Given a surface
F, we are interested in
valued invariants of immersions of
F into
, which are constant on each connected component of the complement of the quadruple point discriminant in
. Such invariants will be called “
q-invariants.” Given a regular homotopy class
, we denote by
the space of all
q-invariants on
A of order
. We show that if
F is orientable, then for each regular homotopy class
A and each
n,
dim(Vn (A) / Vn-1(A) ) £ 1\dim (V_n (A) / V_{n-1}(A) ) \leq 1.
Received June 15, 1999; in final form September 22, 1999 / Published online October 30, 2000