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Write $p(z)=c{\prod }_{j=1}^{2m}(z-a_j)$, including root multiplicities. For $|z|>R$ the convergent logarithm series for every factor $1-a_j/z$ definesChoosing either square root of the nonzero leading coefficient gives the holomorphic square root outside all polynomial rootsIts square is $p$. The even degree makes $z^m$ single-valued, avoiding the winding obstruction that an odd degree would create on the exterior annulus.

For the branch behaving as $z^2$ at infinity, the Laurent series isIt converges uniformly on the circle of radius two. Only the coefficient of $z^{-1}$ contributes to its anticlockwise integral, sofor this branch, and $+\pi i$ for the other. These are the two allowed answers.