Proof of Scott’s conjecture

Abstract:

We give a proof of Conjecture 7 in [2, p. 155] first stated in 1881 by R. F. Scott [4]. It reads as follows: Conjecture 7 (R. F. Scott). Let ${x_1}, \ldots ,{x_n}$ and ${y_1}, \ldots ,{y_n}$ be the distinct roots of ${x^n} - 1 = 0$ and ${y^n} + 1 = 0$, respectively. Let $A$ be the $n \times n$ matrix whose $(i,j)$ entry is $1/({x_i} - {y_i}),i,j = 1, \ldots ,n$. Then \[ \left | {\operatorname {per} (A)} \right | = \left \{ {\begin {array}{*{20}{c}} {n{{(1 \cdot 3 \cdot 5 \cdots (n - 2))}^2}/{2^{n}}}, & {{\text {if}}\;n\;{\text {is }}\operatorname {odd} ,} \\ {0,} & {{\text {if}}\;n\;{\text {is}}\;\operatorname {even} {\text {.}}} \\ \end {array} } \right .\] Actually, our proof gives more, namely an explicit expression for ${\operatorname {per}}(A)$ (see Theorem 2.1).