In this work, we establish conditions under which a family $\{f_t\}$ of functions, possibly with non-isolated singularities, defined on a toric variety gives rise to a Whitney equisingular family of hypersurfaces $\{f^{−1)_t⁡(0)\}$. In a recent and notable work, Eyral and Oka provided an important framework for studying Whitney equisingularity in families with non-isolated singularities in $\mathbb C^n$. Our approach builds directly on their concepts, such as admissibility, local tameness, and the role of Newton boundaries, by extending them to the toric setting. However, because toric varieties may present arbitrary singular sets, these extensions alone are not sufficient. To overcome this, we combine their framework with fundamental properties of Whitney stratifications and finite morphisms, thereby extending the classical theory to the broader context of toric varieties. Moreover, when $\mathbb C^n$ is viewed as a toric variety, our conditions recover exactly those obtained by Eyral and Oka.

This is a joint work with Danilo da Nóbrega Santos