arXiv2026
Polynomial first-order projective differential moment invariants are investigated. The main result establishes an exact correspondence between such moment invariants and symmetric polynomial multipoint relative projective differential invariants of weight $-1$. This correspondence describes each homogeneous component with respect to moment degree and allows previously obtained results on relative projective invariants to be used for systematic construction. It is also proved that passing from the formal algebra of moments to their integral definitions introduces no additional algebraic relations among the constructed invariants. For low moment degrees, the absence of nontrivial invariants of degrees $1$ and $2$ is established, and a complete description of degree $3$ invariants is obtained. The ten cubic differential moments are naturally identified with the coefficients of a ternary cubic, connecting the construction with classical invariant theory. In particular, the Aronhold invariants $S$ and $T$ yield projective differential moment invariants of degrees $4$ and $6$. One of the known moment invariants of Wang et al. is shown to be proportional to $S$. The algebra of polynomial moment invariants is proved not to be finitely generated. The field of rational moment invariants coincides with the field of fractions of the polynomial invariant algebra. For the explicit constructions, compact multipoint representations define the corresponding moment polynomials without requiring their full expansions. These representations provide a basis for algorithmic construction and software computation of the invariants.