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Original Articles

Flattenings and Koszul Young flattenings arising in complexity theory

Pages 4002-4017 | Received 11 Jun 2016, Published online: 31 Jan 2017

References

  • Briand, E. (2010). Covariants vanishing on totally decomposable forms. In: Liaison, Schottky Problem and Invariant Theory. Progress in Mathmetics, Vol. 280. Basel: Birkhäuser Verlag, pp. 237–256 (MR 2664658).
  • Brion, M. (1993). Stable properties of plethysm: On two conjectures of Foulkes. Manuscripta Math. 80(4):347–371 (MR MR1243152 (95c:20056)).
  • Brion, M. (1997). Sur certains modules gradués associés aux produits symétriques. In: Algèbre non commutative, groupes quantiques et invariants (Reims, 1995), Sémin. Congr., Vol. 2. Soc. Math. France, Paris, pp. 157–183 (MR 1601139 (99e:20054)).
  • Catalisano, M. V., Geramita, A. V., Gimigliano, A. (2007). Segre-Veronese embeddings of ℙ1 × ℙ1 × ℙ1 and their secant varieties. Collect. Math. 58(1):1–24 (MR 2310544 (2008f:14069)).
  • Chen, X., Kayal, N., Wigderson, A. (2010). Partial derivatives in arithmetic complexity and beyond. Found. Trends Theor. Comput. Sci. 6(1–2):1–138 (front matter, (2011) MR 2901512).
  • Efremenko, K., Landsberg, J. M., Schenck, H., Weyman, J. (2015). On minimal free resolutions and the method of shifted partial derivatives in complexity theory (CoRR abs/1504.05171).
  • Farnsworth, C. (2016). Koszul-Young flattenings and symmetric border rank of the determinant. J. Algebra 447:664–676 (MR 3427655).
  • Foulkes, H. O. (1950). Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form. J. London Math. Soc. 25:205–209 (MR MR0037276 (12,236e)).
  • Fulton, W., Harris, J. (1991). Representation Theory. Graduate Texts in Mathematics, Vol. 129. New York: Springer-Verlag (A first course, Readings in Mathematics. MR 1153249 (93a:20069)).
  • Gel'fand, I. M., Kapranov, M. M., Zelevinsky, A. V. (1994). Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Boston, MA: Birkhäuser Boston Inc. (MR 95e:14045).
  • Gordon, P. (1894). Das zerfallen der curven in gerade linien. Math. Ann. 45:410–427.
  • Green, E. L. (1978). Complete intersections and Gorenstein ideals. J. Algebra 52(1):264–273 (MR 0480472).
  • Guan, Y. (2015). Brill’s equations as a GL(V)-module (arXiv1508.02293).
  • Gupta, A., Kamath, P., Kayal, N., Saptharishi, R. (2013). Approaching the chasm at depth four. In: Proceedings of the Conference on Computational Complexity (CCC).
  • Gupta, A., Kamath, P., Kayal, N., Saptharishi, R. (2013). Arithmetic circuits: A chasm at depth three. Electron. Colloq. Comput. Complexity 20:26.
  • Hadamard, J. (1897). Mémoire sur l’élimination. Acta Math. 20(1):201–238 (MR 1554881).
  • Hadamard, J. (1899). Sur les conditions de décomposition des formes. Bull. Soc. Math. France 27:34–47 (MR 1504330).
  • Hermite, C. (1854). Sur la theorie des fonctions homogenes a deux indeterminees. Cambridge Dublin Math. J. 9:172–217.
  • Howe, R. (1987). (GLn, GLm) -duality and symmetric plethysm. Proc. Indian Acad. Sci. Math. Sci. 97(1–3):85–109 ((1988) MR MR983608 (90b:22020)).
  • Iarrobino, A., Emsalem, J. (1978). Some zero-dimensional generic singularities; finite algebras having small tangent space. Compos. Math. 36(2):145–188 (MR 515043).
  • Iarrobino, A., Kanev, V. (1999). Power Sums, Gorenstein Algebras, and Determinantal Loci. Lecture Notes in Mathematics, Vol. 1721. Berlin: Springer-Verlag (Appendix C by Iarrobino and Steven L. Kleiman. MR MR1735271 (2001d:14056)).
  • Kadish, H., Landsberg, J. M. (2014). Padded polynomials, their cousins, and geometric complexity theory. Commun. Algebra 42(5):2171–2180 (MR 3169697).
  • Landsberg, J. M. (2012). Tensors: Geometry and Applications. Graduate Studies in Mathematics, Vol. 128. Providence, RI: American Mathematical Society (MR 2865915).
  • Landsberg, J. M. (2015). Geometric complexity theory: An introduction for geometers. Ann. Univ. Ferrara Sez. VII Sci. Math. 61(1):65–117 (MR 3343444).
  • Landsberg, J. M., Manivel, L. (2004). On the ideals of secant varieties of Segre varieties. Found. Comput. Math. 4(4):397–422 (MR MR2097214 (2005m:14101)).
  • Landsberg, J. M., Michalek, M. (2016). On the geometry of border rank algorithms for matrix multiplication and other tensors with symmetry (ArXiv e-prints).
  • Landsberg, J. M., Ottaviani, G. (2013). Equations for secant varieties of Veronese and other varieties. Ann. Mat. Pura Appl. (4) 192(4):569–606 (MR 3081636).
  • Landsberg, J. M., Ottaviani, G. (2015). New lower bounds for the border rank of matrix multiplication. Theory Comput. 11(11):285–298.
  • Landsberg, J. M., Weyman, J. (2007). On the ideals and singularities of secant varieties of Segre varieties. Bull. Lond. Math. Soc. 39(4):685–697 (MR MR2346950).
  • Manivel, L. (1998). Gaussian maps and plethysm. In: Algebraic Geometry (Catania, 1993/Barcelona, 1994). Lecture Notes in Pure and Application Mathematics, Vol. 200. New York: Dekker, pp. 91–117 (MR MR1651092 (99h:20070)).
  • Ranestad, K., Schreyer, F.-O. (2011). On the rank of a symmetric form. J. Algebra 346:340–342 (MR 2842085 (2012j:13037)).
  • Valiant, L. G. (1979). Completeness classes in algebra. In: Proc. 11th ACM STOC, pp. 249–261.

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