On the coefficients of the permanent and the determinant of a circulant matrix: Applications

Colarte, Liena; Mezzetti, Emilia; Miró-Roig, Rosa; Salat, Martí

doi:10.1090/proc/14296

On the coefficients of the permanent and the determinant of a circulant matrix: Applications
HTML articles powered by AMS MathViewer

by Liena Colarte, Emilia Mezzetti, Rosa M. Miró-Roig and Martí Salat PDF

Proc. Amer. Math. Soc. 147 (2019), 547-558 Request permission

Abstract:

Let $d(N)$ (resp., $p(N)$) be the number of summands in the determinant (resp., permanent) of an $N\times N$ circulant matrix $A=(a_{ij})$ given by $a_{ij}=X_{i+j}$ where $i+j$ should be considered mod $N$. This short note is devoted to proving that $d(N)=p(N)$ if and only if $N$ is a prime power. We then give an application to homogeneous monomial ideals failing the Weak Lefschetz property.

References

C. Almeida, Aline V. Andrade, and R.M. Miró-Roig, Gaps in the number of generators of monomial Togliatti systems, Journal of Pure and Applied Algebra. To appear.
Holger Brenner and Almar Kaid, Syzygy bundles on $\Bbb P^2$ and the weak Lefschetz property, Illinois J. Math. 51 (2007), no. 4, 1299–1308. MR 2417428
Richard A. Brualdi and Morris Newman, An enumeration problem for a congruence equation, J. Res. Nat. Bur. Standards Sect. B 74B (1970), 37–40. MR 266850
Pietro De Poi, Emilia Mezzetti, Mateusz Michatek, Rosa Maria Miró-Roig, Eran Nevo, Circulant matrices and Galois-Togliatti systems, arXiv:1808.08387.
D.R. Grayson and M.E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/
Emilia Mezzetti, Rosa M. Miró-Roig, and Giorgio Ottaviani, Laplace equations and the weak Lefschetz property, Canad. J. Math. 65 (2013), no. 3, 634–654. MR 3043045, DOI 10.4153/CJM-2012-033-x
Emilia Mezzetti and Rosa M. Miró-Roig, The minimal number of generators of a Togliatti system, Ann. Mat. Pura Appl. (4) 195 (2016), no. 6, 2077–2098. MR 3558320, DOI 10.1007/s10231-016-0554-y
Emilia Mezzetti and Rosa M. Miró-Roig, Togliatti systems and Galois coverings, J. Algebra 509 (2018), 263–291. MR 3812202, DOI 10.1016/j.jalgebra.2018.05.014
Mateusz Michałek and Rosa M. Miró-Roig, Smooth monomial Togliatti systems of cubics, J. Combin. Theory Ser. A 143 (2016), 66–87. MR 3519816, DOI 10.1016/j.jcta.2016.05.004
Juan C. Migliore, Rosa M. Miró-Roig, and Uwe Nagel, Monomial ideals, almost complete intersections and the weak Lefschetz property, Trans. Amer. Math. Soc. 363 (2011), no. 1, 229–257. MR 2719680, DOI 10.1090/S0002-9947-2010-05127-X
Rosa M. Miró-Roig and Martí Salat, On the classification of Togliatti systems, Comm. Algebra 46 (2018), no. 6, 2459–2475. MR 3778405, DOI 10.1080/00927872.2017.1388813
Irwin Kra and Santiago R. Simanca, On circulant matrices, Notices Amer. Math. Soc. 59 (2012), no. 3, 368–377. MR 2931628, DOI 10.1090/noti804
Nicholas A. Loehr, Gregory S. Warrington, and Herbert S. Wilf, The combinatorics of a three-line circulant determinant, Israel J. Math. 143 (2004), 141–156. MR 2106980, DOI 10.1007/BF02803496
J. Malenfant, On the Matrix-Element Expansion of a Circulant Determinant, arXiv:1502.06012v5
Hugh Thomas, The number of terms in the permanent and the determinant of a generic circulant matrix, J. Algebraic Combin. 20 (2004), no. 1, 55–60. MR 2104820, DOI 10.1023/B:JACO.0000047292.01630.a6
A. Wyn-Jones, Circulants, Chapters 10 and 11. www.circulants.org/circ/circall.pdf.
Eugenio G. di Togliatti, Alcuni esempî di superficie algebriche degli iperspazî che rappresentano un’ equazione di Laplace, Comment. Math. Helv. 1 (1929), no. 1, 255–272 (Italian). MR 1509396, DOI 10.1007/BF01208366
Eugenio Togliatti, Alcune osservazioni sulle superficie razionali che rappresentano equazioni di Laplace, Ann. Mat. Pura Appl. (4) 25 (1946), 325–339 (Italian). MR 23563, DOI 10.1007/BF02418089
L. G. Valiant, The complexity of computing the permanent, Theoret. Comput. Sci. 8 (1979), no. 2, 189–201. MR 526203, DOI 10.1016/0304-3975(79)90044-6