Lower bounds on the Graver complexity of $M$-fold matrices

Abstract: In this paper, we present a construction that turns certain relations on Graver basis elements of an $M$-fold matrix $A<sup>{(M)}$ into relations on Graver basis elements of an $(M+1)$-fold matrix $A<sup>{(M+1)}$. In doing so, we strengthen the bound on the Graver complexity of the $M$-fold matrix $A_{3\times M}$ from $g(A_{3\times M})\geq 17\cdot 2<sup>{M-3}-7$ (Berstein and Onn) to $g(A_{3\times M})\geq 24\cdot 2<sup>{M-3}-21$, for $M\geq 4$. Moreover, we give a lower bound on the Graver complexity $g(A<sup>{(M)})$ of general $M$-fold matrices $A<sup>{(M)}$ and we prove that the bound for $g(A_{3\times M})$ is not tight.