Invariant Sets in Banach Lattices and Linear Differential Equations and Delay

Positivity ◽  
2004 ◽  
Vol 8 (2) ◽  
pp. 127-142 ◽  
Author(s):  
S. Boulite ◽  
H. Bouslous ◽  
L. Maniar
Keyword(s):  
Differential Equations ◽  
Banach Lattices ◽  
Linear Differential Equations ◽  
Invariant Sets ◽  
Linear Differential

scholarly journals Zero Counting and Invariant Sets of Differential Equations

2017 ◽  
Vol 2019 (13) ◽  
pp. 4119-4158
Author(s):  
Gal Binyamini
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Upper Bound ◽  
Linear Differential Equations ◽  
Invariant Sets ◽  
Polynomial Vector ◽  
Algebraic Hypersurface ◽  
Polynomial Differential Equations ◽  
Polynomial Vector Field ◽  
Linear Differential

Abstract Consider a polynomial vector field $\xi$ in ${\mathbb C}^n$ with algebraic coefficients, and $K$ a compact piece of a trajectory. Let $N(K,d)$ denote the maximal number of isolated intersections between $K$ and an algebraic hypersurface of degree $d$. We introduce a condition on $\xi$ called constructible orbits and show that under this condition $N(K,d)$ grows polynomially with $d$. We establish the constructible orbits condition for linear differential equations over ${\mathbb C}(t)$, for planar polynomial differential equations and for some differential equations related to the automorphic $j$-function. As an application of the main result, we prove a polylogarithmic upper bound for the number of rational points of a given height in planar projections of $K$ following works of Bombieri–Pila and Masser.

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