Abstract
We prove that the determinantal complexity of a hypersurface of degree \(d > 2\) is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the \(3 \times 3\) permanent is 7. We also prove that for \(n> 3\), there is no nonsingular hypersurface in \({\mathbb {P}}^n\) of degree d that has an expression as a determinant of a \(d \times d\) matrix of linear forms, while on the other hand for \(n \le 3\), a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
If \(\mathrm{char}(k) = 2\), one checks that the argument of [16, p. 289] applies to \(Q(w)=\sum _{j=2}^m w_{1j}w_{j1}\) using the bilinear form \(Q_0(w, w') = Q(w+w') - Q(w) - Q(w')\).
References
Leslie G. Valiant. Completeness classes in algebra. In Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing (Atlanta, Ga.), pages 249–261. 1979.
Leslie G. Valiant. The complexity of computing the permanent. Theoret. Comput. Sci., 8(2):189–201, 1979.
Thierry Mignon and Nicolas Ressayre. A quadratic bound for the determinant and permanent problem. Int. Math. Res. Not., (79):4241–4253, 2004.
Jin-Yi Cai, Xi Chen, and Dong Li. A quadratic lower bound for the permanent and determinant problem over any characteristic \(\ne 2\). In STOC’08, pages 491–497. ACM, New York, 2008.
Bruno Grenet. An Upper Bound for the Permanent versus Determinant Problem. to appear in Theory of Computing, 2011.
Joachim von zur Gathen. Permanent and determinant. Linear Algebra Appl., 96:87–100, 1987.
Roy Meshulam. On two extremal matrix problems. Linear Algebra and its Applications, 114115:261 – 271, 1989.
Jesko Hüttenhain and Christian Ikenmeyer. Binary Determinantal Complexity. arXiv:1410.8202 [cs.CC], 2015.
David Eisenbud. Personal communication. 2015.
Leonard Eugene Dickson. Determination of all general homogeneous polynomials expressible as determinants with linear elements. Trans. Amer. Math. Soc., 22(2):167–179, 1921.
Arnaud Beauville. Determinantal hypersurfaces. Michigan Math. J., 48:39–64, 2000.
Heinrich Schröter. Nachweis der 27 geraden auf der allgemeinen oberfläche dritter ordnung. J. Reine Angew. Math., 62:265–280, 1863.
Luigi Cremona. Mémoire de géométrie pure sur les surfaces du troisièeme ordre. J. Reine Angew. Math., 68:1–133, 1868.
Hermann Grassmann. Die stereometrischen gleichungen dritten grades, und die dadurch erzeugten oberflächen. J. Reine Angew. Math., 49:49–65, 1855.
Michela Brundu and Alessandro Logar. Parametrization of the orbits of cubic surfaces. Transform. Groups, 3(3):209–239, 1998.
Joe Harris. Algebraic geometry, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.
Acknowledgments
We thank Nicolas Ressayre for raising the question regarding the value of \({{\mathrm{dc}}}(xy^2+yt^2+z^3) \) during the problem discussion at the Geometric Complexity Theory workshop at the Simons Institute in Berkeley in September 2014. During the preparation of this paper, the first author was partially supported by the Australian Research Council Grant DE140101519. The second and third authors were partially supported by the FAPA funds from Universidad de los Andes.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Peter Buergisser.
Rights and permissions
About this article
Cite this article
Alper, J., Bogart, T. & Velasco, M. A Lower Bound for the Determinantal Complexity of a Hypersurface. Found Comput Math 17, 829–836 (2017). https://doi.org/10.1007/s10208-015-9300-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10208-015-9300-x