THE MODULI SPACE OF POLYNOMIAL MAPS AND THEIR HOLOMORPHIC INDICES: I. GENERIC PROPERTIES IN THE CASE OF HAVING MULTIPLE FIXED POINTS

The paper proves that, on each multiplicity stratum (d1,...,dℓ) with ℓ≥2, the holomorphic-index map has generic fiber size (d−1)!/(d−ℓ)! on MCd and (d−2)!/(d−ℓ)! on MPd, with equality precisely when (i) the pairs (d_i,m_i) are all distinct and (ii) no nonempty proper partial sum of the m_i’s vanishes; the cubic all-simple case (d=ℓ=3) is the lone exception for the converse. The proof proceeds by expressing f(z)=z+ρ∏(z−ζ_i)^{d_i}, translating residue constraints into a square linear system built from confluent Vandermonde blocks, and then eliminating auxiliary variables to obtain ℓ−2 equations whose degrees multiply to (d−2)!/(d−ℓ)!, yielding the generic count; the MPd bound is lower by a factor of d−1 via MCd→MPd. All of this matches the candidate solution’s residue/partial-fraction reduction, block–Vandermonde linear algebra, generic finiteness, and the d−1 factor between MCd and MPd. The only correction needed is minor: the paper obtains a complete intersection of ℓ−2 equations rather than a single homogeneous equation on the centered projective coordinates. Otherwise, the approaches are the same and reach the same bounds and genericity/branch-locus description (including the special d=ℓ=3 case) (Main Theorem; matrix identities (3.2)–(3.4); invertibility Proposition 2.4; generic count Proposition 3.24; the (d−1)-to-1 relation in Proposition 3.19(5)) .