### D.6.2 arcpoint_lib

Library:
  arcpoint.lib
Purpose:
  Truncations of arcs at a singular point
Author:
  Nadine Cremer cremer@mathematik.uni-kl.de

Overview:
  An arc is given by a power series in one variable, say t, and truncating it at a positive integer i means cutting the t-powers > i. The set of arcs truncated at order <bound> is denoted Tr(i). An algorithm for computing these sets (which happen to be constructible) is given in [Lejeune-Jalabert, M.: Courbes trac'ees sur un germe d'hypersurface, American Journal of Mathematics, 112 (1990)]. Our procedures for computing the locally closed sets contributing to the set of truncations rely on this algorithm.

Procedures:

> D.6.2.1 nashmult

> determines locally closed sets relevant for computing truncations of arcs over a hypersurface with isolated singularity defined by f. The sets are given by two ideals specifying relations between coefficients of power series in t. One of the ideals defines an open set, the other one the complement of a closed set within the open one. We consider only coefficients up to t^<bound>. Moreover, the sequence of Nash Multiplicities of each set is displayed

> D.6.2.2 removepower |  | modifies the ideal I such that the algebraic set defined by it remains the same: removes powers of variables
> D.6.2.3 idealsimplify |  | further simplification of I in the above sense: reduction with other elements of I. The positive integer <maxiter> gives a bound to the number of repetition steps
> D.6.2.4 equalJinI |  | tests if two ideals I and J are equal under the assumption that J is contained in I. Returns 1 if this is true and 0 otherwise

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