Christie–Dykema–Klep: minimal vanishing sums of roots of unity

---

Library card, especially Proposition 2.3 and Theorem 3.3.

Louis Christie, Kenneth J. Dykema, and Igor Klep, "Classifying minimal vanishing sums of roots of unity," arXiv:2008.11268 (2020).

The paper defines the type of a vanishing sum recursively (Definition 2.4) and, in Theorem 3.3, classifies by hand the types and parities of all minimal vanishing sums of weight at most 16 (76 types); a computer search extends the list to weight 21, which the authors present as conjectural (pp. 1-2). A minimal sum is the circuit-like object relevant to relation hypergraphs: every forbidden relation contains a minimal one. The type and parity bookkeeping expose how larger relations are assembled from prime cycles and smaller vanishing sums.

For E0774 this gives a finite catalogue for testing low-weight portions of a roots-of-unity construction and may suggest bounded local templates for a Ramsey amplification. It concerns nonnegative vanishing sums with multiplicity. A signed $\{-1,0,1\}$ relation is such a sum once each term $-\omega$ is read as the root of unity $-\omega$ (the paper's parity counts these signs, p. 5); its vanishing proper sub-relations are then exactly the vanishing proper sub-sums, so the two minimality notions agree.