The polynomial model in the study of counterexamples to S. Piccard's theorem (Q2715938)

A ruler \(R\) with \(n\) integral marks \(0=r_1<r_2<\cdots <r_n=N\) determines the multiset of \(n(n-1)/2\) distances \(\Delta (R)=\{r_j-r_i: i<j\}\). Suppose \(\Delta (R)\) and \(\Delta (S)\) are sets (i.e., no distance is repeated) and \(\Delta (R)=\Delta (S)\). There is a conjecture saying that then \(S=R\) or \(S=N-R\), unless \(n=6\). For \(n=6\) we have counterexamples, e.g. \(R=(0, 1, 4, 10, 12, 17)\) and \(S=(0, 1, 8, 11, 13, 17)\) (first noted by \textit{G. S. Bloom}). The authors use polynomials \(r(x)=\sum x^{r_i}\) and their factorizations to prove several results relevant to the conjecture.