  We consider the Frobenius algebra of functions on the critical set of the
master function of a weighted arrangement of hyperplanes in $\C^k$ with normal
crossings. We construct two potential functions (of first and second kind) of
variables labeled by hyperplanes of the arrangement and prove that the matrix
coefficients of the Grothendieck residue bilinear form on the algebra are given
by the $2k$-th derivatives of the potential function of first kind and the
matrix coefficients of the multiplication operators on the algebra are given by
the $(2k+1)$-st derivatives of the potential function of second kind. Thus the
two potentials completely determine the Frobenius algebra. The presence of
these potentials is a manifestation of a Frobenius like structure similar to
the Frobenius manifold structure.
  We introduce the notion of an elementary subarrangement of an arrangement
with normal crossings. It turns out that our potential functions are local in
the sense that the potential functions are sums of contributions from
elementary subarrangements of the given arrangement. This is a new phenomenon
of locality of the Grothendieck residue bilinear form and multiplication on the
algebra.
  It is known that this Frobenius algebra of functions on the critical set is
isomorphic to the Bethe algebra of this arrangement. (That Bethe algebra is an
analog of the Bethe algebras in the theory of quantum integrable models.) Thus
our potential functions describe that Bethe algebra too.
