Final answer:
The emperor in Constantinople maintained nominal authority over the Germanic kingdoms that emerged in the west after the fall of the Western Roman Empire in 476 C.E., although this influence gradually diminished.
Step-by-step explanation:
After the year 476 C.E., when the Roman West fell, the Byzantine Empire under the emperor in Constantinople continued to have influence over Western Europe. Even though the Western Roman Empire was no more, the territory was divided into various Germanic kingdoms which recognized the authority of the emperor in Constantinople, at least nominally and until around the seventh century when Byzantine influence waned. Despite the long distance and the emergence of new powers in the West, the acknowledgement of the emperor's authority suggests the continuation of the idea of a unified Roman Empire in some form.
Notably, even the Franks, far away from the Byzantine influence, perceived a genuine threat from a possible Byzantine invasion. The treaties with Constantinople included loopholes that could be exploited, adding an element of unpredictability in their relations. This period marked a transformation in the Roman state, with the Eastern Roman Empire preserving and adapting the Roman legacy within the shifting contours of the medieval world, while the West fragmented into independent kingdoms with evolving allegiances.