On the behavior of the Shannon function of the implementation complexity of monomials system

In this paper, we examined the computational complexity (minimum possible number of operations) of systems of monomials by circuits using two-input composition operation, which can be considered as a generalization of multiplication operation. We found that growth asymptotic of the Shannon function, characterizing the maximum complexity among systems of $p$ monomials of $q$ variables with exponents no more than $K$, given that $pq \log K \rightarrow \infty$ and some additional restrictions, has the form $ \min(p,q) \log_2 K + \frac{pq}{\log_2(pq)}$.