预解度、希尔伯特第13问题与几何
Resolvent degree, Hilbert's 13th Problem and geometry
摘要：
我们发展了由Brauer \cite{Br}引入的预解度理论，以研究多项式根公式的复杂性，并给出希尔伯特第13问题的精确公式。我们将这一理论推广到代数几何中的计数问题，并将其视为有限群的一个内在不变量。作为这一观点的一个应用，我们证明了希尔伯特第13问题，以及他的六次和八次猜想，等价于各种计数几何问题，例如在光滑的三次曲面上求直线或在光滑的平面四次曲面上求双切线的问题。
英文摘要：
We develop the theory of resolvent degree, introduced by Brauer \cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilbert's 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.
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