AI 中文总结
本文研究带代数柯西变换的母体测度，构造泛函𝔇_{P,K}(μ)，证明其连续性与最小值可达性，刻画零缺陷，建立有理情形下缺陷的对偶公式，构造正代数柯西变换并实现零缺陷到母体的过渡。
AI 中文摘要
在对带有代数柯西变换的母体测度的持续研究中，我们将正代数芽f与其不可约方程P(z,w)=0、紧凸集K相关联，构造了泛函𝔇_{P,K}(μ)=∫_K|P(z,𝒞_μ(z))|^{1/d}dA(z)，其中d=deg_w P，该泛函定义在支撑于K且柯西变换在无穷远处具有芽f的正测度集合上。我们证明了𝔇_{P,K}(μ)的连续性及其最小值可达性，刻画了零缺陷，并在P的首项系数零点之外得到估计式μ(D(a,r))≤C₁r + C₂𝔇_{P,K}(μ)/r，其中D(a,r)是以a为中心、半径为r的圆盘。我们在有理情形下建立了缺陷的对偶公式，通过Herglotz理论、Fuss–Catalan和Raney定律、正和、多项式推前及分支图构造了正代数柯西变换。在支撑集的平面零假设下，我们实现了从零缺陷到母体的过渡。
英文摘要
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 \[ \mathfrak D_{P,K}(μ)= \int_K\left|P\bigl(z,\mathcal C_μ(z)\bigr)\right|^{1/d}\,dA(z), \qquad d=\text{deg}_wP, \] on the set of positive measures supported in $K$ and whose Cauchy transform has germ $f$ at infinity. We prove continuity of $\mathfrak D_{P,K}(μ)$ and attainment of its minimum, characterize zero defect, and obtain the estimate \[ μ(D(a,r))\le C_1r+C_2\mathfrak D_{P,K}(μ)/r \] 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.
Comments19 pages
