The Truncated Moment Problem on Reducible Cubic Curves I: Parabolic and Circular Type Relations

In this article we study the bivariate truncated moment problem (TMP) of degree 2 k on reducible cubic curves. First we show that every such TMP is equivalent after applying an affine linear transformation to one of 8 canonical forms of the curve. The case of the union of three parallel lines was solved in Zalar (Linear Algebra Appl 649:186–239, 2022. https://doi.org/10.1016/j.laa.2022.05.008 ), while the degree 6 cases in Yoo (Integral Equ Oper Theory 88:45–63, 2017). Second we characterize in terms of concrete numerical conditions the existence of the solution to the TMP on two of the remaining cases concretely, i.e., a union of a line and a circle y(ay+x²+y²)=0, a∈ R \0\ y ( a y + x 2 + y 2 ) = 0 , a ∈ R \ { 0 } , and a union of a line and a parabola y(x-y²)=0 y ( x - y 2 ) = 0 . In both cases we also determine the number of atoms in a minimal representing measure.