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The existence of valuative interpolation at a singular point

The present paper studies the existence of valuative interpolation on the local ring of an irreducible analytic subvariety at singular points. We firstly develop the concepts and methods of Zhou weights and Tian functions near singular points of irreducible analytic subvarieties. By applying these tools, we establish the necessary and sufficient conditions for the existence of valuative interpolations on the rings of germs of holomorphic functions and weakly holomorphic functions at a singular point. As applications, we characterize the existence of valuative interpolations on the quotient ring of the ring of convergent power series in real variables. We also present separated necessary and sufficient conditions for the existence of valuative interpolations on the quotient ring of polynomial rings with complex coefficients and real coefficients. Furthermore, we show that the conditions become both necessary and sufficient under certain conditions on the zero set of the given polynomials.