Let p be a prime, let n≥ 1, and let F∈ Z_p[X]^n satisfy det JF∈Z_p^×. A vector in Z_p^n is unimodular when the ideal generated by its entries is Z_p. The unimodular conjecture. There exists b∈Z_p^n such that F(b)∈Z_p^n is unimodular. This conjecture is used in the paper through the result that the Jacobian conjecture for C is equivalent to it holding for all but finitely many primes p. Its resolution is not supplied in the given text.

The unimodular conjecture over the p-adic integers

Let $p$ be a prime, let $n\geq 1$ , and let

F\in \mathbb{Z}_p[X]^n

satisfy $\det JF\in\mathbb{Z}_p^\times$ . A vector in $\mathbb{Z}_p^n$ is unimodular when the ideal generated by its entries is $\mathbb{Z}_p$ . The unimodular conjecture. There exists $b\in\mathbb{Z}_p^n$ such that $F(b)\in\mathbb{Z}_p^n$ is unimodular. This conjecture is used in the paper through the result that the Jacobian conjecture for $\mathbb{C}$ is equivalent to it holding for all but finitely many primes $p$ . Its resolution is not supplied in the given text.

Sources & referencesView supporting material

Primary source

Lucas Hamada, Kazuki Kato and Ryo Komiya, “A Tate algebra version of the Jacobian Conjecture”, arXiv:2502.10769 (2025).

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Progress summary

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Partially solved

No proof or counterexample has been found: the conjecture remains open, with only restricted cases known.

The conjecture asks whether every polynomial map over $\mathbb{Z}_p$ with invertible Jacobian determinant takes some value whose entries generate the unit ideal. Essen--Lipton related this question to the Jacobian conjecture over $\mathbb{C}$ : the latter holds exactly when the unimodular conjecture holds for all but finitely many primes.

Known results

$\mathbb{Z}_p$ is $(p-1)$ -unimodular for every prime $p$ .
$\mathbb{Z}_p$ is $3$ -unimodular for every prime $p>3$ , yielding the conjecture for maps of degree at most $3$ for almost all primes.
Additional results cover quasi-Druzkowski maps and give conditional criteria for the full conjecture.

2025 Tate-algebra reformulation

A 2025 paper claims an equivalence between the complex Jacobian conjecture and a Tate-algebra statement over $\mathbb{C}_p$ , using the Essen--Lipton equivalence. This is a reformulation, not a proof or counterexample for the unimodular conjecture itself.

Current status (as of August 2026): The full unimodular conjecture remains open; partial cases are known, and no verified proof or counterexample is reported.

Sources

No solutions have been posted yet.