We study the sets of semistar and star operations on a semilocal Prüfer domain, with an emphasis on which properties of the domain are enough to determine them. In particular, we show that these sets depend chiefly on the properties of the spectrum and of some localizations of the domain; we also show that, if the domain is h -local, the number of semistar operations grows as a polynomial in the number of semistar operations of its localizations.
