Polynomials of small slice rank and strength

Abstract

This paper investigates defining equations for secant varieties of the variety of reducible polynomials, which geometrically encode the notions of strength and slice rank of homogeneous polynomials. We present three main results. First, we reinterpret Ruppert's classical equations for reducible ternary forms in the language of representation theory and we extend them to an arbitrary number of variables. Second, we construct new determinantal equations for polynomials of small strength based on syzygies of their partial derivatives. Finally, we establish a reduction theorem for cubic forms, proving that slice rank two is determined by generic linear sections in $14$ variables; this gives one of the few explicit upper bounds for defining equations for the image of a polynomial map in the framework of noetherianity for polynomial functors.