  This paper makes the first systematic attempt to determine using perturbation
theory the positions of images by gravitational lensing due to arbitrary number
of coplanar masses without any symmetry on a plane, as a function of lens and
source parameters. We present a method of Taylor-series expansion to solve the
lens equation under a small mass-ratio approximation. First, we investigate
perturbative structures of a single-complex-variable polynomial, which has been
commonly used. Perturbative roots are found. Some roots represent positions of
lensed images, while the others are unphysical because they do not satisfy the
lens equation. This is consistent with a fact that the degree of the
polynomial, namely the number of zeros, exceeds the maximum number of lensed
images if N=3 (or more). The theorem never tells which roots are physical (or
unphysical). In this paper, unphysical ones are identified. Secondly, to avoid
unphysical roots, we re-examine the lens equation. The advantage of our method
is that it allows a systematic iterative analysis. We determine image positions
for binary lens systems up to the third order in mass ratios and for arbitrary
N point masses up to the second order. This clarifies the dependence on
parameters. Thirdly, the number of the images that admit a small mass-ratio
limit is less than the maximum number. It is suggested that positions of extra
images could not be expressed as Maclaurin series in mass ratios.
Magnifications are finally discussed.
