## 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 $$\Gamma$$.

### MSC:

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

### Keywords:

Łojasiewicz exponent; Newton polyhedra; analytic functions
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