Landen transforms as families of (commuting) rational self-maps of projective space

TL;DR

This paper studies Landen transforms as families of rational self-maps on projective space, proving their algebraic properties, dominance, and commutativity, and extending them over integers.

Contribution

It establishes that Landen transforms induce dominant rational self-maps of projective space with specific degrees, and shows their extension over Spec(Z) and mutual commutativity.

Findings

Landen transforms induce dominant rational maps of specified degrees.

The maps extend over the integers, maintaining their properties.

The family of maps R_{d,m,0} is commuting for all m.

Abstract

The classical (m,k)-Landen transform F_{m,k} is a self-map of the field of rational functions 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 P^{2d+1}, we prove that F_{m,0} induces a dominant rational self-map R_{d,m,0} of P^{2d+1} of algebraic degree m, and for 0 < k < m, the transform F_{m,k} induces a dominant rational self-map R_{d,m,k} of algebraic degree m of a certain hyperplane in P^{2d+1}. We show in all cases that R_{d,m,k} extends nicely to a map of P^{2d+1} over Spec(Z), and that {R_{d,m,0} : m \ge 0} is a commuting family of maps.