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The combinatorics of a three-line circulant determinant. (English) Zbl 1072.15009

The authors study the polynomial \(\Phi(x,y) = \prod_{j=0}^{p-1} (1-x\omega^j - y\omega^{qj})\), where \(\omega\) is a primitive root of unity. They prove that it is the determinant of the \(p \times p\) circulant matrix with first row \((1, -x, 0, \ldots, 0, -y, 0, \ldots, 0)\) where \(-y\) is in position \(q+1\). The authors also prove, for example, that the monomial \(x^ry^s\) appears in \(\Phi\) if and only if \(p\) divides \(r+sq\).

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
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[1] D’Angelo, J. P., Invariant holomorphic mappings, Journal of Geometric Analysis, 6, 163-179 (1996) · Zbl 0901.32017
[2] J. P. D’Angelo,Number-theoretic properties of certain CR mappings, Preprint, 2003.
[3] Brègman, L. M., Certain properties of nonnegative matrices and their permanents, Doklady Akademii Nauk SSSR, 211, 27-30 (1973) · Zbl 0293.15010
[4] Codenotti, B.; Crespi, V.; Resta, G., On the permanent of certain (0, 1) Toeplitz matrices, Linear Algebra and its Applications, 267, 65-100 (1997) · Zbl 0891.65049
[5] Codenotti, B.; Resta, G., Computation of sparse circulant permanents via determinants, Linear Algebra and its Applications, 355, 15-34 (2002) · Zbl 1017.65044 · doi:10.1016/S0024-3795(02)00330-0
[6] Davis, P. J., Circulant Matrices (1979), New York-Chichester-Brisbane: Wiley-Interscience, New York-Chichester-Brisbane · Zbl 0418.15017
[7] G. P. Egorychev,Reshenie problemy van-der-Wardena dlia permanentov, Institute of Physics im. L. V. Kirenskogo, USSR Academy of Sciences, Siberian Branch, preprint IFSO-13M, Krasnoiarsk, 1980. · Zbl 0438.15010
[8] Falikman, D. I., A proof of van der Waerden’s conjecture on the permanent of a doubly stochastic matrix, Matematicheskie Zametki, 29, 931-938 (1981) · Zbl 0475.15007
[9] McIlroy, M., Number theory in computer graphics, inThe Unreasonable Effectiveness of Number Theory (Orono, ME), Proceedings of Symposia in Applied Mathematics, 46, 105-121 (1992) · Zbl 0773.11001
[10] Minc, H., Upper bounds for permanents of (0, 1)-matrices, Bulletin of the American Mathematical Society, 69, 789-791 (1963) · Zbl 0116.25202
[11] Schrijver, A., A short proof of Minc’s conjecture, Journal of Combinatorial Theory, Series A, 25, 80-83 (1978) · Zbl 0391.15006 · doi:10.1016/0097-3165(78)90036-5
[12] Schrijver, A., Counting 1-factors in regular bipartite graphs, Journal of Combinatorial Theory, Series B, 72, 122-135 (1998) · Zbl 0905.05060 · doi:10.1006/jctb.1997.1798
[13] Voorhoeve, M., A lower bound for the permanents of certain (0, 1)-matrices, Indagationes Mathematicae, 41, 83-86 (1979) · Zbl 0401.05005
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