2014 / December Volume 9 No.4
Landen Transforms as Families of (Commuting) Rational Self-Maps of Projective Space
| Published Date |
2014 / December
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|---|---|
| Title | Landen Transforms as Families of (Commuting) Rational Self-Maps of Projective Space |
| Author | |
| Keyword | |
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| Pagination | 547-584 |
| 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.
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| AMS Subject Classification |
14E05, 37P05
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| Received |
2013-12-28
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| Accepted |
2014-01-14
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