Published September 29, 2026 | Version 3
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Arbitrary-Lag Obstruction to Positive Strong Shift Equivalence over Z+[t] and the Polynomial Baker Problem

Description

Nonnegative integer matrices encode shifts of finite type, and Williams showed that topological conjugacy is characterized by strong shift equivalence (SSE), generated by elementary factorizations A = RS, B = SR with nonnegative factors. Baker's matrices

Ak = (1, k+1; k, 1),     Bk = (1, k(k+1); 1, 1)

have long provided concrete test cases for SSE, with recent computational work of Jeandel treating small values of k.

Replacing k by a variable t gives the polynomial Baker pair A(t), B(t) over ℤ+[t]. We prove that these matrices are not strong shift equivalent over ℤ+[t], at arbitrary finite lag and with intermediate matrices of arbitrary finite size. This polynomial result does not by itself decide SSE for any individual integer specialization of t.

The proof connects the Baker pair by explicit positive elementary factorizations to 3×3 matrices C, E4. Using Williams's classical implication that an SSE of lag n induces a shift equivalence of lag n, a hypothetical chain yields the standard intertwining and power identities. Solving the resulting intertwining equations reduces the problem to

(1 + t)x2 − ty2 = (1 − t − t2)k.

We classify its coefficientwise nonnegative solutions, derive a constant-term bound, and obtain an obstruction polynomial Rh,m. Exact coefficient formulas force a negative coefficient in every admissible case, contradicting positivity. The proof is algebraic and independent of finite-lag or finite-state enumeration.

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Related works

Cites
Preprint: 10.5281/zenodo.22936613 (DOI)
Preprint: 10.5281/zenodo.22949732 (DOI)

Dates

Copyrighted
2026-09-29
Copyright © 2026 Kevin L. Merdy

References

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