Conway–Jones: trigonometric Diophantine equations and vanishing sums

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Held copy and library card.

J. H. Conway and A. J. Jones, "Trigonometric diophantine equations (On vanishing sums of roots of unity)," Acta Arithmetica 30 (1976), no. 3, 229--240.

Conway and Jones develop a finite procedure for describing rational linear relations among roots of unity and classify vanishing sums of length at most nine. Their main quantitative result for the present problem is Theorem 5: if a minimal vanishing sum has length $l$ and reduced exponent $r$, then

$$l \ge 2 + \sum_{p\mid r}(p-2).$$

Thus every additional prime appearing in a minimal relation has an explicit support cost. Theorem 4 (p. 234) splits a vanishing sum $S$ that is not similar to $1+\omega+\cdots+\omega^{r-1}$ ($r$ prime, $\omega$ a primitive $r$th root) into two vanishing sums $S'+S''$ with $l(S')\le l(S)$, $r(S')<r(S)$, $l(S'')<l(S)$, $r(S'')\le r(S)$, and Theorem 6 gives concrete normal forms through length nine.

For E0774, Theorem 5 is a sharp way to rule out short signed relations whose reduced exponent contains too many or too-large prime factors. It can support a prime-coordinate construction or an analysis of the $T_{195}$ test case. It does not by itself control how many mutually interacting minimal relations occur in a finite set, so a proportional extraction or coloring argument still needs an additional combinatorial step.