Its powers have poles of order formula_40 and so on. Therefore, such a formula_1 has the gap sequence  In general if the gap sequence is  the weight of the Weierstrass point is  This is introduced because of a counting theorem: on a Riemann surface the sum of the weights of the Weierstrass points is formula_45  For example, a hyperelliptic Weierstrass point, as above, has weight formula_46 Therefore, there are (at most) formula_47 of them. The formula_48 ramification points of the ramified covering of degree two from a hyperelliptic curve to the projective line are all hyperelliptic Weierstrass points and these exhausts all the Weierstrass points on a hyperelliptic curve of genus formula_11. Further information on the gaps comes from applying Clifford's theorem. Multiplication of functions gives the non-gaps a numerical semigroup structure, and an old question of Adolf Hurwitz asked for a characterization of the semigroups occurring. A new necessary condition was found by R.-O. Buchweitz in 1980 and he gave an example of a subsemigroup of the nonnegative integers with 16 gaps that does not occur as the semigroup of non-gaps at a point on a curve of genus 16 (see ). A definition of Weierstrass point for a nonsingular curve over a field of positive characteristic was given by F. K. Schmidt in 1939.