Infimum Microlensing Amplification of the Maximum Number of Images of $n$-point Lens Systems

The total amplification of a source inside a caustic curve of a binary lens is no less than 3. Here we show that the infimum amplification 3 is satisfied by a family of binary lenses where the source position is at the mid-point between the lens positions independently of the mass ratio which parameterizes the family. We present a new proof of an underlying constraint that the total amplification of the two positive images is bigger than that of the three negative images by one inside a caustic. We show that a similar constraint holds for an arbitrary class of $n$-point lens systems for the sources in the `maximal domains'. We introduce the notion that a source plane consists of {\it graded caustic domains} and the `maximal domain' is the area of the source plane where a source star results in the maximum $n^2+1$ images. We show that the infimum amplification of a three point lens is 7, and it is bigger than $n^2+1-n$ for $n\ge 4$. This paper has raised many interesting and very basic questions such as ``whether lensing is a physical process, a mathematical process, or both" since it was submitted for publication a year ago. The result is the addition of 8 page appendix, and that is the reason of this replacement. We hope that the future authors wouldn't have to pay so long for using the elegant Jacobian matrix in complex coordinate basis. (They are neither wrong nor funny!!)