  This paper investigates the low-rank matrix completion (LRMC) problem from a
generic vantage point. Unlike most existing work that has focused on recovering
a low-rank matrix from a subset of the entries with specified values, the only
information available here is just the pattern (i.e., positions) of observed
entries. To be precise, given is an $n\times m$ pattern matrix ($n\le m$) whose
entries take fixed zero, unknown generic values, and missing values that are
free to be chosen from the complex field, the question of interest is whether
there is a matrix completion with rank no more than $n-k$ ($k\ge 1$) for almost
all values of the unknown generic entries, which is called the generic low-rank
matrix completion (GLRMC) associated with $({\cal M},k)$, and abbreviated as
GLRMC$({\cal M},k)$. Leveraging an elementary tool from the theory on systems
of polynomial equations, we give a simple proof for genericity of this problem,
which was first observed by $Kir\'aly et al.$, that is, depending on the
pattern of the observed entries, the answer is either yes or no, for almost all
realizations of the unknown generic entries. We provide necessary and
sufficient conditions ensuring feasibility of GLRMC$({\cal M},1)$. Aso, we
provide a sufficient condition and a necessary condition (which we conjecture
to be sufficient too) for the general case (i.e., $k\ge 1$). In the following,
two randomized algorithms are presented to determine upper and lower bounds for
the generic minimum completion rank. Our approaches are based on the algebraic
geometry theory, and a novel basis preservation principle discovered herein.
Finally, numerical simulations are given to corroborate the theoretical
findings and the effectiveness of the proposed algorithms.
