Weierstrass point

In mathematics, a Weierstrass point P {displaystyle P} on a nonsingular algebraic curve C {displaystyle C} defined over the complex numbers is a point such that there are more functions on C {displaystyle C} , with their poles restricted to P {displaystyle P} only, than would be predicted by the Riemann–Roch theorem. In mathematics, a Weierstrass point P {displaystyle P} on a nonsingular algebraic curve C {displaystyle C} defined over the complex numbers is a point such that there are more functions on C {displaystyle C} , with their poles restricted to P {displaystyle P} only, than would be predicted by the Riemann–Roch theorem. The concept is named after Karl Weierstrass. Consider the vector spaces where L ( k P ) {displaystyle L(kP)} is the space of meromorphic functions on C {displaystyle C} whose order at P {displaystyle P} is at least − k {displaystyle -k} and with no other poles. We know three things: the dimension is at least 1, because of the constant functions on C {displaystyle C} ; it is non-decreasing; and from the Riemann–Roch theorem the dimension eventually increments by exactly 1 as we move to the right. In fact if g {displaystyle g} is the genus of C {displaystyle C} , the dimension from the k {displaystyle k} -th term is known to be