Context: The Eastern Roman Empire – today conventionally named the Byzantine Empire, a name not in use during its own time – became increasingly influenced by Greek culture after the 7th century, when Emperor Heraclius (AD 575 - 641) decided to make Greek the empire's official language. Certainly from then on, but likely earlier, the Roman and Greek cultures were virtually fused into a single Greco-Roman world. Although the Latin West recognized the Eastern Empire's claim to the Roman legacy for several centuries, after Pope Leo III crowned Charlemagne, king of the Franks, as the "Roman Emperor" on 25 December 800, an act which eventually led to the formation of the Holy Roman Empire, the Latin West started to favour the Franks and began to refer to the Eastern Roman Empire largely as the Empire of the Greeks (Imperium Graecorum).

Question: When did the Byzantine Kingdom come under the impact of the Greeks. 
Answer: influenced by Greek culture after the 7th century,

Question: Who ruled Rome during this time period ?
Answer: Emperor Heraclius (AD 575 - 641)

Question: What decision did he make that influenced the culture in an attempt to make things more homogeneous ?
Answer: decided to make Greek the empire's official language.

Question: What made the Eastern Empire be known as the Imperium Graecorum ?
Answer: Pope Leo III crowned Charlemagne, king of the Franks, as the "Roman Emperor"

Question: Who was favored by those who lived in the west and mainly used another form of well know verbal language ?
Answer: the Latin West started to favour the Franks

Question: When did the Byzantine Kingdom not come under the impact of the Greeks?
Answer: Unanswerable

Question:  Who didn't rule Rome during this time period?
Answer: Unanswerable

Question: What decision did he make that influenced the culture in an attempt to make things less homogeneous?
Answer: Unanswerable

Question:  What made the Eastern Empire be known as the non-Imperium Graecorum?
Answer: Unanswerable

Question: Who was favored by those who lived in the west and mainly used another form of not well known verbal language?
Answer: Unanswerable