Lonely Runners over Function Fields: Quantized Phase--Riesz product

Abstract

Let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over ${\mathbb{F}}_q$ whose associated codimension-$k$ partial-circulant kernels cover the full coefficient space. Chow and Rimani'c conjectured that $C_k(q)=1+q+\cdots +q^k$. We disprove the unrestricted conjecture by constructing thirteen monic polynomials over ${\mathbb{F}}_2$ whose $k=3$ kernels cover ${\mathbb{F}}_2^7$; in particular, $C_3(2)\le 13<15$. For a general covering family of size $N=q^k+S$ and ${\mathbb{F}}_q$-linear rank $d$, we prove $S\gg d^{2/3}{\left(\frac{\log (2q)}{\log (eN{q}^k/S)}\right)}^{2/3}$. Consequently, for every fixed $k\ge 2$ and all sufficiently large $q$, $C_k(q)\ge q^k+c_k{q}^{2/3}$. When $k=2$, an integer-multiplicity refinement of the second-moment covering argument yields ${\mathrm{liminf}}_{q\to \infty }(C_2(q)-q^2)/q\ge {\tilde{c}}_2$, where ${\tilde{c}}_2$ is an explicit one-variable variational constant with numerical value ${\tilde{c}}_2=0.5829944375\dots$. We also classify triples admitting two independent low-degree polynomial syzygies and prove a conditional packet-free lower bound of size $q^k+(1/2-o(1))q^{k-1}$.

25 pages, no figures. Work-in-progress formalization and related code will be developed and maintained at https://github.com/hxypqr/lonely-runners-function-fields. Corrected the historical attribution in the Introduction. No mathematical changes