scholarly journals Hilbert-Type Dimension Polynomials of Intermediate Difference-Differential Field Extensions

2020 ◽  
pp. 64-79
Author(s):  
Alexander Levin
Keyword(s):  
Field Extensions ◽  
Differential Field ◽  
Hilbert Type

Valued Differential Fields

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Asymptotic Behavior ◽  
Residue Field ◽  
Field Extension ◽  
Field Extensions ◽  
Differential Field ◽  
Differential Fields ◽  
Dominant Part

This chapter deals with valued differential fields, starting the discussion with an overview of the asymptotic behavior of the function vsubscript P: Γ‎ → Γ‎ for homogeneous P ∈ K K{Y}superscript Not Equal To. The chapter then shows that the derivation of any valued differential field extension of K that is algebraic over K is also small. It also explains how differential field extensions of the residue field k give rise to valued differential field extensions of K with small derivation and the same value group. Finally, it discusses asymptotic couples, dominant part, the Equalizer Theorem, pseudocauchy sequences, and the construction of canonical immediate extensions.

scholarly journals Multivariable dimension polynomials and new invariants of differential field extensions

2001 ◽  
Vol 27 (4) ◽  
pp. 201-214 ◽  
Author(s):  
Alexander B. Levin
Keyword(s):  
Field Extension ◽  
Differential Polynomials ◽  
Finitely Generated ◽  
Field Extensions ◽  
Differential Field ◽  
Differential Dimension ◽  
Characteristic Sets ◽  
Differential Dimension Polynomial ◽  
Dimension Polynomial

We introduce a special type of reduction in the ring of differential polynomials and develop the appropriate technique of characteristic sets that allows to generalize the classical Kolchin's theorem on differential dimension polynomial and find new differential birational invariants of a finitely generated differential field extension.

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