  A perfect cuboid is a rectangular parallelepiped whose edges, whose face
diagonals, and whose space diagonal are of integer lengths. The second cuboid
conjecture specifies a subclass of perfect cuboids described by one Diophantine
equation of tenth degree and claims their non-existence within this subclass.
This Diophantine equation is called the cuboid characteristic equation. It has
two parameters. The linear region is a domain on the coordinate plane of these
two parameters given by certain linear inequalities. In the present paper
asymptotic expansions and estimates for roots of the characteristic equation
are obtained in the case where both parameters tend to infinity staying within
the linear region. Their applications to the cuboid problem are discussed.
