The Chow–Rimanić covering number Cₖ(2): exact values by degree window, and the optimal families

Abstract

Chow and Rimanić introduced a function-field analogue of the lonely runner conjecture: for a finite family F of nonzero polynomials over 𝔽_q one sets δ(F)=sup_αmin_(f ∈ F)nm(α f) and lets Cₖ(q) be the least size of a family with δ(F)<q⁻ᵏ; they conjectured Cₖ(q)=Qₖ(q)=1+q+…+qᵏ. Hu has recently refuted this at (q,k)=(2,3) with thirteen polynomials over 𝔽₂ found by a cross-entropy search, giving 9 ≤ C₃(2) ≤ 13<15, and asked for the exact value of C₃(2). We settle that question inside the degree window in which the counterexample lives: no twelve nonzero polynomials over 𝔽₂ of degree at most four have loneliness less than 2⁻³, so the thirteen are optimal among all speeds of degree at most four. We then classify the optimal families in that window: there are exactly three, they carry the same covering-multiplicity distribution, and coefficient reversal carries the published family to the second and fixes the third. For k=2 we compute the value restricted to degrees at most four and at most five and find the conjectured 7 in both — past the reach of the Chow–Rimanić small-degree theorem, which at q=2, k=2 reaches only degree two. Every lower bound is an exhaustive search whose branching rule is proved complete rather than assumed, and every result proved here is machine-checked in Lean 4; the two results taken from the literature are marked as such.

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