Local degrees of a polynomial map on the riemann sphere

Here is the sphere model and its orientation. The map $s(z)=(2\mathrm{Re}z,2\mathrm{Im}z,\mathrm{\mid }z{\mathrm{\mid }}^2-1)\mathrm{/}(1+\mathrm{\mid }z{\mathrm{\mid }}^2)$ lies on the unit sphere, and its inverse off the north pole is $(x,y,t)\mapsto (x+iy)\mathrm{/}(1-t)$. Both formulas are continuous and inverse by substitution. Moreover $s(z)$ approaches the north pole exactly as $\mathrm{\mid }z\mathrm{\mid }\to \mathrm{\infty }$, giving the topology with neighborhoods of infinity containing $\{\mathrm{\mid }z\mathrm{\mid }>R\}\cup \{\mathrm{\infty }\}$. Declare the finite coordinate $z$ positive and use $w=1\mathrm{/}z$ near infinity. At $z_0\mathrm{\ne }0$, writing $z=z_0+u$, the transition difference is $1\mathrm{/}(z_0+u)-1\mathrm{/}{z}_0=-u\mathrm{/}(z_0(z_0+u))$. On a sufficiently small disk its nonzero factor is homotopic through nonzero factors to $-1\mathrm{/}{z}_0^2$. Multiplication by a nonzero complex constant is a rotation followed by a positive dilation, homotopic through invertible real maps to the identity. Thus the two charts give compatible local orientations; no orientation-reversal is hidden at infinity.

The leading-term estimate gives $\mathrm{\mid }P(z)\mathrm{\mid }\ge \mathrm{\mid }{a}_d\mathrm{\mid }\mathrm{\mid }z{\mathrm{\mid }}^d\mathrm{/}2$ for all sufficiently large $\mathrm{\mid }z\mathrm{\mid }$: divide the sum of lower-degree terms by $\mathrm{\mid }z{\mathrm{\mid }}^d$, which tends to zero. It proves continuity at infinity. For a finite $a$, polynomial division yields $P(a+u)-P(a)=u^{m}q(a+u)$ with $m\ge 1$ and $q(a)\mathrm{\ne }0$. Shrink the disk until $\mathrm{\mid }q(a+u)-q(a)\mathrm{\mid }<\mathrm{\mid }q(a)\mathrm{\mid }$. The homotopy $u^m((1-s)q(a+u)+sq(a))$ has zero only at $u=0$ for every $0\le s\le 1$. Uniform boundedness of the factors permits a source disk mapping into one fixed target coordinate disk. Hence this is a homotopy of punctured local pairs to $q(a)u^m$.

For a disk $D$ centered at zero, the pair exact sequence [F3] identifies $H_2(D,D\setminus \{0\};\mathbb{Z})$ with $H_1(D\setminus \{0\};\mathbb{Z})$; radial deformation identifies the latter with the counterclockwise circle generator. The normalized map of a small circle for $c{u}^m$, $c\mathrm{\ne }0$, is a rotation times $e^{it}\mapsto e^{imt}$. The latter has $m$ preimages of $1$. Near each preimage its angular formula is $t\mapsto mt$, homotopic through positive linear slopes to $t\mapsto t$, so its local degree is $+1$. Applying [F1] in dimension one gives angular degree $m$. Rotation acts trivially by its rotation homotopy. Naturality [F4] of the disk pair boundary therefore gives local degree $m$ for $c{u}^m$, and step 2.1 gives the claimed finite local multiplicities.

In the coordinates $w=1\mathrm{/}z$ and $v=1\mathrm{/}P(z)$ at the two infinities, the map is $v=w^d\mathrm{/}(a_d+a_{d-1}w+\cdots +a_0{w}^d)$, extended by $v(0)=0$. Its denominator is nonzero on a small disk. The same nonvanishing-factor homotopy and disk-boundary calculation give local degree $d$ at infinity. The value is positive with the compatible chart orientations of step 1.1.

For any finite $b$, [F2] applied to the degree-$d$ polynomial $P-b$ says its distinct roots have positive multiplicities summing to $d$. They form the entire finite fibre, since infinity maps to infinity. By step 3.1 and [F1], $\mathrm{deg}\widehat{P}$ is the sum of these local degrees, hence $d$. This also agrees with applying [F1] to the singleton fibre of infinity and step 4.1. Repeated roots and $d=1$ require no change.