Coppersmith–Steinberger: entry sums of cyclotomic arrays

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Held copy and library card, especially Theorems 1 and 3.

Don Coppersmith and John Steinberger, "On the entry sum of cyclotomic arrays," Integers 6 (2006), #A26; the held copy is the Zenodo deposit, doi:10.5281/zenodo.8275405.

For a nonnegative integer cyclotomic array, the total entry sum is a nonnegative integer combination of the side lengths. 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. 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.

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.