A certain graph obtained from a set of several points on a Riemann surface
Let \(M\) be a compact Riemann surface of genus \(g\geq 2,\) and let \(P_1,P_2,\dots{},P_n\) be distinct points on \(M.\) Then define the Weierstrass gap set \(G(P_1,P_2,\dots{},P_n)\) \(=\{(\gamma _1,\gamma _2,\dots{},\gamma _n)\in(N)^n\) such that there does not exist a meromorphic function \(f\) on \(M\) whose pole divisor \((f)_\infty\) is \(\gamma _1P_1+\dots{}+\gamma _nP_n\},\) where \(N\) is the set of non-negative integers. When \(n=1\), \(G(P_1)\) is the set of Weierstrass gaps at \(P_1.\) \textit{S. J. Kim} [Arch. Math. 62, No. 1, 73-92 (1994; Zbl 0815.14020)] has given formulas for \( \#G(P_1,P_2)\) and shown the inequalities \(\frac{g^2+3g}2\leq \#G(P_1,P_2)\leq \frac{3g^2+g}2.\) In this paper the author starts studying Riemann-Roch graphs \(D^{(n)}\) and their gap sets \(G^{(n)}\) in general. In particular he proves that \(\#G^{(n)}\geq \binom{n+g}g-1.\) He shows the upper bound of \(\#G^{(3)}\leq g(7g^2+6g+5)/6\) and there is a unique graph satisfying \(\#G^{(3)}= g(7g^2+6g+5)/6.\)
scientific article; zbMATH DE number 1347380