AI 中文总结
本文证明Arnold边界奇点的判别式自然出现在积分几何中，作为体积函数的不规则集，并通过Milnor纤维丛变体和偶Petrovsky类刻画其有限叶性，为欧几里得域代数可积性提供新障碍。
AI 中文摘要
本文证明了由V.~I.~Arnold引入的边界奇点的判别式自然地出现在积分几何中。它们表示体积函数的不规则集。该函数将仿射超平面映射到欧几里得空间中由该超平面切割出的域的部分体积。对于孤立边界奇点，边界的作用被证明与射影空间中的无穷远超平面相关。定义并研究了适用于当前问题的Milnor纤维丛的变体。引入了局部偶Petrovsky类的类似物并进行了计算。证明了仅当这些类平凡时，体积函数才是局部有限叶的。对于简单边界奇点——即在分类中没有模的奇点——列出了判别式补集的此类分量（拟腔隙）。所得结果建立了欧几里得空间中域的代数可积性的新障碍。
英文摘要
It is shown that the discriminants of boundary singularities introduced by V.~I.~Arnold naturally arise in integral geometry. They represent the irregularity sets of the volume function. This function maps an affine hyperplane to the volume of the part of a domain in the Euclidean space cut off by this hyperplane. For isolated boundary singularities, the role of the boundary is shown to be related to the hyperplane at infinity in the projective space. Versions of the Milnor fiber bundle adapted to the present problem are defined and studied. Analogs of the local even Petrovsky class are introduced and computed. It is proved that the volume function is locally finite-sheeted only if these classes are trivial. Such components of the complements to the discriminants (quasilacunas) are listed for simple boundary singularities --- those having no moduli in the classification. The obtained results establish new obstructions to the algebraic integrability of domains in a Euclidean space.
