# Multiplicative complexity of a bilinear form over a commutative ring

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## Abstract

We characterize the class of Noetherian commutative rings K such that the multiplicative complexity of a bilinear form over K coincides with its rank. The asymptotic behaviour of the multiplicative complexity of bilinear forms from one special class over the polynomial rings is described, and in particular it is shown that there is no finite upper bound for the difference between the multiplicative complexity of a bilinear form from this class and the rank of this form. The relationship between the multiplicative complexity of a bilinear form over a ring K and homological properties of the ring is explained.

## Keywords

Bilinear Form Commutative Ring Integral Domain Polynomial Ring Polynomial Multiplication
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## Copyright information

© Springer-Verlag Berlin Heidelberg 1981