $m$, the existentially closed models of the theory of fields with $m$ commuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential…
In contrast with QFT, classical field theory can be formulated in a strict mathematical way if one defines even classical fields as sections of smooth fiber bundles. Formalism of jet manifolds provides the conventional language of dynamic…
We estimate the proportion of function fields satisfying certain conditions which imply a function-field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a…
Field theory is an area in physics with a deceptively compact notation. Although general purpose computer algebra systems, built around generic list-based data structures, can be used to represent and manipulate field-theory expressions,…
We give in this paper an isomorphism theorem between derived functors over categories of modules.There is a nice class of categories that gives examples in which this theorem applies for a special construction. This leads us to a new…
In this paper we consider the very wide class of varieties of representations of Lie algebras over the field k, which has characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the…
Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove the following result related to the Tate Conjecture: for a smooth, projective,…
We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.
We develope a difference calculus analogous to the differential geometry by translating the forms and exterior derivatives to similar expressions with difference operators, and apply the results to fields theory on the lattice [Ref. 1]. Our…
The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of…
We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of "tame" differential fields. We state several…
We develop a geometric theory for difference equations with a given group of automorphisms. To solve this problem we extend the class of difference fields to the class of absolutely flat simple difference rings called pseudofields. We prove…
Minor technical changes. Section 4 improved.
We study Grothendieck rings (in the sense of logic) of fields. We prove the triviality of the Grothendieck rings of certain fields by constructing definable bijections which imply the triviality. More precisely, we consider valued fields,…
The algebras of interacting "Lie random fields" that were introduced in J. Math. Phys. 48, 122302 (2007) are developed further. The conjecture that the vacuum vector defines a state over a Lie random field algebra is proved. The difference…
The Modular Isomorphism Problem asks, if an isomorphism between modular group algebras of finite $p$-groups over a field $F$ implies an isomorphism of the group bases. We explore the differences of knowledge on the problem when $F$ is…
In this paper we prove that there are exactly eight function fields, up to isomorphism, over finite fields with class number one.
This paper deals with properties of the algebraic variety defined as the set of zeros of a "deficient" sequence of multivariate polynomials. We consider two types of varieties: ideal-theoretic complete intersections and absolutely…