Recognition of unimodal map germs from the plane to the plane by invariants

Abstract

In this paper, we characterize the classification of unimodal maps from the plane to the plane with respect to [Formula: see text]-equivalence given by Rieger in terms of invariants. We recall the classification over an algebraically closed field of characteristic [Formula: see text]. On the basis of this characterization, we present an algorithm to compute the type of the unimodal maps from the plane to the plane without computing the normal form and also give its implementation in the computer algebra system Singular.

On the Computation of the Codimension of Map Germs Using the Lie Algebra Associated with a Restricted Left–Right Group

“…In the history of the theory of singularities of map germs, A-equivalence has been the most natural equivalence among map germs from the view point of differential topology. Group A, the tangent space to the orbit under the action of this group and its codimension play an important role in the classification of map germs (see [1][2][3][4][5][6][7][8][9][10][11][12]). In [13], the authors gave an algorithm to compute the codimension of map germs under an A-equivalence.…”

Contact unimodal map germs from the plane to the plane

“…Rieger gave the classification of all A −simple and A −unimodal map germs from the plane to the plane of corank at most 1 in [12,13]. These classifications are characterized in [3,5,11] in terms of certain invariants. In [9], Dimca and Gibson gave the classification of all map germs from the plane to the plane of Boardman symbol (2,1) and (2,2) with respect to K -equivalence.…”