An Efficient Algorithm for Determining...

Polarized morphisms are important objects in various mathematical and computational fields, and their composition is a fundamental operation. In this talk, we present a new algorithm to efficiently compute the composition relationship of two polarized morphisms using the concept of dynamical canonical height. Our approach utilizes the properties of dynamical...

On Sub-semigroup Generated by Finitely...

In the realm of category theory, an endomorphism represents a map that preserves an object's structure. This opens the door to exploring the (semi)-group structure of $\mbox{Hom}(X)$, the collection of such endomorphisms. A pivotal concept in this exploration is the Tits alternative, which classifies linear groups generated by two elements....

Introduction of SageMath

Polarized morphisms are important objects in various mathematical and computational fields, and their composition is a fundamental operation. In this talk, we present a new algorithm to efficiently compute the composition relationship of two polarized morphisms using the concept of dynamical canonical height. Our approach utilizes the properties of dynamical...

On Tits Alternative and its...

In the realm of category theory, an endomorphism represents a map that preserves an object's structure. This opens the door to exploring the (semi)-group structure of $\mbox{Hom}(X)$, the collection of such endomorphisms. A pivotal concept in this exploration is the Tits alternative, which classifies linear groups generated by two elements....

Why is it so hard...

We study the composition relations of rational functions, focusing on two main aspects. First, we establish a stronger version of the Tits alternative for endomorphisms of the projective line, proving that for almost all rational functions $f$ of degree greater than 3, the semigroup generated by $f$ and any fixed...

Periodicity and Dynamical Systems of...

In this talk, I will present our recent investigation into the dynamical properties of Dickson polynomials over finite fields. We'll explore the periodicity and structural behavior of iterated sequences defined by \( [D_n(x, \alpha) \mod (x^q - x)]_n \), where \( D_n(x, \alpha) \) is a Dickson polynomial of the...

從陶哲軒的十分鐘談起：AI時代下數學本質的變與不變

Overgroups of the arboreal representation...

Consider a number field $K$ and a rational function $f$ of degree greater than 1 over $K$. By taking preimages of $\alpha\in K$ under successive iterates of $f$, an infinite $d$-ary tree $T_\infty$ rooted at $\alpha$ can be constructed. An edge is assigned between two preimages $x$ and $y$ if...

Catching a spider: Automating the...

The Spider Algorithm is a well-known topological tool used to study the dynamics of polynomials and rational functions. However, implementing this algorithm computationally requires a system capable of distinguishing valid homotopic paths from tangled, intersecting ones. In this talk, we explore a computational framework that translates the continuous topological constraints...

Explicit Arithmetic Distance Motivated by...

Motivating by the study of Tits alternative in arithmetic dynamics, we investigate the arithmetic distance between distinct rational maps $f,\ g: \mathbb{P}^1(\mathbb{Q})\to \mathbb{P}^1(\mathbb{Q})$ defined by $$\hat{\Delta}(f,g)=\sup_{x\in \mathbb{P}^1(\mathbb{Q})}|\hat{h}_f(x)-\hat{h}_g(x)|$$ Moreover, when $g$ is monomial, the arithmetic distance is invariant under conjugation by elements of $\operatorname{PGL}_2$. This motivate the definition of the arithmetic...