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Proof of Scott's conjecture


Author: D. Svrtan
Journal: Proc. Amer. Math. Soc. 87 (1983), 203-207
MSC: Primary 15A15; Secondary 12D99
DOI: https://doi.org/10.1090/S0002-9939-1983-0681822-9
MathSciNet review: 681822
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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

$\displaystyle \left\vert {\operatorname{per} (A)} \right\vert = \left\{ {\begin... ...{if}}\;n\;{\text{is}}\;\operatorname{even} {\text{.}}} \\ \end{array} } \right.$

Actually, our proof gives more, namely an explicit expression for $ {\operatorname{per}}(A)$ (see Theorem 2.1).


References [Enhancements On Off] (What's this?)

  • [1] C. W. Borchardt, Bestimmung der symmetrischen Verbindungen vermittelst ihrer erzeugenden Funktion, Monatsb. Akad. Wiss. Berlin 1888 (1855), 165-171; or Crelle's J. 53 (1855), 193-198; or Gesammelte Werke, 97-105.
  • [2] Henryk Minc, Permanents, Encyclopedia of Mathematics and its Applications, vol. 6, Addison-Wesley Publishing Co., Reading, Mass., 1978. With a foreword by Marvin Marcus. MR 504978
  • [3] Thomas Muir, A treatise on the theory of determinants, Revised and enlarged by William H. Metzler, Dover Publications, Inc., New York, 1960. MR 0114826
  • [4] R. F. Scott, Mathematical notes, Messenger of Math. 10 (1881), 142-149.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0681822-9
Keywords: Permanent, circulant, roots of $ \pm$ unity, Viète formulas
Article copyright: © Copyright 1983 American Mathematical Society