Estimation of Lojasiewicz exponents and Newton polygons
Estimation of Lojasiewicz exponents and Newton polygons
复制标题
Lojasiewicz 指数和牛顿多边形的估计
DOI：
10.1007/bf01389274
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发表时间：
1981
影响因子：
3.1
通讯作者：
B. Lichtin
中科院分区：
文献类型：
--
作者：
B. Lichtin
通过引入牛顿多边形f +(f)，简化了复解析映射芽f （C”，0）—，（C, 0）的孤立奇异拓扑不变量的计算。对于那些相对于其多边形“非简并”的芽f，可以通过“对偶多边形”构造对C”的修正。这使得我们可以将f置于原点的原像中任意一点的邻域内的法向交叉形式。如果不变量可以用单项式的指数来表示，那么标准的方法就是用多边形的几何特征来解释这些指数。用这种方法，Varcenko计算了一个非简并胚的单算子[-9]的zeta函数和振荡指标[10]。尚未计算是一个不变的“C o程度的充分性”v f我的地图胚芽。这个数字是最小的整数v满足性质的订单大于v不要修改拓扑类型的多项式组成的这些术语在f(订单最多诉更准确地说,让阵线是多项式组成的所有条款与订单最多v f (v th飞机f(0)。那么对于任意的胚芽g，其所有项的阶至少为v+ 1， f~+ g在拓扑上等价于fv(0)，作为C '上映射的胚芽。vi是具有这个性质的最小整数，fvs是“c0充分射流”一个自然的愿望是用F+(F)来表示y。
Computation of topological invariants of an isolated singularity for a complex analytic map germ f:(C", 0)--,(C, 0), has been facilitated by the introduction of the Newton polygon F+(f) of f. For those germs f which are" non-degenerate" with respect to their polygon, a modification of C" can be constructed by" dualizing the polygon." This allows one to place f in a normal crossing form in a neighborhood of any point in the preimage of the origin in C". If the invariant can be expressed when f is in this monomial form in terms of the exponents of the monomials then the standard procedure is to interpret these exponents in terms of the geometrical characteristics of the polygon. In this way, Varcenko has computed the zeta function of the monodromy operator [-9] and the oscillatory index [10] for a nondegenerate germ. One invariant as yet not computed is the" C o degree of sufficiency" v I for the map germ f. This number is the smallest integer v satisfying the property that terms of order greater than v do not modify the topological type of the polynomial consisting of those terms in f of order at most v. More precisely, let fv be the polynomial consisting of all terms in f with order at most v (" v th jet of f at 0"). Then for any germ g all of whose terms have order at least v+ 1, f~+ g is topologically equivalent to fv at 0, as germs of maps on C". v I is the smallest integer with this property, fvs is a" C o sufficient jet." A natural desire is to express vy in terms of F+(f).