Effective Strassmann Certificates for Local $p$-adic Dynamical Mordell--Lang Interpolants

Farrukh Mukhamedov

Effective Strassmann Certificates for Local $p$-adic Dynamical Mordell--Lang Interpolants

T0 review · 0 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read For p-adic maps close to the identity, the exact Strassmann index of an interpolated orbit is certifiable from finitely many orbit values.

desk verdict A clean, well-scoped methods paper: exact finite certificates for the Strassmann index under a standard congruence hypothesis, with honest limits; worth a serious refereeing. read the letter →

arxiv 2607.14339 v1 pith:IJS5XI6H submitted 2026-07-15 math.DS

Farrukh Mukhamedov This is my paper · ORCID

classification math.DS MSC 37P5537P2011S8030G0614G20

keywords DynamicalMordell–Langp-adicinterpolationStrassmannindexfinitecertificatebinomialexpansionrestrictedpowerseriesresidue-classzoomingmaps

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that the exact zero-counting index of a p-adic analytically interpolated orbit can be certified from finitely many orbit values alone, without ever computing the full power series. Under the hypothesis that the map is the identity plus a perturbation divisible by p^q with q(p−1)>1, the finite certificate is both exact and terminating when the interpolant is nonzero. A finite-precision version shows that orbit values known only modulo p^R are enough, giving a rigorous stopping rule for approximate computations. The same machinery yields sharper bounds by zooming into residue classes of time, a gcd-based certificate for targets defined by several equations, and—in dimension one—an exact identification of the bound with the number of roots of the target in a p-adic ball. Applications include certified intersections of power-map orbits with finite target sets and a uniform return statement for roots of unity.

What carries the argument

The load-bearing mechanism is the valuation-raising divisibility Δ^m(A)⊆p^{qm}A, where Δ=T−I is the difference of the pullback by f and the identity; it makes every binomial-basis coefficient B_m=(Δ^mF)(a) divisible by p^{qm}. Converting the binomial series Σ B_m binom(z,m) to ordinary power series via signed combinatorial conversion coefficients s(m,j)/m! introduces factorial denominators, which are controlled by the formula v_p(m!) = (m−s_p(m))/(p−1); from this the threshold λ_{p,q}(M) arises. The same divisibility reappears in the one-dimensional contact theorem, where the arc Γ_a(z)=f^z(a) is shown to be an analytic isomorphism Z_p→a+p^{v_p(f(a)−a)}Z_p, turning the index into a root coun

What would settle it

Compute the certificate for a range of maps satisfying the hypothesis (e.g., p=3, f(x)=x+3x^2, a=1, F(x)=x−1) using the algorithm in Section 5. If any case with H_F≠0 returns a certified index different from the value obtained by directly computing the ordinary power series to high precision, or if the stopping test never succeeds for some nonzero interpolant, the central claim fails. For the torsion application, search for a root of unity ζ and a map f(x)=x+p^qΦ(x) with f(ζ)≠ζ such that f^n(ζ)^N=1 for some n≥1; the theorem asserts no such triple exists.

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Extended reading notes

Core claim

The paper proves that, for f(x)=x+p^qΦ(x) with q(p−1)>1, the Strassmann index of H_F(z)=F(f^z(a))—the p-adic zero bound for the orbit hitting a target—is exactly recoverable from finite data: if the minimal valuation α_M of the truncated coefficients C_j^(M) (built from orbit values y_0,…,y_M) satisfies α_M < λ_{p,q}(M), then SI(H_F) is the largest j≤M attaining that minimum; and for nonzero H_F the condition holds for all large M. A finite-precision variant requires only orbit values modulo p^R. The paper further shows that zooming into a time residue class replaces f by f^{p^h} and strengthens the tail by h powers of p, that multiequation targets reduce to a one-variable gcd certificate, a

Load-bearing premise

The entire certificate rests on the standing hypothesis f(x)=x+p^qΦ(x) with q(p−1)>1, which guarantees that each pullback difference raises p-adic valuations by q; if a global dynamical system cannot be reduced to this local form after passing to residue classes and iterates—especially under ramified or non-étale behavior—the certificates do not apply.

Editorial extensions

If this is right

A computation that has found SI(H_F) ordinary zeros of the interpolant is certified complete: no further orbit value can hit the target.
Finite-precision orbit data (values modulo p^R) yield rigorous stopping rules, so approximate p-adic arithmetic can be used without losing exact zero bounds.
Residue-class zooming gives a branch-and-bound strategy in which difficult time classes are split into p subclasses with progressively stronger certificate tails.
For multiequation targets, the arc-gcd certificate gives a zero bound that can be strictly smaller than the bound from any single defining equation.
In dimension one, the certificate is optimal: the certified index equals the root count of the target in the orbit ball, so hits are exhausted exactly when the root count is reached.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A practical implementation of the certificate could serve as a terminating search routine for exponential Diophantine equations of the type α^{r^n}=β, where the theorem certifies when all solutions have been found; the paper gives the bound but leaves the algorithmic packaging implicit.
The arc-ideal viewpoint points toward a finite-precision Weierstrass-gcd algorithm for the pulled-back equations; the paper explicitly notes this as future work, and the certificate would be the verification step for such a routine.
Because the divisibility Δ^m(A)⊆p^{qm}A is the only input from the map's structure, the method may generalize to controlled non-étale models if contracting directions can be handled by valuation growth rather than analytic interpolation; the paper identifies this as the main open boundary.
In dimension one, the sharpness of the certificate means that the number of orbit hits to a finite set is computed exactly once the orbit ball and the target's roots are known—no infinite search remains.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short take: this is a solid effective-methods paper. It doesn't try to prove new global Dynamical Mordell–Lang; it makes the local zero-bound step explicit and certifiable. What's new: the finite-data certificate (Theorem 4.2) for the exact Strassmann index, the finite-precision analogue (Theorem 5.1), residue-class zooming (Theorem 6.3), the arc-gcd bound (Proposition 7.1), and the root-of-unity avoidance result (Theorem 11.1). I checked the main proof chain: the tail bound follows from the divisibility Δ^m(A)⊆p^{qm}A plus Legendre's formula, and the strict inequality in the certificate rules out both the analytic tail and numerical error. The logic is clean and the claims are scoped with the standing hypothesis f(x)=x+p^qΦ(x), q(p−1)>1.

The paper is careful about what it does not do. Section 14 explicitly says the certificates don't apply if a global system cannot be reduced to this local identity-congruent form; the arc-gcd result is a correctness theorem but no certified gcd routine is supplied (acknowledged in Remark 7.2 and Section 14); and identically-zero interpolants need a separate zero-detection step (Remark 4.4 and the algorithm note). These are honest boundaries, not hidden gaps. The main soft spot is the applicability boundary itself—it is load-bearing, though not an internal inconsistency. The geometric root-count interpretation is dimension-one only; in higher dimensions the certificate still computes the index but without the ball picture.

The math looks sound: the derivations are explicit, and there's no fitting to conclusions. The citation pattern is appropriate—standard p-adic black boxes plus contextual comparison with static-equation work, not circular. For a reader working in p-adic or computational dynamics, this is directly useful: it gives stopping criteria and finite checks that were not spelled out before. For others, it's a well-executed but narrow toolbox paper.

Recommendation: send it to a serious referee. It deserves referee time, and I'd expect acceptance after minor revision—mainly tightening presentation and making the applicability boundary and the separate zero-detection step more prominent. I'd cite it if I worked on effective p-adic DML or orbit-intersection computations. I'd also bring it to our reading group.

Referee Report

0 major / 3 minor

Summary. The paper develops effective certificates for the Strassmann index of p-adic analytic interpolants H_F(z)=F(f^z(a)) arising in local p-adic Dynamical Mordell–Lang. Under the standing hypothesis f(x)=x+p^qΦ(x) with q(p−1)>1, the Mahler coefficients B_m lie in p^{qm}Z_p, yielding a tail bound λ_{p,q}(M). Theorem 4.2 shows that if the minimal valuation of truncated ordinary coefficients C_j^(M) is strictly below λ, then the Strassmann index is exactly the largest index attaining that minimum; termination holds for H_F≠0. Finite-precision (Thm 5.1), residue-class zooming (Thm 6.3), arc-ideal gcd (Prop 7.1), one-shot (Prop 8.1), first-order escape (Prop 8.3), and one-dimensional contact/root-count results (Thms 9.3, 9.6) are established. Applications include certified bounds for power-map orbits and root-of-unity avoidance.

Significance. If correct, the paper turns Strassmann's existence theorem into a checkable stopping rule for local orbit-intersection computations. The proofs are explicit, the error bounds are quantitative, and the finite-precision theorem addresses practical computation. The paper is careful to state the conditional nature of the results and the limitations (Section 14). The applications to power maps and root-of-unity avoidance are concrete and demonstrate the usefulness of the framework.

minor comments (3)

[§3, Cor. 3.3; §7, Eq. (7.1)] The notation A^d_{Z_p} for affine d-space conflicts with the Tate algebra A=Z_p⟨x_1,...,x_d⟩ defined in the introduction. Using the same letter for both objects is confusing; I recommend writing \mathbb{A}^d_{\mathbb{Z}_p} for the ambient affine space.
[§5, Algorithm] The algorithm's 'inconclusive' output is correctly described as possibly meaning either insufficient M/R or H_F≡0, with a separate zero-detection step needed. Since Theorems 4.2 and 5.1 assume H_F≠0, it would help to state explicitly near Theorem 1.1 that the certificates do not provide a decision procedure for the identically-zero case.
[§9.3] The comparison with static p-adic equations is interesting but somewhat digressive; it cites [5,6] for context only. Consider condensing it into a remark so that the main line of the section flows more directly.

Circularity Check

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No significant circularity: the certificate theorems are conditional and derive from the standing congruence via explicit valuation bounds; self-citations are comparison-only.

full rationale

The derivation chain is self-contained and non-circular. The central input is the standing hypothesis f(x)=x+p^q Phi(x) with q(p-1)>1, which gives the divisibility Delta^m(A) subseteq p^{qm}A in (3.3). From this divisibility, Lemma 4.1 derives explicit lower bounds on the tails of the ordinary power-series coefficients of H_F, using only Stirling numbers and the valuation of factorials. Theorem 4.2 then shows that if the minimum valuation alpha_M of the finite truncated coefficients satisfies alpha_M < lambda_{p,q}(M), the Strassmann index is exactly the largest index attaining that minimum. This is not a definitional identification: alpha_M is computed from finite orbit values, lambda_{p,q}(M) is an independently derived tail threshold, and the equality SI(H_F)=JM is proved, not assumed. The finite-precision Theorem 5.1 adds an explicit numerical error bound R - V_M, again derived from factorial denominators. Termination (Corollary 4.3) follows from the fact that the true minimum is eventually separated from the tail. The paper explicitly relies on standard external facts (Poonen, Strassmann, Mahler, Weierstrass preparation) rather than on the author's own prior results; the only self-citations, [5,6], are explicitly identified in Section 9.3 as comparison-only and are not used in the proofs. Section 14 concedes that the certificates do not apply when a global DML instance cannot be reduced to the identity-congruent local form; this is an applicability boundary, not an internal circularity. No prediction or derived quantity is fitted to the desired conclusion, and no load-bearing claim reduces to a self-citation chain.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper introduces no fitted free parameters; all numerical thresholds are explicitly derived quantities. It relies on standard p-adic theorems (Strassmann, Mahler, Poonen, Weierstrass) plus the standing identity-congruence hypothesis, which is an acknowledged applicability boundary rather than an ad hoc postulate.

assumptions (9)

standard math Strassmann's theorem: a nonzero restricted power series over a complete non-archimedean field has at most SI(H) zeros in the valuation ring.
Used as the basic zero-counting mechanism (Standard fact 2.1).
standard math Mahler expansion and binomial-coefficient integrality on Z_p.
Basis for representing H_F in the binomial basis and converting via signed Stirling numbers (Standard fact 2.2).
standard math Poonen's interpolation theorem: for f(x)=x+p^q Φ(x), q(p−1)>1, the iterates extend analytically in the time variable.
Supplies the interpolant H_F and the analytic arc Γ_a; load-bearing black box (Standard fact 2.3).
standard math Legendre formula: v_p(m!) = (m − s_p(m))/(p−1).
Controls factorial denominators in the Mahler-to-ordinary coefficient conversion (Standard fact 2.4).
standard math Weierstrass preparation: the Weierstrass degree equals the number of zeros in the closed unit disc over C_p, and analytic automorphisms preserve this degree.
Gives the root-counting interpretation in dimension one (Standard fact 2.5).
standard math The one-variable Tate algebra Q_p⟨z⟩ is a principal ideal domain.
Justifies the arc-gcd generator in Section 7 (Standard fact 2.6).
standard math One-variable p-adic inverse function theorem: U(z)≡uz mod p with u a unit implies U is an analytic automorphism of the closed unit disc.
Used to show Γ_a parametrizes the whole orbit ball in dimension one (Standard fact 2.7).
standard math Root-of-unity separation: v_p(ρ−1) ≤ 1/(p−1) for a nontrivial root of unity ρ.
Used in the root-of-unity avoidance application (Standard fact 2.8).
domain assumption Standing hypothesis: f(x)=x+p^q Φ(x), Φ∈A^d, q(p−1)>1.
The source of the divisibility Δ^m(A)⊆p^{qm}A; all certificates depend on it, and Section 14 states that non-étale/ramified systems may not satisfy it.

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Cite this review

Pith. "Pith review of Effective Strassmann Certificates for Local $p$-adic Dynamical Mordell--Lang Interpolants." pith.science (2026). https://pith.science/paper/IJS5XI6H

@misc{pith2026260714339,
  author       = {Pith},
  title        = {Pith review of: Effective Strassmann Certificates for Local $p$-adic Dynamical Mordell--Lang Interpolants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IJS5XI6H}},
  note         = {Machine review of arXiv:2607.14339}
}

abstract

The $p$-adic method for Dynamical Mordell--Lang often reduces a residue class of an orbit to the zero set of a locally analytic interpolating function. This paper assumes the standard interpolation, Strassmann, Mahler, and Weierstrass tools, and studies the effective local zero-bound problem that remains after interpolation: certifying the Strassmann index of the resulting one-variable analytic function. We give finite certificates for this index, including finite-data, finite-precision, refined-tail, adaptive residue-class, one-shot, and first-order escape criteria. Since the Strassmann index is a rigorous upper bound for zeros in $\Zp$ and, through Weierstrass preparation, a root count on the closed disc over $\Cp$, these certificates give checkable stopping criteria for local orbit-intersection computations. A residue-class zooming principle replaces a congruence class of times by the iterate $f^{p^h}$, gaining $h$ additional powers of $p$ in the certificate tails. We also introduce an arc-ideal viewpoint for target varieties defined by several equations, replacing a chosen hypersurface bound by the one-variable gcd of all defining equations along the interpolated orbit. In dimension one, the method identifies the certified Strassmann index with the corresponding local Weierstrass root count in the orbit ball. Applications include certified bounds for intersections of non-fixed power-map orbits with finite target sets, and root-of-unity avoidance for maps tangent to the identity at torsion units.

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Reference graph

Works this paper leans on

10 extracted references · 2 linked inside Pith

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

J. P. Bell, D. Ghioca, and T. J. Tucker,The dynamical Mordell–Lang problem for étale maps, American Journal of Mathematics132(2010), no. 6, 1655–1675; arXiv:0808.3266

arXiv 2010

[2] [2]

Bosch, U

S. Bosch, U. Güntzer, and R. Remmert,Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry, Grundlehren der Mathematischen Wissenschaften, vol. 261, Springer-Verlag, Berlin, 1984

1984

[3] [3]

Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd ed., Graduate Texts in Mathematics, vol

N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd ed., Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York, 1984

1984

[4] [4]

Mahler,An interpolation series for continuous functions of ap-adic variable, Journal für die reine und angewandte Mathematik199(1958), 23–34

K. Mahler,An interpolation series for continuous functions of ap-adic variable, Journal für die reine und angewandte Mathematik199(1958), 23–34

1958

[5] [5]

Mukhamedov, B

F. Mukhamedov, B. Omirov, and M. Saburov,On cubic equations overp-adic fields, International Journal of Number Theory10(2014), no. 5, 1171–1190

2014

[6] [6]

Mukhamedov and M

F. Mukhamedov and M. Saburov,On equationxq = a over Qp, Journal of Number Theory133(2013), no. 1, 55–58

2013

[7] [7]

Poonen, p-adic interpolation of iterates, Bulletin of the London Mathematical Society46(2014), no

B. Poonen, p-adic interpolation of iterates, Bulletin of the London Mathematical Society46(2014), no. 3, 525–527

2014

[8] [8]

A. M. Robert,A Course inp-adic Analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000

2000