[dalke]

It's clear you aren't a mathematician. They work with "typical integers" like 10^9000 all the time, and certainly not as a trivial scaling of doubles.

There are 9763 significant digits in that calculation. They wrote it in shorter form because the actual digits were irrelevant. The numbers they got from Mathematica had the wrong sign and were off six orders of magnitude. There's no reason to list more digits to show that's the case.

Mathematica promises arbitrary precision. Here's the documentation. http://reference.wolfram.com/language/ref/Det.html . It says "Use exact arithmetic to compute the determinant", for the construction given in the paper. The also submitted a bug report, and got the statement "It does appear there is a serious mistake on the determinant operation you mentioned."

Yes, of course random algorithms can end up with different answers. The determinate definition is not random, the documentation for Det doesn't say it uses a random implementation, non-random methods to compute it are well known. Again, the vendor says it's wrong - why are you disagreeing with practicing mathematicians and the vendor? Why do you think you know enough about the topic to be able to judge what "odd" means, in the context of this sort of field?