2025/09/16 by Faustin Adiceam, Adiceam, Faustin, Dzmitry Badziahin +1
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2509.12826
Given a prime p, the p-adic Littlewood Conjecture stands as a well-known arithmetic variant of the celebrated Littlewood Conjecture in Diophantine Approximation. In the same way as the latter, it admits a natural function field analogue depending on the choice of an irreducible polynomial P(t) with coefficients in a field \mathbbK. This analogue is referred to as the P(t)-adic Littlewood Conjecture (P(t)-LC for short). P(t)-LC is proved to fail for any choice of irreducible polynomial P(t) over any ground field \mathbbK with characteristic ℓ ≡ 3\pmod4. The counterexample refuting it is shown to present a local arithmetic obstruction emerging from the fact that -1 is not a quadratic residue modulo a prime ℓ≡ 3\pmod4. The theory developed elucidates and generalises all previous approaches towards refuting the conjecture. They were all based on the computer-assisted method initiated by Adiceam, Nesharim and Lunnon (2021) which has been able to establish that P(t)-LC fails in some small characteristics (essentially up to 11). This computer-assisted method is, however, unable to provide a general statement as it relies on ad hoc computer verifications which, provided they terminate, refute P(t)--LC in a given characteristic. This limitation is overcome by exhibiting an arithmetic obstruction to the validity of P(t)--LC in infinitely many characteristics. The existence of arithmetic obstructions within the context of P(t)--LC leaves the remaining case of odd characteristics ℓ≡ 1\pmod4 dependent on their determination. This is shown to hold in an effective and explicit way.