On the classification of complex multi-germs of corank one and codimension one

Houston, Kevin

doi:10.7146/math.scand.a-14953

Cited by 37 publications

(28 citation statements)

References 12 publications

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“…Parts (ii) and (iii) follow from the fact that if we complexify the k-jet, then we get a complex analytic corank 1 A e -codimension 1 map-germ and the results are true in this case, see Theorem 3.1 of [5].…”

Section: Proof Of Theorem 31 Parts (Ii) and (Iii)mentioning

confidence: 89%

“…Since f t is primitive for all t the A-orbit of f t is open in its K-orbit, as can be seen from the proof of Lemma 3.6 of [5]. As the family members are all K-equivalent this implies Mather's condition (a).…”

mentioning

confidence: 81%

“…The form is the same as the complex form, see Theorem 3.1 of [5], except that the sum of squares can have negative coefficients. Remark 3.3.…”

Section: Remark 32mentioning

confidence: 99%

“…Recent work has been done in the classification of A e -codimension 1 singularities in [1], [2] and [4]. In [5] A e -codimension 1 corank 1 multi-germs f : (C n , S) → (C p , 0) with n < p and S a finite set have been classified. Explicit normal forms for the mono-germs are given and constructions involving three operations on these mono-germs are used to produce the multi-germs.…”

mentioning

confidence: 99%

“…This is because one of the processes used to create A e -codimension 1 multi-germs from unfolding other codimension 1 germs, the process of binary concatentation (see [2] or [5]), depends on the unfoldings used. Nonetheless, it seems likely that real multi-germs have a form similar to that in the complex case as described in Theorem 6.4 of [5].…”

mentioning

confidence: 99%

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On the classification of real mono-germs of corank one and codimension one

Houston

2004

Geometric Singularity Theory

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Section: Proof Of Theorem 31 Parts (Ii) and (Iii)mentioning

confidence: 89%

mentioning

confidence: 81%

“…The form is the same as the complex form, see Theorem 3.1 of [5], except that the sum of squares can have negative coefficients. Remark 3.3.…”

Section: Remark 32mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On the classification of real mono-germs of corank one and codimension one

Houston

2004

Geometric Singularity Theory

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The extra-nice dimensions

2021

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We define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in $$C^{\infty }(N\times [0,1],P)$$ C ∞ ( N × [ 0 , 1 ] , P ) is dense if and only if the pair of dimensions $$(\dim N, \dim P)$$ ( dim N , dim P ) is in the extra-nice dimensions. This result is parallel to Mather’s characterization of the nice dimensions as the pairs (n, p) for which stable maps are dense. The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have $${\mathscr {A}}_e$$ A e -codimension 1 hyperplane sections. They are also related to the simplicity of $${\mathscr {A}}_e$$ A e -codimension 2 germs. We give a sufficient condition for any $${\mathscr {A}}_e$$ A e -codimension 2 germ to be simple and give an example of a corank 2 codimension 2 germ in the nice dimensions which is not simple. Then we establish the boundary of the extra-nice dimensions. Finally we answer a question posed by Wall about the codimension of non-simple maps.

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Singularities, Bifurcations and Catastrophes

Montaldi

2021

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Suitable for advanced undergraduates, postgraduates and researchers, this self-contained textbook provides an introduction to the mathematics lying at the foundations of bifurcation theory. The theory is built up gradually, beginning with the well-developed approach to singularity theory through right-equivalence. The text proceeds with contact equivalence of map-germs and finally presents the path formulation of bifurcation theory. This formulation, developed partly by the author, is more general and more flexible than the original one dating from the 1980s. A series of appendices discuss standard background material, such as calculus of several variables, existence and uniqueness theorems for ODEs, and some basic material on rings and modules. Based on the author's own teaching experience, the book contains numerous examples and illustrations. The wealth of end-of-chapter problems develop and reinforce understanding of the key ideas and techniques: solutions to a selection are provided.

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On the classification of complex multi-germs of corank one and codimension one

Abstract: Corank one multi-germs f : (C n , S) → (C p , 0), n < p, of A e -codimension one are classified. The mono-germs are given in an explicit normal form and the multi-germs are described in terms of augmentations, and concatenations of mono-germs and a special bi-germ.

Cited by 37 publications

References 12 publications

On the classification of real mono-germs of corank one and codimension one

On the classification of real mono-germs of corank one and codimension one

The extra-nice dimensions

Singularities, Bifurcations and Catastrophes

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