On the asymptotic behavior of Shannon-type functions characterizing the computing complexity of systems of monomials

In this paper, we examined the computational complexity of systems of monomials for some models that allow multiple use of intermediate results, such as composition circuits and multiplication circuits.
For these models, we studied Shannon-type functions that characterize the maximum computational complexity of systems of monomials with exponents not exceeding the corresponding elements of a given matrix $A$. We found that for composition circuits, under the condition of unlimited growth of the maximum of matrix elements, this function grows asymptotically as the binary logarithm of the maximum absolute value (without regard to the sign) of the term from the determinant of the matrix $A.$ Using generalized circuits as an auxiliary model, we transferred this result (under some restrictions) to the model of multiplication circuits.