Useful Facts about RLCT
نویسنده
چکیده
In this paper, we will be working with rings and ideals of real analytic functions. Given x ∈ R, let Ax be the ring of real-valued functions f : R → R that are analytic at x. When x = 0 is the origin, it is useful to think of A0 as a subring of the formal power series ring R[[ω1, . . . , ωd]] = R[[ω]] which consists of power series which are convergent in some neighborhood of the origin. For all x ∈ R, Ax is isomorphic to A0 via a translation. Given a subset Ω ⊂ R, let AΩ denote the ring of real functions analytic at each point x ∈ Ω. Locally, each function can be represented as a power series centered at x. Given f ∈ AΩ, we define the analytic variety VΩ(f) = {ω ∈ Ω : f(ω) = 0} while for an ideal I ⊂ AΩ, we set VΩ(I) to be the intersection of VΩ(f) over all f ∈ I. Let ∇f denote the gradient of f and ∇f its Hessian. Given a symmetric matrix A ∈ Rn×n, we write A 0 if A is positive definite, and A 0 if A is positive semidefinite. Let Ω be a compact subset of R. Let us assume that Ω is semianalytic, i.e. Ω = {x ∈ R : g1(x) ≥ 0, . . . , gl(x) ≥ 0} is defined by real analytic inequalities. Here, the functions gi(x) only have to be real analytic at points on the boundary where they are active. Let I = 〈f1, . . . , fr〉 be the ideal generated by functions f1, . . . , fr in the ring AΩ. Let φ be nearly analytic, i.e. φ is a product φaφs of functions where φa is real analytic on Ω and φs is a smooth and positive on Ω. Definition 1.1 ([4],[6, §7.1]). The following definitions of the real log canonical threshold RLCTΩ(I;φ) = (λ, θ) are equivalent.
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تاریخ انتشار 2012