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

For a rational diagonal G of N/Q over the p-adic integers with Q(0) a unit, the minimal automaton for the coefficients modulo pα has pO(α) states for fixed input and prime: the exponent is linear in α. Main theorem. 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. Then the p-kernel of the coefficients of G modulo pα has size Kα(G) ≤ p(|J|+1)α−1 + (pα − 1)/(p − 1), where J is an explicit finite support set determined by the Newton polytopes of N and Q, and |J| ≤ ∏(hi + 1), where hi is the larger of the degrees of N and Q in the i-th variable. Comparison. The general upper bounds of Rowland and Yassawi have exponents that are cubic in α for Furstenberg series and of order αm+1 for diagonals in m variables. Method. A δ-decomposition with synchronous Cartier sections and a count of compatible histories; no smoothness or ordinarity assumption is needed. The transitions are given explicitly. Consequences. Bounds for Furstenberg series. For the Furstenberg series F of bx2y2 + (1 + ax + dx2)y + cx2 with odd a, b, c, d: 2α+1 ≤ Kα(F) ≤ 26α−1 + 2α − 1. An appendix proves that the zero orbit in the Rowland–Yassawi automaton for x2y2 + (1 + x + x2)y + x2 has exactly 2α+1 + α + 1 states, resolving their conjecture. Open question. Exact computations motivate, but do not prove, the conjecture lim α−1 log2 Kα(F) = 3 for this polynomial. Files. The paper (PDF, 22 pages) and its LaTeX source with ancillary code, exact data and reproduction instructions (ZIP).