Sharpness of Risler's upper bound for the total curvature of an affine real algebraic hypersurface

2007 ◽  
Vol 62 (2) ◽  
pp. 393-394
Author(s):  
Stepan Yu Orevkov
Keyword(s):  
Upper Bound ◽  
Total Curvature ◽  
Algebraic Hypersurface

ON THE GAUSSIAN CURVATURE OF MAXIMAL SURFACES AND THE CALABI–BERNSTEIN THEOREM

2001 ◽  
Vol 33 (4) ◽  
pp. 454-458 ◽  
Author(s):  
LUIS J. ALÍAS ◽  
BENNETT PALMER
Keyword(s):  
Minkowski Space ◽  
Upper Bound ◽  
Gaussian Curvature ◽  
Total Curvature ◽  
Local Geometry ◽  
Maximal Surface ◽  
Bernstein Theorem ◽  
New Approach ◽  
Maximal Surfaces

In this paper, a new approach to the Calabi–Bernstein theorem on maximal surfaces in the Lorentz– Minkowski space L3 is introduced. The approach is based on an upper bound for the total curvature of geodesic discs in a maximal surface in L3, involving the local geometry of the surface and its hyperbolic image. As an application of this, a new proof of the Calabi–Bernstein theorem is provided.

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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