On finiteness theorems of polynomial functions

Let d be a positive integer. We show a finiteness theorem for semialgebraic RL triviality of a Nash family of Nash functions defined on a Nash manifold, generalising Benedetti-Shiota's finiteness theorem for semialgebraic RL equivalence classes appearing in the space of real polynomial functions of degree not exceeding d. We also prove Fukuda's claim, Theorem 1.3, and its semialgebraic version Theorem 1.4, on the finiteness of the local R types appearing in the space of real polynomial functions of real polynomial function germs of degree not exceeding d.