In the present paper we solve the following different but interrelated problems: (a)~the moment problem on Riemann surfaces, (b) the vanishing problem for polynomial Abelian integrals of dimension zero on the projective plane, (c) the vanishing problem for polynomial hyperelliptic Abelian integrals.
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Lubomir Gavrilov, Fedor Pakovich, Moments on Riemann surfaces and hyperelliptic Abelian integrals. Comment. Math. Helv. 89 (2014), no. 1, pp. 125–155DOI 10.4171/CMH/314