  This paper presents a definition for local linearizations of rational
matrices and studies their properties. This definition allows us to introduce
matrix pencils associated to a rational matrix that preserve its structure of
zeros and poles in subsets of any algebraically closed field and also at
infinity. Moreover, such definition includes, as particular cases, other
definitions that have been used previously in the literature. In this way, this
new theory of local linearizations captures and explains rigorously the
properties of all the different pencils that have been used from the 1970's
until 2019 for computing zeros, poles and eigenvalues of rational matrices.
Particular attention is paid to those pencils that have appeared recently in
the numerical solution of nonlinear eigenvalue problems through rational
approximation.
