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Hasse-type upper bound for reduced orbit length of qP over finite fields

Establish that for every orbit γ of the discrete Painlevé mapping (x,y,t) ↦ (t/(x−s^{-1}y), s x/y, s t) over a finite field F_q, where s ∈ F_q^* has multiplicative order r, the reduced orbit length satisfies #(γ)/r ≤ q + 2√q + 1.

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Background

The paper studies the arithmetic dynamics of the q-difference Painlevé equation qP(A_7{(1)}) defined by (x,y,t) ↦ (t/(x−s{-1}y), s x/y, s t) over finite fields F_q. Over finite fields, all orbits are periodic, and the parameter s has finite multiplicative order r dividing q−1, which also divides the orbit length.

Motivated by extensive computations (over 200M orbits for 2 ≤ q ≤ 499), the authors conjecture that the reduced orbit length #(γ)/r obeys a Hasse-type upper bound analogous to bounds for rational points on elliptic curves, even in this non-autonomous setting.

References

Our main computational results are summarised in Conjecture \ref{conj:numerical}. These conjectures arise from the study of orbits of Equation eq:qp1 over $\mathbb{F}q$, with the prime power $q$ ranging from $2$ to $499$. Let $\gamma$ denote any orbit of Equation eq:qp1 over $\mathbb{F}{q}$ and suppose $r$ is the multiplicative order of $s\in\mathbb{F}_{q}*$. Then the following results hold true. The number of points $#(\gamma)$ satisfies \begin{equation}\label{eq:conj_I_bound} #(\gamma)/r\leq q+2\sqrt{q}+1. \end{equation}

eq:conj_I_bound:

#(γ)/r≤q+2q+1.\#(\gamma)/r\leq q+2\sqrt{q}+1.

eq:qp1:

(x,y,t)↦(x‾,y‾,t‾),{x‾=tx−s−1y,y‾=s xy,t‾=s t.(x,y,t)\mapsto (\overline{x},\overline{y},\overline{t}),\qquad \begin{cases} \displaystyle\overline{x}=\frac{t}{x-s^{-1}y}, & \\ \displaystyle\overline{y}=\frac{s\, x}{y}, & \\ \displaystyle\overline{t}=s\,t. & \end{cases}

— Arithmetic dynamics of a discrete Painlevé equation (2508.18578 - Joshi et al., 26 Aug 2025) in Conjecture 1.2.A, Section 1.2 (Main results)