13 research outputs found
Lascar and Morley ranks differ in differentially closed fields
We note here, in answer to a question of Poizat, that the Morley and Lascar
ranks need not coincide in differentially closed fields. We approach this
through the (perhaps) more fundamental issue of the variation of Morley rank in
families
Algebraic relations between solutions of Painlev\'e equations
We calculate model theoretic ranks of Painlev\'e equations in this article,
showing in particular, that any equation in any of the Painlev\'e families has
Morley rank one, extending results of Nagloo and Pillay (2011). We show that
the type of the generic solution of any equation in the second Painlev\'e
family is geometrically trivial, extending a result of Nagloo (2015).
We also establish the orthogonality of various pairs of equations in the
Painlev\'e families, showing at least generically, that all instances of
nonorthogonality between equations in the same Painlev\'e family come from
classically studied B{\"a}cklund transformations. For instance, we show that if
at least one of is transcendental, then is
nonorthogonal to if and only if or . Our results have concrete interpretations
in terms of characterizing the algebraic relations between solutions of
Painlev\'e equations. We give similar results for orthogonality relations
between equations in different Painlev\'e families, and formulate some general
questions which extend conjectures of Nagloo and Pillay (2011) on transcendence
and algebraic independence of solutions to Painlev\'e equations. We also apply
our analysis of ranks to establish some orthogonality results for pairs of
Painlev\'e equations from different families. For instance, we answer several
open questions of Nagloo (2016), and in the process answer a question of Boalch
(2012).Comment: This manuscript replaces and greatly expands a portion of
arXiv:1608.0475
On algebraic relations between solutions of a generic Painleve equation
We prove that if y" = f(y,y',t,\alpha, \beta,..) is a generic Painleve
equation (i.e. an equation in one of the families PI-PVI but with the complex
parameters \alpha, \beta,.. algebraically independent) then any algebraic
dependence over C(t) between a set of solutions and their derivatives
(y_1,..,y_n,y_1',..,y_n') is witnessed by a pair of solutions and their
derivatives (y_i,y_i',y_j,y_j'). The proof combines work by the Japanese school
on "irreducibility" of the Painleve equations, with the trichomoty theorem for
strongly minimal sets in differentially closed fields.Comment: 23 page
Stable domination and independence in algebraically closed valued fields
We seek to create tools for a model-theoretic analysis of types in
algebraically closed valued fields (ACVF). We give evidence to show that a
notion of 'domination by stable part' plays a key role. In Part A, we develop a
general theory of stably dominated types, showing they enjoy an excellent
independence theory, as well as a theory of definable types and germs of
definable functions. In Part B, we show that the general theory applies to
ACVF. Over a sufficiently rich base, we show that every type is stably
dominated over its image in the value group. For invariant types over any base,
stable domination coincides with a natural notion of `orthogonality to the
value group'. We also investigate other notions of independence, and show that
they all agree, and are well-behaved, for stably dominated types. One of these
is used to show that every type extends to an invariant type; definable types
are dense. Much of this work requires the use of imaginary elements. We also
show existence of prime models over reasonable bases, possibly including
imaginaries
The Logarithmic Derivative and Model-Theoretic Analysability in Differentially Closed Fields
This thesis deals with internal and analysable types, mainly in the context of the stable theory of differentially closed fields. Two main problems are dealt with: the construction of types analysable in the constants with specific properties, and a criterion for a given analysable type to be actually internal to the constants.
For analysable types, the notion of canonical analyses is introduced. A type has a canonical analysis if all its analyses of shortest length are interalgebraic. Given a finite sequence of ranks, it is constructed, in the theory of differentially closed field, a type analysable in the constants such that it admits a canonical analysis and each step of the analysis is of the given rank. The construction of such a type starts from the well-known example of δ(logδx)=0, whose generic type is analysable in the constants in 2 steps but is not internal to the constants. Along the way, techniques for comparing analyses in stable theories are developed, including in particular the notions of analyses by reductions and by coreductions.
The property of the logδ function is further studied when the following question is raised: given a type internal to the constants, is its preimage under logδ, which is 2-step analysable in the constants, ever internal to the constants? The question is answered positively, and a criterion for when the preimage is indeed internal is proposed. Partial results are proven for this conjectured criterion, namely the cases where the group of automorphisms (the binding group) of the given internal type is additive, multiplicative, or trivial. In particular, the conjecture is resolved for generic types of equations of the form δx=f(x) where f is a rational function over the constants. It is discovered that the related problem where logδ is replaced by δ is significantly different, and the analogue of the conjecture fails in this case.
Also included in this thesis are two examples asked for in the literature: internality of a particular twisted D-group, and a 2-step analysable set with independent fibres