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A partial differential equation

$$ \tag{* }
F ( t, x, Du )  =  0
$$

where the function  $  F( t, x, q) $
satisfies the following condition: The roots of the polynomial

$$ 
\sum _ {| \alpha | = m }

\frac{\partial  F ( t, x, Du ) }{\partial  q _  \alpha  }

\lambda ^ {\alpha _ {0} } \xi ^ {\alpha  ^  \prime  }
$$

are real for all real  $  \xi $, 
and there exist  $  \xi \neq 0 $, 
$  t $, 
$  x $, 
and  $  Du $
for which some of the roots either coincide or the coefficient of  $  \lambda  ^ {m} $
vanishes. Here  $  t $
is an independent variable which is often interpreted as time;  $  x $
is an  $  n $-
dimensional vector  $  ( x _ {1} \dots x _ {n} ) $; 
$  u ( t, x) $
is the unknown function;  $  \alpha $
and  $  \alpha  ^  \prime  $
are multi-indices,  $  \alpha = ( \alpha _ {0} \dots \alpha _ {n} ) $, 
$  \alpha  ^  \prime  = ( \alpha _ {1} \dots \alpha _ {n} ) $; 
$  Du $
is a vector with components

$$ 
D  ^  \alpha  u  = \ 

\frac{\partial  ^ {| \alpha | } u }{\partial  t ^ {\alpha _ {0} }
\partial  x _ {1} ^ {\alpha _ {1} } \dots
\partial  x _ {n} ^ {\alpha _ {n} } }
 ;
$$

only derivatives of an order not exceeding  $  m $
enter in equation (*); the  $  q _  \alpha  $
are the components of a vector  $  q $; 
$  \xi $
is an  $  n $-
dimensional vector  $  ( \xi _ {1} \dots \xi _ {n} ) $; 
and  $  \xi ^ {\alpha  ^  \prime  } = \xi _ {1} ^ {\alpha _ {1} } \dots \xi _ {n} ^ {\alpha _ {n} } $.

See also [[Degenerate partial differential equation|Degenerate partial differential equation]] and the references given there.