Curves, dynamical systems, and weighted point counting

Curves, dynamical systems, and weighted point counting.

Suppose X is a (smooth projective irreducible algebraic) curve over a finite field k. Counting the number of points on X over all finite field extensions of k will not determine the curve uniquely. Actually, a famous theorem of Tate implies that two such curves over k have the same zeta function (i.e., the same number of points over all extensions of k) if and only if their corresponding Jacobi...

Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}. For a class F of boolean functions and a class G of graphs, an (F ,G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transit...

Point Counting on Elliptic and Hyperelliptic Curves

observational dynamical systems

چکیده در این پایاننامه ابتدا فضاهای متریک فازی را به صورت مشاهدهگرایانه بررسی میکنیم. فضاهای متریک فازی و توپولوژی تولید شده توسط این متریک معرفی شدهاند. سپس بر اساس فضاهایی که در فصل اول معرفی شدهاند آشوب توپولوژیکی، مینیمالیتی و مجموعههای متقاطع در شیوههای مختلف بررسی شده- اند. در فصل سوم مفهوم مجموعههای جاذب فازی به عنوان یک مفهوم پایهای در سیستمهای نیم-دینامیکی نسبی، تعریف شده است. ...

Point Counting in Families of Hyperelliptic Curves

Let EΓ be a family of hyperelliptic curves defined by Y 2 = Q(X,Γ), where Q is defined over a small finite field of odd characteristic. Then with γ in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve Eγ by using Dwork deformation in rigid cohomology. The complexity of the algorithm is O(n) and it needs O(n) bits...