A note on time functions associated with effectively hyperbolic double characteristics

TL;DR

This paper examines the geometric properties of effectively hyperbolic singular points with real characteristic roots on one side of t=0, highlighting differences from the case where roots are real on both sides.

Contribution

It provides a detailed analysis of the geometrical aspects of effectively hyperbolic singular points with unilateral real characteristic roots.

Findings

Identifies key differences in geometric behavior between unilateral and bilateral real characteristic roots.

Clarifies the structure of singular points over t=0 in effectively hyperbolic cases.

Enhances understanding of hyperbolic singularities in differential equations.

Abstract

Geometrical aspects of effectively hyperbolic singular points over $t=0$ are discussed assuming that the characteristic roots are real only on the one side $t\geq 0$. In particular, the difference from the case that the characteristic roots are real in both sides $t\geq 0$ and $t<0$.