On two four term arithmetic progressions with equal product

We investigate when two four-term arithmetic progressions have an equal product of their terms. This is equivalent to studying the (arithmetic) geometry of a non-singular quartic surface. It turns out that there are many polynomial parametrizations of such progressions, and it is likely that there exist polynomial parametrizations of every positive degree. We find all such parametrizations for degrees 1 to 4, and give examples of parametrizations for degrees 5 to 10.