Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Horizontal gradient of polynomial functions for the standard Engel
  structure on $\mathbb R^4$   We investigate the set $V_f$ of horizontal critical points of a polynomial
function $f$ for the standard Engel structure defined by the 1-forms
$\omega_3=dx_3-x_1dx_2,$ $\omega_4=dx_4-x_3dx_2$, endowed with the
sub-Riemannian metric $g_{SR}=dx_1^2+dx_2^2$. For a generic polynomial, we show
that the intersection of any fiber of $f$ and $V_f$ does not contain a
horizontal curve. Then we prove that each trajectory of the horizontal gradient
of $f$ approaching the set $V_f$ has a limit.

24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom