On finiteness theorems of polynomial functions

TL;DR

This paper establishes finiteness theorems for semialgebraic triviality and local types of Nash and polynomial functions, extending previous results to broader classes and confirming Fukuda's conjecture.

Contribution

It generalizes Benedetti-Shiota's finiteness theorem to Nash families and proves Fukuda's finiteness conjecture for polynomial function germs.

Findings

Finiteness of semialgebraic RL triviality for Nash functions.

Finiteness of local R types in polynomial function spaces.

Confirmation of Fukuda's finiteness conjecture.

Abstract

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.