On completely decomposable defining equations of finite sets in Pⁿ

Let (Formula presented.) be the homogeneous ideal of a finite set Γ in (Formula presented.) of d points in linearly general position. In 1972, Saint-Donat proved that if (Formula presented.) then (Formula presented.) is generated by completely decomposable quadratic polynomials. Later, R. Treger generalized this result by showing that (Formula presented.) is generated by forms of degree (Formula presented.) if (Formula presented.) for some (Formula presented.). This paper aims to generalize Saint-Donat’s work in a different direction by proving that the degree m piece of (Formula presented.) can be generated by completely decomposable forms of degree m if and only if (Formula presented.). Communicated by Daniel Erman.