Application of separability and independence notions for proving lower bounds of circuit complexity

This note consists of two independent parts. In the first part the concept of an $(m,c)$-system for a set of linear forms is introduced, and a lower bound is obtained for the algebraic complexity of the computation of $(m,c)$-systems on algebraic circuits of a special form. In the second part, the notion of an $l$-independent set of boolean functions is introduced and a lower bound is obtained for a certain complexity measure for circuits of boolean functions computing $l$-independent sets. As a corollary it is shown that the standard algorithm for multiplying matrices or polynomials may be realized by a circuit of boolean functions in a way that is optimal with respect to a selected complexity measure.