Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Mappings from R^3 to R^3 and signs of swallowtails   Let M be an oriented 3-manifold. For a generic f \in C^ \infty(M,R^3), there
is a discrete set of swallowtail critical points. In that case, at any
swallowtail point p there exists a well-oriented coordinate system centered at
p, and a coordinate system centered at f(p), such that locally f has the form
f_\pm(x,y,z)=(\pm xy+x^2 z+x^4,y,z), so one may associate with p a sign
I(f,p)\in \{\pm 1\}. A geometric definition of the sign associated with a
swallowtail was recently introduced by Goryunov. We shall show how to compute
the number of swallowtail points having the positive/negative sign, in the case
where f : R^n \rightarrow R^n is a polynomial mapping, in terms of signatures
of quadratic forms.

24
The categories of this paper are: dg-ga
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg