Łojasiewicz exponents, the integral closure of ideals and Newton polyhedra. (English) Zbl 1040.32024

Bivià-Ausina, Carles

J. Math. Soc. Japan 55, No. 3, 655-668 (2003).

The paper deals with the growth order of analytic functions (both real and complex) of \(n\) variables. The main objective is the so called Łojasiewicz exponent \(l(h,I)\) of a germ \(h\) with respect to an ideal \(I\). If \(h\) is a germ at the origin, \(0 \in V (I) \subset V(h)\) and \(I= \langle g_1, \dots , g_s \rangle \), then \[ l(h,I)= \inf \{ \theta : \exists \text{open }U \ni 0 \;\exists C>0 \;\forall x \in U, \;| h(x)| ^\theta \leq C \sup_i | g_i(x)| \}. \] This notion extends for ideals. If \(J=\langle h_1, \dots , h_t \rangle\) then \[ l(J,I)=\max_j\{ l(h_j,I) \}. \] The main goal of the paper is to show that under certain nondegeneracy conditions \(l\) can be estimated in terms of Newton polyhedra and weights associated to their \(n-1\) dimensional faces. Namely the author shows that:
If the system of generators \(g_1, \dots , g_s\) of the ideal \(I\) is adapted to the Newton polyhedron \(\Gamma\) then \[ l(x^k,I) \leq \max_j \frac{ \max_i \{l_j(g_i)\}}{l_j(x^k)}, \] where \(x^k\) is a monomial, \(x^k=x_1^{k_1} \cdot \dots \cdot x_n^{k_n} \), and \(l_j\) is a weight associated to the \(j\)th \(n-1\)-dimensional face of \(\Gamma\).
If the system of generators \(g_1, \dots , g_s\) is strongly adapted, \(J=\langle h_1, \dots , h_t \rangle\), \(\{0\}=V(I) \subset V(J)\), then \[ l(J,I) \leq \max_j \frac{ \max_i \{l_j(g_i)\}}{\min_i \{l_j(h_i) \}}. \] System \(g_1, \dots , g_s\) is called adapted to \(\Gamma\) if for every compact face of \(\Gamma\) the corresponding leading parts \[ q(g_1), \dots q(g_s), \] have no common zeros in \((K\setminus \{0\})^n\).
The adapted system of \(g_i\)’s is called strongly adapted if \(\Gamma\) is intersecting all coordinate axes and for every coordinate subspace \(x_{j_1}= \dots =x_{j_k}=0\) the system of restricted \(g_i\)’s is adapted to the restriction of \(\Gamma\).

Reviewer: Piotr Jaworski (Warszawa)

MSC:

32S05	Local complex singularities
58K20	Algebraic and analytic properties of mappings on manifolds

Keywords:

Łojasiewicz exponent; Newton polyhedra; analytic functions