Chow–Rimanić polynomial-kernel covering conjecture

For every prime power $q$ and integer $k\ge 0$ , 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. The Chow–Rimanić conjecture asserts that $C_k(q)=1+q+\cdots+q^k=\frac{q^{k+1}-1}{q-1}$ .

This conjecture is false in general: the source constructs a covering over $\mathbb{F}_2$ with $k=3$ and $13$ polynomials, giving $C_3(2)\le 13<15$ .

Progress summary

Partially solved

The conjecture’s proposed exact formula is false over finite fields, but related asymptotic questions remain open.

The conjecture predicted the unrestricted covering number $C_k(q)=1+q+\cdots+q^k$ . That prediction has now been disproved over finite fields, while refined conditional and asymptotic versions remain under investigation.

August 2026 disproof

A new arXiv work, Lonely Runners over Function Fields: Quantized Phase--Riesz Product, disproves the unrestricted formula and establishes new asymptotic lower bounds. Thus the exact conjectured value is no longer viable, but the available report does not claim to settle the remaining refined questions.

Current status (as of August 2026): The unrestricted formula $C_k(q)=1+q+\cdots+q^k$ is disproved over finite fields; conditional and refined asymptotic problems remain open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

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

additive-combinatorics finite-fields

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