Linear-exponent bounds for the automatic complexity of rational diagonals modulo prime powers

Abstract

Let $p$ be a prime and let $G$ be the diagonal of $N/Q$ in $m$ variables over the $p$-adic integers, with $Q(0)$ a unit. We prove that the $p$-kernel of the coefficients of $G$ modulo $p^α$ has size $K_α(G)\le p^{(|J|+1)α-1}+(p^α-1)/(p-1)$, where $J$ is an explicit finite support set determined by the Newton polytopes of $N$ and $Q$, and $|J|\le\prod_i(h_i+1)$ for $h_i=\max(°_{z_i}N,°_{z_i}Q)$. Thus the minimal automaton has $p^{O(α)}$ states for fixed input and prime. The general upper bounds of Rowland and Yassawi have exponents that are cubic in $α$ for Furstenberg series and of order $α^{m+1}$ for diagonals in $m$ variables. The proof combines a $δ$-decomposition with synchronous Cartier sections and a count of compatible histories; it requires no smoothness or ordinarity assumption. We give explicit transitions and derive bounds for Furstenberg series. For the Furstenberg series $F$ of $bx^2y^2+(1+ax+dx^2)y+cx^2$ with odd $a,b,c,d\in\mathbb{Z}_2$, we prove $K_α(F)\ge9\cdot8^α/(64α-32)$ and $\lim_{α\to\infty}α^{-1}\log_2K_α(F)=3$. This exponent result rests on primitive reachability, parameter-independent normalized window frames, and an overlap estimate certified in a finite quotient of a rational Cartier lattice. An appendix also proves that the zero orbit in the Rowland-Yassawi automaton for $x^2y^2+(1+x+x^2)y+x^2$ has exactly $2^{α+1}+α+1$ states, resolving their orbit-size conjecture.