# Arnold's problem on monotonicity of the Newton number for surface singularities

## TL;DR

This paper provides a complete characterization of when the Newton number remains monotonic for surface singularities, solving Arnold's problem from 1982 by relating Newton polyhedra and their associated invariants.

## Contribution

It introduces a simple condition based on Newton polyhedra that characterizes the equality of Newton numbers for surface singularities, addressing Arnold's longstanding problem.

## Key findings

- Characterization of Newton number equality in terms of Newton polyhedra
- Complete solution to Arnold's problem for surface singularities
- Conditions for monotonicity of the Newton number

## Abstract

According to the Kouchnirenko theorem, for a generic (precisely non-degenerate in the Kouchnirenko sense) isolated singularity $f$ its Milnor number $\mu (f)$ is equal to the Newton number $\nu (\Gamma_{+}(f))$ of a combinatorial object associated to $f$, the Newton polyhedron $\Gamma_+ (f)$. We give a simple condition characterising, in terms of $\Gamma_+ (f)$ and $\Gamma_+ (g)$, the equality $\nu (\Gamma_{+}(f)) = \nu (\Gamma_{+}(g))$, for any surface singularities $f$ and $g$ satisfying $\Gamma_+ (f) \subset \Gamma_+ (g)$. This is a complete solution to an Arnold's problem (1982-16) in this case.