Bivariate truncated moment sequences with the column relation XY = X m + q ( X ), with q of degree m −1

When the algebraic variety associated with a truncated moment sequence is finite, solving the moment problem follows a well-defined procedure. However, moment problems involving infinite algebraic varieties are more complex and less well-understood. Recent studies suggest that certain bivariate moment sequences can be transformed into equivalent univariate sequences, offering a valuable approach for solving these problems. In this paper, we address the truncated moment problem for certain rational plane curves. For curves of general degree, we derive an equivalent Hankel positive semidefinite completion problem. For cubic curves, we solve this problem explicitly, thereby resolving the truncated moment problem for one of the four types of irreducible cubic curves, up to affine linear equivalence. For the quartic case, we reduce the completion problem to a feasibility question involving a system of inequalities in three variables. In the cubic case, we also determine the Carathéodory number of the associated moment sequences, complementing recent results on real plane cubics and affine plane curves.