Algebraic defect and positive mother body measures

Continuing the study of mother-body measures with algebraic Cauchy transform, we associate with a positive algebraic germ $f$, its irreducible equation $P(z,w)=0$, and a compact convex set $K$ the functional $${\mathfrak{D}}_{P,K}(\mu )={\int }_K{\mid P(z,{\mathcal{C}}_{\mu }(z))\mid }^{1\mathrm{/}d}dA(z),d={\text{deg}}_{w}P,$$ on the set of positive measures supported in $K$ and whose Cauchy transform has germ $f$ at infinity. We prove continuity of ${\mathfrak{D}}_{P,K}(\mu )$ and attainment of its minimum, characterize zero defect, and obtain the estimate $$\mu (D(a,r))\le C_{1}r+C_2{\mathfrak{D}}_{P,K}(\mu )\mathrm{/}r$$ away from the zero set of the leading coefficient of $P$, where $D(a,r)$ is the disk of radius $r$ centered at $a$. We establish a dual formula for the defect in the rational case and construct positive algebraic Cauchy transforms by Herglotz theory, Fuss–Catalan and Raney laws, positive sums, polynomial pushforwards, and branch graphs. The passage from zero defect to a mother body is made under the planar-null hypothesis of the support.