  The paper is devoted to the investigation of finite dimensional commutative
nilpotent (associative) algebras N over an arbitrary base field of
characteristic zero. Due to the lack of a general structure theory for algebras
of this type (as opposed to the semi-simple case) we associate various objects
to every N which encode the algebra structure. Our main results are in the
subclass of algebras having 1-dimensional annihilator, that is, are maximal
ideals of Gorenstein algebras of finite vector dimension > 1. Associated
structural objects are then, for instance, a class of mutually affinely
equivalent algebraic hypersurfaces S in N, and a class of so-called
nil-polynomials p, whose degree is the nil-index of N. Then N can be
reconstructed from S and even from the quadratic plus cubic part of p. If the
algebra N is graded the hypersurface S is affinely homogeneous. The paper
closes with an example of an N of dimension 23 and nil-index 5, for which S is
not affinely homogeneous.

Commutative Algebra