We consider an isolated plane curve singularity and its associated Eisenbud an Neumann diagram. We give an algorithm to compute the maximal spectral value on the diagram and we show that the singularity is topologically equivalent to another singularity for which the maximal spectral value is given by the point $(1,1)$ in the plane of the Newton polygon. From the almost additivity on the splice components of the diagram we compute the sum of the square of the spectral values. This formula with the previous result on the maximal spectral value give us the Hertling conjecture as a corollary.
本文代表了一个步骤，在我们的程序走向证明皮尔斯-伯克霍夫猜想。20世纪80年代，J. Madden证明了环A $is equivalent to a statement about an arbitrary pair of points $ α，β \in \sper \ A $ and their separating ideal $ <α，β> $; we refer to this statement as the Local Pierce-Birkhoff conjecture at $ α，β $. In this paper, for each pair $（α，β）$ with $ ht的Pierce-Birkhoff猜想（<α，β>）= \dim A $, we define a natural number, called complexity of $（α，β）$. Complexity 0 corresponds to the case when one of the points $ α，β $ is monomial; this case was already settled in all dimensions in a preceding paper. Here we introduce a new conjecture, called the Str…
