Landen Transforms as Families of (Commuting) Rational Self-Maps of Projective Space
by Michael Joyce   Shu Kawaguchi   Joseph H. Silverman  

Vol. 9 No. 4 (2014) P.547~P.584

ABSTRACT

The classical $(m,k)$-Landen transform $\mathfrak{F}_{m,k}$ is a self-map of the field of rational functions $\mathbb{C}(z)$ obtained by forming a weighted average of a rational function over twists by $m$'th roots of unity. Identifying the set of rational maps of degree $d$ with an affine open subset of $\mathbb{P}^{2d+1}$, we prove that $\mathfrak{F}_{m,0}$ induces a dominant rational self-map $\mathfrak{R}_{d,m,0}$ of $\mathbb{P}^{2d+1}$ of algebraic degree $m$, and for $1 \le k < m $, the transform $\mathfrak{F}_{m,k}$ induces a dominant rational self-map $\mathfrak{R}_{d,m,k}$ of algebraic degree $m$ of a certain hyperplane in $\mathbb{P}^{2d+1}$. We show in all cases that $\mathfrak{R}_{d,m,k}$ extends nicely to $\mathbb{P}^{2d+1}_\mathbb{Z}$, and that $\{\mathfrak{R}_{d,m,0} : m\ge0\}$ is a commuting family of maps.

KEYWORDS
Landen transform, commuting rational maps, algebraic dynamical systems

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 14E05, 37P05

MILESTONES

Received: 2013-09-02
Revised : 2013-12-28
Accepted: 2014-01-14

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