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

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 \(\Gamma\).


32S05 Local complex singularities
58K20 Algebraic and analytic properties of mappings on manifolds
Full Text: DOI