to have power over someone or something: to rule a province or a kingdom, to get the better of a victim, or to place another person in charge.
This Aramaic verb is not about titles. It asks who actually has the power in a situation, and it answers with whoever is on top at the end of the story. That is why the book of Daniel can use one and the same verb for an empire that governs the whole earth and for a fire that fails to touch three men.
The range that follows from this is wider than it first looks. A bronze kingdom rules the earth, a foreign exile is promoted to rule as third in Babylon, lions get the better of the men thrown down to them, and fire has no power at all over the bodies it was heated for. Kings, animals, and flames are all measured by the same test: did they in fact prevail?
Twice the verb runs the other direction and says that someone was put in power by another: Nebuchadnezzar over every living creature, Daniel over the province of Babylon. Both times the power comes down from someone higher up, which is exactly the argument the book is making about who hands out authority in the first place.
The rest of this word family fills in the picture. The adjective (שַׁלִּיט) says of a person that he is the one with the say, and the noun (שָׁלְטָן) names the ruling power itself as something that can be given, taken back, or made to last forever. This verb is the moment that power is actually exercised.