I know that Roman Empire built some of the most beautiful things for the epoch and also had an incredible culture (law, poetry, ...).
However, my question is: Why there are there no known Roman mathematicans from the Roman Empire? All the famous roman were emperors, soldiers, lawyers, but not mathematicians.
The Roman Empire did not consist of Latin-speaking "Romans" only. It included Greece, North Africa and parts of what we now call the Middle East. Probably, a majority of the inhabitants of the Roman Empire were Greek-speaking. And there were mathematicians among them. Menelaus, Ptolemy and Diophantus were living in the Roman Empire. Some of them could be Roman citizens, some not. We know practically nothing about their lives. Most likely all were Roman citizens after the citizenship was granted to all residents of the Empire.
So the correct questions would be:
Why did Roman mathematicians write in Greek?
Why did most of them prefer Northern Africa for their residence?
And the answer is that there was a tradition among the Greeks, in Greek language, especially in the Middle East, especially in Alexandria. Tradition means schools, libraries, scientific environment. This environment was created by the Greeks in Hellenistic times, and partially survived almost to the end of the Roman Empire.
By the way, many Roman works in engineering, applied science (including military science!) were also mostly written in Greek. See a related discussion here: Roman engineers
Greek was the language of science in the Roman Empire. Similar to how Latin was the language of Science in Europe in the 16th to 18th centuries.
EDIT. We really know nothing about Diophantus, except the books signed with this Greek-sounding name. Where did he live? When did he live? His masterpiece looks completely isolated (nothing remotely similar before or after or around his time). Could he create all this out of nothing? Very improbable.
EDIT2. The question implicitly assumes that science and mathematics (in the modern sense) are necessary (or common, or desirable) activities of a civilization.
Far from it. There were no mathematicians in Rome. But there were also no mathematicians in the Parthian Empire, India or China at that time. Mathematics (in the modern narrow sense) was invented by the Greeks, flourished in the Hellenistic world, then declined and vanished, as an activity, and very little of it spread elsewhere. Why should one expect it in Rome, or in Londinium or in the Parthian Empire? This was just some marginal activity of the Greek culture.
It is only recently (since the 17th century) that mathematics became important and mathematical research spread widely.
In ancient Rome doing calculations with Roman numerals was a very cumbersome procedure which impeded the development of mathematics. Interestingly, ancient Rome also had no prominent scientists or philosophers as compared to ancient Greece. Probably science, mathematics and philosophy were not highly regarded in Roman society similarly to ancient China. These activities were left to Greek (slaves), who were also frequently used as teachers.
