# Uniform stable radius, L\^e numbers and topological triviality for line singularities

## TL;DR

This paper demonstrates that for families of complex polynomial functions with line singularities, a uniform stable radius ensures the invariance of L extsuperscript{e} numbers and, in higher dimensions, guarantees topological triviality of the family.

## Contribution

It establishes the link between uniform stable radius and invariance of L extsuperscript{e} numbers, leading to topological triviality in higher dimensions for line singularities.

## Key findings

- L extsuperscript{e} numbers are independent of parameter t under a uniform stable radius.
- Families of weighted homogeneous line singularities have a uniform stable radius when nearby fibers are uniformly non-singular.
- In dimensions n ≥ 5, constant L extsuperscript{e} numbers imply topological triviality for line singularity families.

## Abstract

Let $\{f_t\}$ be a family of complex polynomial functions with line singularities. We show that if $\{f_t\}$ has a uniform stable radius (for the corresponding Milnor fibrations), then the L\^e numbers of the functions $f_t$ are independent of $t$ for all small $t$. In the case of isolated singularities --- a case for which the only non-zero L\^e number coincides with the Milnor number --- a similar assertion was proved by M. Oka and D. O'Shea. By combining our result with a theorem of J. Fern\'andez de Bobadilla --- which says that families of line singularities in $\mathbb{C}^n$, $n\geq 5$, with constant L\^e numbers are topologically trivial --- it follows that a family of line singularities in $\mathbb{C}^n$, $n\geq 5$, is topologically trivial if it has a uniform stable radius. As an important example, we show that families of weighted homogeneous line singularities have a uniform stable radius if the nearby fibres $f_t^{-1}(\eta)$, $\eta\not=0$, are "uniformly" non-singular with respect to the deformation parameter $t$.