On the Entry Sum of Cyclotomic Arrays

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Don Coppersmith and John Steinberger, "On the Entry Sum of Cyclotomic Arrays," Integers: Electronic Journal of Combinatorial Number Theory 6 (2006), #A26. Zenodo deposit: https://doi.org/10.5281/zenodo.8275405 (the deposit lists the second author as "Steinberg, John"; the article prints "John Steinberger").

Local reading copy. A Markdown reading copy sits beside the PDF. No notice is printed in the file; the Zenodo record (https://zenodo.org/record/8275405, read 2026-10-02) states the rights "Creative Commons Attribution 4.0 International" with open access, the Creative Commons Attribution 4.0 license.

Research digest

For a nonnegative integer cyclotomic array, the total entry sum is a nonnegative integer combination of the side lengths (Theorem 1). Under the standard square-free correspondence, fibers are regular prime polygons and the theorem recovers the Lam–Leung restriction on the weight of a nonnegative vanishing sum (Theorem 3: the coefficient sum of a vanishing sum of $n$th roots of unity with nonnegative integer coefficients is a nonnegative integer combination of the primes dividing $n$). The paper also emphasizes that nonnegative arrays of dimension three or more need not be positive sums of fibers, and that minimal ones can have entries superpolynomially large compared to the volume of the array (citing its reference [9]).

This supplies numerical obstructions for positive relations in a prime-coordinate construction and warns against assuming a fiber decomposition. E0774 needs signed $\{0,\pm1\}$ relations, so entry-sum arithmetic alone is insufficient; apply it only after separating a signed relation into positive and negative sides and tracking the loss of information.

Bears on. E0774.