Relation between rank and multiplicative complexity of a bilinear form over a commutative Noetherian ring

The concept of multiplicative complexity of a bilinear form is introduced for a commutative Noetherian ring. Rings are described for which the multiplicative complexity coincides with the rank for all forms. It is shown that for regular rings of dimension 3 the multiplicative complexity can exceed the rank by an arbitrarily large number.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 86, pp. 66–81, 1979.