Multplicities and degree relative to a set (Q6609494)

Let \(f\) be an analytic function-germ at \(0 \in \mathbb{R}^{n}\). A basic invariant of \(f\) is the order of its Taylor series. A relative analog of this notion for an analytic subset germ \((S, 0)\) of \((\mathbb{R}^{n} ,0)\) is the following: the largest exponent \(q\) such that the inequality \(\left \vert f(x)\right \vert \leq M\| x \|^{q}\) holds on \((S ,0)\) for some constant \(M\). It is called the multiplicity of \(f\) relative to \((S, 0)\) and denoted by \(\operatorname{mult}_{S}f\). \N\NThe main result is the existence of ``testing'' curves for polynomials of degree \(d\) i.e. there is a semialgebraic curve \(\Gamma_{d}\) (depending only on \(S\)) such that for any polynomial \(P_{d}\) of degree \(d\) the equality holds \(\operatorname{mult}_{S}P_d=\operatorname{mult}_{\Gamma_{d}}P_d\). Analogous results are given for semi-algebraic sets \(S\) at infinity.