Bound for the Weierstrass weights of points on a smooth plane algebraic curve (Q1881436)

Let \(C\) be a smooth plane curve of degree \(n \geq 3\). The following upper bound for the Weierstrass weight \(w(P)\) of any point \(P \in C\) is proved. Theorem 1. \(w(P) \leq \frac{1}{24}(n-1)(n-2)(n-3)(n+4)\) with equality if and only if \(P\) is a {total inflection point} (i.e. the {intersection number} at \(P\), \((C . L)_P\), of \(C\) and \(L\), the tangent line of \(C\) at \(P\), is exactly \(n\)). Moreover if \(C\) has an involution, a lower bound for the weights of fixed points of the involution is obtained, namely Theorem 2. \(w(P) \leq (n-1)(n-3)/8\) if \(n\) is odd, \(w(P) \leq (n-2)(n-4)/8\) if \(n\) is even. Furthermore all possible {Weierstrass gap sequences} and weights of fixed points of the involution are obtained for \(n=5, \;6\) (Prop. 1 and Prop. 2). Four examples, concerning these last two results are collected in the last section of the paper.