阿贝尔积分矩阵行列式的一个显式公式
An explicit formula for the determinant of the Abelian integral matrix
摘要：
本文考虑了n+1>1次两复变数的多项式h（x，y）具有一般的高次齐次部分.其非奇异水平曲线h（x，y）=t的第一个同调群的秩为n*n。对于变量空间中的每一个1-形式和同调群的生成元，我们可以把该形式沿生成元的（阿贝尔）积分联系起来。阿贝尔积分是t中的多值函数。对于n*n个单项1-形式的固定标准元组，我们考虑t中的多值方阵函数，其元素是沿着生成元的形式的阿贝尔积分。它的行列式不依赖于同调群中生成元的选择（直到改变符号，这对应于反转方向的生成元系统的改变）。1999年Yu.S.Ilyashenko证明了阿贝尔积分矩阵的行列式是t中n*n次多项式，其零点是h的临界值。我们给出了行列式的一个显式公式。
英文摘要：
We consider a polynomial h(x,y) in two complex variables of degree n+1>1 with a generic higher homogeneous part. The rank of the first homology group of its nonsingular level curve h(x,y)=t is n*n. To each 1- form in the variable space and a generator of the homology group one associates the (Abelian) integral of the form along the generator. The Abelian integral is a multivalued function in t. For a fixed canonic tuple of n*n monomial 1- forms we consider the multivalued square matrix function in t whose elements are the Abelian integrals of the forms along the generators. Its determinant does not depend on the choice of the generators in the homology group (up to change of sign, which corresponds to change of generator system that reverses orientation). In 1999 Yu.S.Ilyashenko proved that the determinant of the Abelian integral matrix is a polynomial in t of degree n*n whose zeroes are the critical values of h. We give an explicit formula for the determinant.
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