Landen Transforms as Families of (commuting) Rational Self-maps of Projective Space
نویسندگان
چکیده
The classical (m, k)-Landen transform Fm,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, we prove that Fm,0 induces a dominant rational self-map Rd,m,0 of P 2d+1 of algebraic degree m, and for 1 ≤ k < m, the transform Fm,k induces a dominant rational self-map Rd,m,k of algebraic degree m of a certain hyperplane in P . We show in all cases that Rd,m,k extends nicely to P 2d+1 Z , and that {Rd,m,0 : m ≥ 0} is a commuting family of maps.
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