Addition-multiplication chains with one small step

An addition-multiplication chain of length $l$ for a positive integer $n$ is a monotonic increasing sequence $1=a_0<a_1<\dots <a_l=n$ of positive integers, such that for each $1\le i\le l,$ we have $a_i=a_j+a_k\text{or}{a}_i=a_j\times a_k,$ where $0\le k\le j\le i-1.$ In this paper, we establish upper bounds for each element in any AM-chain based on the number of squaring steps and the distribution of non-squaring steps preceding the given element. Moreover, we get additional upper bounds independent of the distribution of non-squaring steps. These bounds yield a new upper bound for the number of non-squaring steps in any AM-chain. Finally, we determine all AM-chains that include exactly one small step and, as a consequence, compute the shortest lengths ${\ell }_{AM}(.)$ of certain integers.