Related papers: Elimination with applications to singularities in …
Updated version. Includes comments on the advances in the field from the Kyoto workshop on Resolution of Singularities in Positive Characteristic, December 2008. The article surveys the theory of kangaroo points as they appear in the…
We introduce and study a new class of differential fields in positive characteristic. We call them separably differentially closed fields and demonstrate that they are the differential analogue of separably closed fields. We prove several…
Partial descriptions of the Universe are presented in the form of linear equations considered in the free (full, super) Fock space. The universal properties of these equations are discussed. The closure problem caused by computational and…
Many types of point singularity have a topological index, or 'charge', associated with them. For example the phase of a complex field depending on two variables can either increase or decrease on making a clockwise circuit around a simple…
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…
This paper builds fundamental perfect fields of positive characteristic and shows the structure of perfect fields that a field of positive characteristic is a perfect field if and only if it is an algebraic extension of a fundamental…
We discuss the problems of incompleteness and inexpressibility. We introduce almost self-referential formulas, use them to extend set theory, and relate their expressive power to that of infinitary logic. We discuss the nature of proper…
For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit…
We use group representation theory to give algebraic formulae to compute complete transversals of singularities of vector fields, either in the nonsymmetric or in the reversible equivariant contexts. This computation produces normal forms…
Assume that there exists a hypersurface singularity which cannot be resolved by iterated monoidal transformations in positive characteristic. We show that in the set of defining functions of hypersurface singularities which cannot be…
We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the…
We explain how to use computer experiments over finite fields to gain heuristic information about the solution set of polynomial equations in characteristic zero. These are notes of a tutorial I gave at the NATO Advanced Study Institute on…
We discuss Hironaka's theorem on resolution of singularities in charactetistic 0 as well as more recent progress, both on simplifying and improving Hironaka's method of proof and on new results and directions on families of varieties,…
We investigate the discrete Painleve II equation over finite fields. We treat it over local fields and observe that it has a property that is similar to the good reduction over finite fields. We can use this property, which seems to be an…
The author surveys Galois theory of function fields with non-zero caracteristic and its relation to the structure of finite permutation groups and matrix groups.
Starting with a general discussion, a program is sketched for a quantization based on dilations. This resolving-power quantization is simplest for scalar field theories. The hope is to find a way to relax the requirement of locality so that…
The present research deals with generalizations of the Salem function with arguments defined in terms of certain alternating expansions of real numbers. The special attention is given to modelling such functions by systems of functional…
We introduce a new version of arithmetic in all finite types which extends the usual versions with primitive notions of extensionality and extensional equality. This new hybrid version allows us to formulate a strong form of extensionality,…
We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can…
The first part is expository: it explains how finite fields may be used to prove theorems on infinite fields by a reduction mod p process. The second part gives a variant of P.Smith's fixed point theorem which applies in any characteristic.