Given a hyperelliptic hyperbolic surface S of genus g ≥ 2, we find bounds on the lengths of homologically independent loops on S. As a consequence, we show that for any λ ∈ (0, 1) there exists a constant N(λ) such that every such surface has at least \(\left\lceil {\lambda \cdot {2 \over 3}g} \right\rceil \) homologically independent loops of length at most N(λ), extending the result in [Mu] and [BPS]. This allows us to extend the constant upper bound obtained in [Mu] on the minimal length of non-zero period lattice vectors of hyperelliptic Riemann surfaces to almost \({2 \over 3}g\) linearly independent vectors.

给定一个超椭圆双曲曲面S ，且属g ≥ 2，我们发现S上同调独立环的长度有界。因此，我们证明对于任何 λ ε (0, 1) 都存在一个常数N (λ)，使得每个这样的表面至少有\(\left\lceil {\lambda \cdot {2 \over 3}g } \right\rceil \)长度至多为N (λ) 的同源独立环，将结果扩展为 [Mu] 和 [BPS]。这使我们能够将超椭圆黎曼曲面的非零周期晶格向量的最小长度上以 [Mu] 为单位获得的常数上限扩展到几乎\({2 \over 3}g\)个线性无关向量。