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''in the complex domain,

$$ \tag{1 }
\sum_{j=0} ^ { n } 
P _ {j} ( z) w ^ {( n - j ) }  =  0 ,\ \ 
P _ {0} ( z)  =  1
$$

''

The number  $  r = k + 1 $, 
where

$$ 
k  =  \max _ {1 \leq  j \leq  n }  
\frac{n _ {j} }{j}
 ,
$$

if the coefficients in equation

are series which are convergent for large  $  | z | $:

$$ 
P _ {j} ( z)  = \ 
\sum _ { m = - \infty } ^ { {n _ j } } p _ {jm} z  ^ {m} ,\ \ 
j = 1 \dots n .
$$

The concept of rank is used only when  $  z = \infty $
is a [[Singular point|singular point]] of the differential equation . The rank of the differential equation is also called the rank of the singular point  $  z = \infty $. 
If this point is a [[Regular singular point|regular singular point]], then  $  r = 0 $; 
if it is an [[Irregular singular point|irregular singular point]], then  $  r > 0 $. 
The number  $  k $
is called the subrank. The rank of the equation is an integer or a rational number. If the subrank is rational with denominator  $  q \geq  2 $, 
then the subrank of the equation obtained from

by the change of variable  $  z = \zeta  ^ {q} $
is an integer. The rank of the equation is invariant with respect to a change of variable of the form  $  z = \zeta \phi ( \zeta ) $, 
with  $  \phi $
a holomorphic function at the point  $  \zeta = \infty $
that is non-zero at this point.

The concept of the rank of an equation is used in investigating the structure of the solutions to equation

with a singular point at infinity. Let  $  Q ( z) $
be a polynomial of degree  $  p $, 
let

$$ 
\Psi ( \zeta )  = \sum_{m=0} ^  \infty \psi _ {m} \zeta  ^ {-m}
$$

be a formal series, and let  $  s \geq  1 $
be an integer. The series

$$ \tag{2 }
w  =  e ^ {Q ( z  ^ {1/s} ) } z  ^  \rho  \Psi ( z  ^ {1/s} )
$$

is a normal (subnormal, respectively) series of order  $  p / s $
if  $  s = 1 $(
$  s \geq  2 $, 
respectively). A solution to equation

which is represented by a normal (subnormal) series, convergent in a neighbourhood of  $  z = \infty $, 
is called a normal (subnormal) solution of the same order (see [[#References|[2]]], [[#References|[3]]]).

The order of a normal (subnormal) solution does not exceed the rank of the equation; this is true also for formal solutions of the form (2). If the rank  $  r $
of equation

is an integer, then

has at least one formal solution of the form (2) of order  $  r $. 
The substitution  $  w ( z) = e ^ {Q ( z) } u ( z) $
does not alter the rank of the equation. If the subrank is  $  k = p / q $, 
where  $  p , q $
are mutually prime integers and  $  q \geq  2 $, 
then the equation has no less than  $  q $
formal solutions of the form (2) of order  $  r $.

A Hamburger equation is an equation

with rational coefficients which has exactly two singular points: a regular one  $  z = 0 $
and an irregular one  $  z = \infty $. 
For a Hamburger equation one can obtain sufficient conditions for it to have normal solutions; when  $  n = 2 $
there are necessary and sufficient conditions for the existence of normal and subnormal solutions (see [[#References|[2]]]).

One also introduces the concept of rank in the case when equation

has a finite singular point (see [[#References|[2]]], [[#References|[3]]]).

In the case of a linear system of  $  n $
ordinary differential equations in the complex domain,

$$ \tag{3 }
w  ^  \prime   =  z  ^ {r} A ( z) w ,
$$

where  $  r \geq  - 1 $
is an integer and the matrix-function  $  A ( z) $
is holomorphic at  $  z = \infty $
and  $  A ( \infty ) \neq 0 $, 
the number  $  r + 1 $
is called the rank of the system (3) or the rank of the singular point  $  z = \infty $, 
the number  $  r $
is its subrank (see [[#References|[4]]]–[[#References|[6]]]). If  $  r = - 1 $, 
then the point  $  z = \infty $
is a regular singular point; in contrast to a scalar equation , the point  $  z = \infty $
can be a regular singular point if  $  r > - 1 $(
see [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,   "Sur les intégrales irregulières des équations linéaires"  ''Acta Math.'' , '''8'''  (1866)  pp. 295–344</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.Ya. Latysheva,   N.I. Tereshchenko,   G.S. Orel,   "Normally regular solutions and their applications" , Kiev  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Kamke,   "Handbuch der gewöhnliche Differentialgleichungen" , Chelsea, reprint  (1947)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  W. Wasov,   "Asymptotic expansions for ordinary differential equations" , Interscience  (1965)</TD></TR></table>

====Comments====
For the word rank sometimes the term grade is used. The following can be proved, see [[#References|[a1]]]. For every  $  r $
a solution  $  w( z) $
exists for which

$$ 
\lim\limits _ {| z | \rightarrow \infty }  | z |  ^ {-} r  \mathop{\rm log}  | w( z) |  >  0 ,
$$

except for a finite number of directions.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hille,   "Lectures on ordinary differential equations" , Addison-Wesley  (1969)</TD></TR></table>