Based rational-map moduli of sigma-model lumps (${\dim }_{\mathbb{C}}{\mathcal{R}}_k^{\ast }=2k$)

Fix the value $w(\infty )=0$ for positive-degree sigma-model lumps. Their holomorphic rational maps can be written $p/q$, where $q$ is monic of degree $k$, $p$ has degree at most $k-1$, and the polynomials have no common zero. The coefficients of $p$ and the nonleading coefficients of $q$ give $2k$ complex parameters; coprimality is an open condition. The topological degree is $k$, obtained by counting generic preimages with multiplicity. Allowing a common factor lowers the degree after cancellation and can describe a degeneration rather than another smooth point of this degree-$k$ parameter space.