The extended fixed-point index conjecture for polynomial maps

Let $f$ be a polynomial map of degree $d$, and let $\mathrm{Fix}(f)$ be its set of fixed points, considered without multiplicity in this conjecture. For $\zeta\in\mathrm{Fix}(f)$, define the fixed-point multiplicity $m(f,\zeta)$ and the holomorphic index

$$\iota(f,\zeta):=\frac{1}{2\pi\sqrt{-1}}\oint_{|z-\zeta|=\epsilon}\frac{dz}{z-f(z)}.$$

The index is invariant under biholomorphic transformations, equals $1/(1-f'(\zeta))$ at a simple fixed point, and satisfies

$$\sum_{\zeta\in\mathrm{Fix}(f)}m(f,\zeta)=\deg f,\qquad \sum_{\zeta\in\mathrm{Fix}(f)}\iota(f,\zeta)=0.$$

Let $\widetilde{P}_d$ be the parameter space of such maps and let $\widetilde{\Phi}_d$ send $f$ to the collection $\bigl([\iota(f,\zeta),m(f,\zeta)]\bigr)_{\zeta\in\mathrm{Fix}(f)}$.

Extended fixed-point index conjecture. The map $\widetilde{\Phi}_d$ is finite, and the analogous results to the main theorems hold for $\widetilde{\Phi}_d$ for every parameter value, without exception.

This conjecture extends the multiplier-data construction to parameter values outside the generic space $V_d$, including maps with multiple fixed points. The supplied text gives no resolution status.