Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence of Nontrivial Negative Resonances for Polynomial Ordinary
  Differential Equations With Painlev\'e Property   The Painlev\'e classification is one of the central problems in analytics
theory of differential equations rooted in the XIX century. Although it saw
many significant advances in analyzing certain classes of equations, the
classification still remains an open problem especially for the higher-order
equations. One of the main classical methods of Painlev\'e analysis is based on
considering the resonance numbers corresponding to the possible indices of
arbitrary coefficients in the Laurent expansion of the general solution in a
neighborhood of a movable singularity. Complex and non-integer values of
resonance numbers point out to existence of the movable critical singularities
and positive integer numbers could be used to construct the said general
solution. Also the equation always possesses at least one negative resonance
number of $-1$ which corresponds to an arbitrary position of a movable pole.
However our understanding of the role of nontrivial negative resonances
different from $-1$ remains limited in spite of certain recent methodological
advances related to it. And though in the lower-order classifications built so
far such equations with nontrivial negative resonances have rather been a
special case, the result of present work demonstrates that negative resonances
are in fact common for the higher degree ordinary differential equations with
Painlev\'e property. Specifically we'll prove that their presence is the
necessary condition of the Painlev\'e property for the equations with degree of
the leading terms higher than two.

14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con