• Corpus ID: 239009420

When any three solutions are independent

@inproceedings{Freitag2021WhenAT,
  title={When any three solutions are independent},
  author={James Freitag and R'emi Jaoui and Rahim Moosa},
  year={2021}
}
Given an algebraic differential equation of order > 1, it is shown that if there is any nontrivial algebraic relation amongst any number of distinct nonalgebraic solutions, along with their derivatives, then there is already such a relation between three solutions. This is deduced as an application of the following model-theoretic result: Suppose p is a stationary nonalgebraic type in the theory of differentially closed fields of characteristic zero; if any three distinct realisations of p are… 

References

SHOWING 1-10 OF 35 REFERENCES
Mehrfach transitive Operationen algebraischer Gruppen
In dieser Arbeit werden alle algebraisehen Gruppen bestimmt, die fiber einem algebraiseh abgesehlossenen K6rper k definiert sind und die auf den rationalen Punkten einer Variet~t biregul/~r und
Bounding nonminimality and a conjecture of Borovik-Cherlin
Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is
Relative internality and definable fibrations
We first elaborate on the theory of relative internality in stable theories, focusing on the notion of uniform relative internality (called collapse of the groupoid in an earlier work of the second
Generic planar algebraic vector fields are disintegrated
In this article, we study model-theoretic properties of algebraic differential equations of order $2$, defined over constant differential fields. In particular, we show that the existentially closed
Internality of logarithmic-differential pullbacks
A criterion in the spirit of Rosenlicht is given, on the rational function f(x), for when the planar vector field defined by x'=f(x) and y'=xy admits a pair of algebraically independent first
Isolated types of finite rank: an abstract Dixmier–Moeglin equivalence
Suppose T is a totally transcendental first-order theory and every minimal non-locally-modular type is nonorthogonal to a nonisolated minimal type over the empty set. It is shown that a finite rank
Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups
We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory,
Bi-algebraic geometry and the André-Oort conjecture
Hypertranscendence of solutions of Mahler equations
The last years have seen a growing interest from mathematicians in Mahler functions. This class of functions includes the generating series of the automatic sequences. The present paper is concerned
Algebraic independence of generic Painlevé transcendents: PIII and PVI
We prove that if y"=f(y,y',t) is a generic Painleve equation from the class III and VI, and if y_1,...,y_n are distinct solutions, then y_1,y_1',...,y_n,y_n' are algebraically independent over C(t).
...
1
2
3
4
...