Résumé : In this paper we study the persistent and tangential center problems in dimension zero. These problems are motivated by the study of planar vector fields. The question there is when a center persists after deformation, or when it persists up to first order. The zero-dimensional tangential center problem is the problem of identical vanishing of Abelian integrals on zero-dimensional cycles. Looking at the behavior of cycles at infinity, we introduce the notions of balanced, unbalanced and totally unbalanced cycles. We solve the zero-dimensional polynomial tangential center problem for totally unbalanced cycles. Pakovich and Muzychuk solved the polynomial moment problem and showed that the solution is a sum of composition terms. One can say that a weak composition conjecture holds in this case. We show that the polynomial moment problem can be seen as a special case of the tangential center problem on a totally unbalanced cycle. We also study the center problem on some balanced cycles. We provide examples where even the weak composition conjecture for the tangential center problem does not hold. Pakovich improved the result by showing that the above sum can be written with at most three terms. We show that even in the totally unbalanced case, when the weak composition conjecture holds, one cannot in general reduce the sum to three terms. We give some applications of our results to planar tangential center problems in hyper-elliptic equations and generalized Van de Pol 's equations.