Monodromy of complete intersections and surface potentials

TL;DR

This paper investigates the monodromy groups associated with Newtonian potentials of algebraic hypersurfaces, revealing conditions under which the attraction force is algebraic or non-algebraic, with implications for the geometry of these potentials.

Contribution

It analyzes the monodromy of algebraic hypersurfaces' potentials, establishing when the attraction force is algebraic or non-algebraic based on surface degree and dimension.

Findings

Force is algebraic for $n=2$ or $d=2$ outside the surface.

Force is non-algebraic for generic surfaces with $d extgreater 2$, $n extgreater 2$, and $n+d extgreater 8$.

Later work removed the $n+d extgreater 8$ restriction.

Abstract

Following Newton, Ivory and Arnold, we study the Newtonian potentials of algebraic hypersurfaces in $R^n$. The ramification of (analytic continuations of) these potential depends on a monodromy group, which can be considered as a proper subgroup of the local monodromy group of a complete intersection (acting on a {\em twisted} vanishing homology group if $n$ is odd). Studying this monodromy group we prove, in particular, that the attraction force of a hyperbolic layer of degree $d$ in $R^n$ coincides with appropriate algebraic vector-functions everywhere outside the attracting surface if $n=2$ or $d=2$, and is non-algebraic in all domains other than the hyperbolicity domain if the surface is generic and $d\ge 3$ and $n\ge 3$ and $n+d \ge 8$. (Later, W. Ebeling has removed the last restriction $d+n \ge 8$).