Hi,
Almost certainly ivory. But I'd rather hold it to decide.
No one, to my knowledge, has come up with a synthetic that reproduces the Schreger lines characteristic of elephant and mammoth ivory. The elephant Schreger lines are at >115 degree angle, as seen in these balls. Mammoth at 90 degrees. Careful examination shows areas where the lines are off from parallel. Any synthetic likely would be perfect.
You can use a UV ("Black Light") and look for fluorescence, which is characteristic of most plastics and resins. Then the hot pin test - ivory chars, plastics melt, smoke and smell.
Density is also important. When I tested a synthetic cue ball on an electron micropobe I came up with lots of barium, undoubtedly barium sulfate used to increase the ball density. Ivory has a density of 1.85; most plastics are less dense than water (they are made from oil, and oil floats on water). Typical pool ball density is 1.7. Plastics can be dense - the teflon in my labware is 2.2, and very expensive. The phenolic resins used for billiard balls have a density around 1.3.
Take care,
Nick