221 research outputs found
Nonstandard methods for bounds in differential polynomial rings
Motivated by the problem of the existence of bounds on degrees and orders in
checking primality of radical (partial) differential ideals, the nonstandard
methods of van den Dries and Schmidt ["Bounds in the theory of polynomial rings
over fields. A nonstandard approach.", Inventionnes Mathematicae, 76:77--91,
1984] are here extended to differential polynomial rings over differential
fields. Among the standard consequences of this work are: a partial answer to
the primality problem, the equivalence of this problem with several others
related to the Ritt problem, and the existence of bounds for characteristic
sets of minimal prime differential ideals and for the differential
Nullstellensatz.Comment: 18 page
Diophantine approximation and deformation
We associate certain curves over function fields to given algebraic power
series and show that bounds on the rank of Kodaira-Spencer map of this curves
imply bounds on the exponents of the power series, with more generic curves
giving lower exponents. If we transport Vojta's conjecture on height inequality
to finite characteristic by modifying it by adding suitable deformation
theoretic condition, then we see that the numbers giving rise to general curves
approach Roth's bound. We also prove a hierarchy of exponent bounds for
approximation by algebraic quantities of bounded degree
On function field Mordell-Lang and Manin-Mumford
We present a reduction of the function field Mordell-Lang conjecture to the
function field Manin-Mumford conjecture, in all characteristics, via model
theory, but avoiding recourse to the dichotomy theorems for (generalized)
Zariski structures.
In this version 2, the quantifier elimination result in positive
characteristic is extended from simple abelian varieties to all abelian
varieties, completing the main theorem in the positive characteristic case.
In version 3, some corrections are made to the proof of quantifier
elimination in positive characteristic, and the paper is substantially
reorganized.Comment: 21 page
THE NEW-OLD COSMOLOGY
The recently discovered physics discipline of cryodynamics, sister discipline to thermodynamics, enables a new picture of the cosmos. A stationary, infinite, eternal, fractal cosmos that can be called the Clifford-Zwicky-Mandelbrot (CZM) cosmos emerges. Many elements of the currently accepted cosmology are put up for replacement in a 12-point list
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