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 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|≤∏i(hi+1) for hi=max(°ziN,°ziQ). Thus the minimal automaton has pO(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 αm+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 bx2y2+(1+ax+dx2)y+cx2 with odd a,b,c,d∈ℤ2, we prove Kα(F)≥9·8α/(64α−32) and limα→∞α−1log2Kα(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 x2y2+(1+x+x2)y+x2 has exactly 2α+1+α+1 states, resolving their orbit-size conjecture.
Related
- Rational Area Profiles of Regular Polygons Cut by Two Congruent Diagonals
- An example of an automatic sequence with non-regular abelian complexity
- Parity-Word Complexity and Exponent Growth in Non-Bounded Collatz Orbits
- Bayesian Tensor Train Kernel Machines with Automatic Model Complexity Inference
- The exponent 1/4 in three complexity bounds: integer bases, continued fractions, and the (3/2)n steering word
BibTeX
@misc{shvets2026linearexponent,
title = {{Linear-exponent bounds for the automatic complexity of rational diagonals modulo prime powers}},
author = {Alex Shvets},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01481},
eprint = {2611.01481},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01481},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Linear-exponent bounds for the automatic complexity of rational diagonals modulo prime powers
AU - Alex Shvets
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01481
UR - https://arxiv.org/abs/2611.01481
AB - 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)≤ 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|≤∏_i(h_i+1) for h_i=max(°_z_iN,°_z_iQ). 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∈ℤ_2, we prove K_α(F)≥9·8^α/(64α−32) and lim_α→∞α^−1log_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.
ER -