The paper develops second Thomae theorem in hyperelliptic case. The main formula, called general Thomae formula, provides expressions for values at zero of the lowest non-vanishing derivatives of theta functions with singular characteristics of arbitrary multiplicity in terms of branch points and period matrix. We call these values derivative theta constants. First and second Thomae formulas follow as particular cases. Some further results are derived. Matrices of second derivative theta constants (Hessian matrices of zero-values of theta functions with characteristics of multiplicity two) have rank three in any genus. Similar result about the structure of order three tensor of third derivative theta constants is obtained, and a conjecture regarding higher multiplicities is made. As a byproduct, a generalization of Bolza formulas are deduced.

中文翻译：

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奇异半周期的通导 Thomae 公式

该论文在超椭圆情况下发展了第二个托迈定理。主要公式称为通用托马公式，它提供了在分支点和周期矩阵方面具有任意重数的奇异特征的 theta 函数的最低非零导数的零处值的表达式。我们将这些值称为导数 theta 常数。第一个和第二个 Thomae 公式遵循特殊情况。得出了一些进一步的结果。二阶导数 theta 常数矩阵（具有多重性 2 特征的 theta 函数零值的 Hessian 矩阵）在任何属中都排在第三位。对于三阶导数θ常数的三阶张量的结构，得到了类似的结果，并对更高的多重性进行了猜想。作为副产品，推导出了 Bolza 公式的推广。