There is a problem with the timing, too, which could also be an artifact of the book having been redacted long years after the event.That about sums up the historical and philological issues. The only thing I would add is that the lack of Greek language or other Greek influence in the book may be more a cultural than a chronological feature. Most of the Hebrew sectarian texts from the Dead Sea Scrolls are similarly devoid of Greek influence even though they were written well into the Hellenistic period. They just didn't like Greek culture. Esther too may have been written in the Hellenistic period by a writer with a similar aversion.
The Book of Esther says Mordechai was exiled from Judah with King Jeconiah: "Now in Shushan the palace there was a certain Jew, whose name was Mordechai… Who had been carried away from Jerusalem with the captivity which had been carried away with Jeconiah king of Judah, whom Nebuchadnezzar the king of Babylon had carried away" (Esther 2:5-6).
But King Jeconiah was expelled from Judah by Nebuchadnezzar 130 years before Xerxes ascended the throne. Surely a contemporary writer like Mordechai would have known that.
And then there’s the language of the book. On the one hand, the fact that no Greek influence made it into Esther is strong evidence that the book was written before the Achaemenid Dynasty was destroyed by Alexander the Great in 330 B.C.E., marking the start of the Hellenistic period.
On the other hand, the form of the Hebrew of the book, and, even more so, the form of the many Persian loanwords embedded in it, both indicate that the book was likely written toward the end of the Achaemenid Dynasty.
Taken together, the evidence of the vagueness about the king and the timeline problems, and the language, indicate that the redaction was by a Jewish scribe writing in Shushan in the middle of the 4th century B.C.E., about events that apparently happened more than a century before.
The rest of the article summarizes various interesting scholarly speculations about the book and is worth a read.