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The moduli space of polynomial maps and their fixed-point multipliers: II. Improvement to the algorithm and monic centered polynomials
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 11 / November 2023
- Published online by Cambridge University Press:
- 03 February 2023, pp. 3777-3795
- Print publication:
- November 2023
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- Article
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We consider the family
$\mathrm {MC}_d$ of monic centered polynomials of one complex variable with degree 
$d \geq 2$, and study the map 
$\widehat {\Phi }_d:\mathrm {MC}_d\to \widetilde {\Lambda }_d \subset \mathbb {C}^d / \mathfrak {S}_d$ which maps each 
$f \in \mathrm {MC}_d$ to its unordered collection of fixed-point multipliers. We give an explicit formula for counting the number of elements of each fiber 
$\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$ for every 
$\bar {\unicode{x3bb} } \in \widetilde {\Lambda }_d$ except when the fiber 
$\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$ contains polynomials having multiple fixed points. This formula is not a recursive one, and is a drastic improvement of our previous result [T. Sugiyama. The moduli space of polynomial maps and their fixed-point multipliers. Adv. Math. 322 (2017), 132–185] which gave a rather long algorithm with some induction processes.
