The Codimension of a singularity

[Kreizhn]

This may seem like a foolish question, but I can't seem to find the answer anywhere. Also, please forgive the question if it is ambiguous but the context in which it arises is not clear to me:

There is a mapping $H(x,p,\cdot): \mathbb R \to \mathbb R$ with x,p fixed, which attains its maxima at K distinct points $u_k, k \in\left\{1,\ldots, K\right\}$. Each point $u_k$ is a critical point with a singularity of codimension $c_k$.

What is the codimension of a singularity?

I believe the author plans on later generalizing this for a mapping $H:T^*M\times\mathbb R \to \mathbb R$ for smooth mfld M, so if you could explain it in that context it would be helpful.

 

[Kreizhn]

Perhaps the fact that the manifolds are R and that the word singularity is used is what is throwing me off.

I know that a regular point means that the pushforward is surjective. So is the codimension of a critical point the dimension of the relative complement of the image of the pushforward?

 

[mathwonk]

surely the author defines his own terms.

 

[Kreizhn]

If they did, it was very subtly mentioned. I've been reading this book from the beginning and have not seen any mention of it. I shall go back and look closer.

The thing is, it does not seem to be an obscure term. I quick search of google scholar, for example, yields many papers that talk about "Codimension-n singularities" where n seems to be most often one, two, or three. Unfortunately, the papers often seem to define the codimension based on some obscure sets or assume that the reader already has knowledge of singularity codimension. For this reason I was hoping that perhaps I was just unaware of existing terminology.

 

[Kreizhn]

Also, while I have found this book to be generally very valuable, I have found it to be very poorly written. It is

"Singular Trajectories and their Role in Control Theory" by Bonnard and Chyba

The book is great for people who already have a working background knowledge in the field, but there is a dearth of definitions.