发表机构
Óbuda University（欧比达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对平面调和映射在雅可比变号产生焦散时，证明重数加权矩$J_n^{\pm}$仍存在闭式表达式，即某显式原函数的全变差，且断点与阶数无关，从而跨越焦散仅需有限额外求值点。
AI 中文摘要
当平面调和映射的雅可比行列式保持恒定符号时，有向曲面元素与重数加权曲面元素仅相差该符号，因此两个自然的矩层级一致。文献中最近的分类结果正是在这一假设下陈述的；本文则超越了这一假设。一旦雅可比行列式改变符号，临界圆会获得一条焦散作为其像，平面的一部分会被以相反方向多次覆盖，且两个层级分道扬镳：将径向权重对向曲面元素$n_z$积分会使重叠的片层相互抵消，从而给出度加权矩$J_n$；而将权重对$|n_z|$积分则使每个片层都正向计数，从而给出重数加权矩$J_n^{\pm}$。这两者分别是同一映射的经典度公式和面积公式，后者以重数函数（Banach指示线）加权。正是第二个族——即焦散所威胁的那个族——其闭式表达式不再自动成立。我们的主要结果是它仍然存在闭式：$J_n^{\pm}$是同一个显式原函数的全变差，而该原函数的端点差给出$J_n$。因此，跨越焦散仅需额外付出有限个求值点，且这些点对所有阶数$n$都是相同的；对于单调轮廓，至多存在一个内部断点，即临界半径本身。承载这一结果的族是双单项式平面调和映射$F(w)=w+iC\bar w^k$，其中$w=p(u)e^{iv}$，它作为傅里叶扰动曲面的平面投影出现。在度加权一侧，$J_n$对每个$n\ge 0$都有仅含边界的闭式表达式，零阶项即为有向$z$通量。这两个加权积分都不是新对象，焦散也已在调和三项式的文献中被定位过；新的贡献在于跨越焦散时重数加权层级的闭式求值。
英文摘要
While the Jacobian of a planar harmonic mapping keeps a constant sign, the oriented and the multiplicity-weighted surface elements agree up to that sign, and the two natural moment hierarchies coincide. The nearest classification results in the literature are stated under exactly this hypothesis; this paper works past it. Once the Jacobian changes sign, the critical circle acquires a caustic as its image, part of the plane is covered several times with opposite orientations, and the hierarchies part company: integrating a radial weight against the oriented surface element $n_z$ lets overlapping sheets cancel, giving the degree-weighted moments $J_n$, while integrating against $|n_z|$ lets every sheet count positively, giving the multiplicity-weighted moments $J_n^{\pm}$. These are the classical degree and area formulas for the same map, the second weighted by the multiplicity function (Banach indicatrix). It is the second family, the one the caustic threatens, for which a closed form is no longer automatic. Our main result is that it has one anyway: $J_n^{\pm}$ is the total variation of the very same explicit primitive whose endpoint difference gives $J_n$. Crossing the caustic therefore costs only finitely many extra evaluation points, and they are the same points for every order $n$; for a monotone profile there is at most one interior breakpoint, at the critical radius itself. The family carrying this is the two-monomial planar harmonic mapping $F(w)=w+iC\bar w^k$, $w=p(u)e^{iv}$, arising as the planar projection of a Fourier-perturbed surface. On the degree-weighted side $J_n$ has a boundary-only closed form for every $n\ge 0$, the zeroth being the oriented $z$-flux. Neither weighted integral is a new object, and the caustic has been located before in the literature on harmonic trinomials; what is new is the closed evaluation of the multiplicity-weighted hierarchy across it.
Comments29 pages. Comments welcome
