  There are two restriction maps of the logarithmic modules of plane
arrangements in a three dimensional vector space. One is the Euler restriction
and the other is the Ziegler restriction. The dimension of the cokernel of the
Ziegler restriction map of logarithmic derivation modules has been well-studied
for the freeness of hyperplane arrangements after Yoshinaga's celebrated
criterion for freeness, which connects the second Betti number and the
splitting type (exponents). However, though the Euler restriction has a longer
history than the Ziegler restriction, the cokernel and its dimension of the
Euler restriction have not been studied at all.
  The aim of this article is to study the cokernel and dimension of the Euler
restriction maps in terms of combinatorics, more explicitly, the characteristic
polynomial. We give an upper bound of that cokernel, and show the formula for
that if the arrangement is free.
