Polynomial joint first-order differential projective invariants

Abstract

We study polynomial absolute and relative joint first-order differential projective invariants for configurations of $n$ points in the plane. The main approach is based on passing to a homogeneous vector--covector representation, which reduces the problem to the classical invariant theory of the group $SL(3,\mathbb C)$. The algebra of polynomial absolute invariants is described and shown to be generated by cyclic invariants. For polynomial relative invariants of weight $-1$, a graded description is obtained, their finite generation as a module over the algebra of absolute invariants is proved, and an explicit finite generating system is constructed. For $n=3$, the corresponding module is shown to be free of rank $1$. For $n=4$, a minimal homogeneous generating system consisting of $39$ elements is constructed. The obtained results complement the rational theory of joint projective differential invariants by its polynomial counterpart and provide an algebraic foundation for the further construction of projectively invariant integral characteristics.