Vanishing Abelian integrals for balanced and unbalanced cycles

Let $F(x,y)=y^2+f(x)$ with $f\in\mathbb{C}[x]$, let $\kappa\in\mathbb{C}[x]$, and let $g(x)$ be the function associated to $\kappa$ as in the source. Let $\gamma(t)$ be a cycle of $F$; its associated zero-cycle is $\phi(\gamma(t))$, and call it trivial, balanced, unbalanced, or totally unbalanced according to the corresponding zero-cycle. For polynomial decompositions, define

$$H_k(x,y)=(h_k(x),y).$$

Vanishing-integral conjecture. (i) If $\gamma(t)$ is totally unbalanced, then

$$\int_{\gamma(t)}\kappa(x)y\,dx\equiv 0$$

if and only if there exist $f_k,h_k\in\mathbb{C}[x]$ and functions $g_k$ analytic in a neighborhood of infinity such that

$$f=f_k\circ h_k\quad (k=1,\ldots,s),\qquad g=g_1\circ h_1+\cdots+g_s\circ h_s,$$

and every cycle $H_k(\gamma)$ is trivial. (ii) If $\gamma(t)$ is unbalanced, then the vanishing condition stated in the source, namely

, holds if and only if there exist $f_k,h_k\in\mathbb{C}[x]$ and functions $g_k$ analytic in a neighborhood of infinity satisfying the same decomposition conditions, and for every $k$ the projected cycle $H_k(\gamma)$ is either trivial or balanced and

\int_{h_k(\phi(\gamma(t)))}g_k\equiv 0.

These conditions are proposed as an analogue of the cited monodromy theorem, characterizing vanishing Abelian integrals on zero-dimensional cycles through polynomial decompositions and trivial or balanced projected cycles. The parser supplies no resolution evidence; the claim is therefore recorded as open. The source uses undefined references to $g$,

, and the precise indexing of the decompositions, so those details should be checked against the paper.