Skip to main content
Geometry and Complexity Theory
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 1
  • Cited by
    This (lowercase (translateProductType product.productType)) has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Seynnaeve, Tim 2018. Plethysm and fast matrix multiplication. Comptes Rendus Mathematique, Vol. 356, Issue. 1, p. 52.

  • Export citation
  • Recommend to librarian
  • Recommend this book

    Email your librarian or administrator to recommend adding this book to your organisation's collection.

    Geometry and Complexity Theory
    • Online ISBN: 9781108183192
    • Book DOI:
    Please enter your name
    Please enter a valid email address
    Who would you like to send this to *
  • Buy the print book

Book description

Two central problems in computer science are P vs NP and the complexity of matrix multiplication. The first is also a leading candidate for the greatest unsolved problem in mathematics. The second is of enormous practical and theoretical importance. Algebraic geometry and representation theory provide fertile ground for advancing work on these problems and others in complexity. This introduction to algebraic complexity theory for graduate students and researchers in computer science and mathematics features concrete examples that demonstrate the application of geometric techniques to real world problems. Written by a noted expert in the field, it offers numerous open questions to motivate future research. Complexity theory has rejuvenated classical geometric questions and brought different areas of mathematics together in new ways. This book will show the beautiful, interesting, and important questions that have arisen as a result.

Refine List
Actions for selected content:
Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Send to Kindle
  • Send to Dropbox
  • Send to Google Drive
  • Send content to

    To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to .

    To send content items to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

    Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

    Find out more about the Kindle Personal Document Service.

    Please be advised that item(s) you selected are not available.
    You are about to send

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
[ABV15] J., Alper, T., Bogart, and M., Velasco, A lower bound for the determinantal complexity of a hypersurface, to appear in FOCM 17 (2017), no. 3, pp. 829–836.
[AFLG15] Andris, Ambainis, Yuval, Filmus, and François, Le Gall, Fast matrix multiplication: limitations of the Coppersmith-Winograd method (extended abstract), Symposium on Theory of Computing, ACM, New York, 2015, pp. 585–593. MR 3388238
[AFT11] Boris, Alexeev, Michael A., Forbes, and Jacob, Tsimerman, Tensor rank: some lower and upper bounds, 26th Annual Conference on Computational Complexity, IEEE, Los Alamitos, CA, 2011, pp. 283–291. MR 3025382
[AH95] J., Alexander and A., Hirschowitz, Polynomial interpolation in several variables, J., Algebraic Geom. 4 (1995), no. 2, 201–222. MR 96f:14065
[Ahl78] Lars V., Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1978.
[AJ15] N.R., Aravind and Pushkar S., Joglekar, On the expressive power of read-once determinants, Fundamentals of computation theory, Lecture Notes in Comput. Sci., vol. 9210, Springer, Cham, 2015, pp. 95–105. MR 3440499
[Alp17] Levent, Alpoge, Square-root cancellation for the signs of Latin squares, Combinatorica 37 (2017), no. 2, 137–142. MR 3638338
[AR03] Elizabeth S., Allman and John A., Rhodes, Phylogenetic invariants for the general Markov model of sequence mutation, Math. Biosci. 186 (2003), no. 2, 113–144. MR 2 024 609
[Aro58] S., Aronhold, Theorie der homogenen Functionen dritten Grades von drei Veranderlichen, J. Reine Angew. Math. 55 (1858), 97–191. MR 1579064
[AS81] A., Alder and V., Strassen, On the algorithmic complexity of associative algebras, Theoret. Comput. Sci. 15 (1981), no. 2, 201–211. MR MR623595 (82g:68038)
[AS13] V.B., Alekseev and A.V., Smirnov, On the exact and approximate bilinear complexities of multiplication of 4 × 2 and 2 × 2 matrices, Proc. Steklov Inst. Math. 282 (2013), no. 1, 123–139.
[AT92] N., Alon and M., Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992), no. 2, 125–134. MR 1179249
[Atk83] M.D., Atkinson, Primitive spaces of matrices of bounded rank. II, J. Austral. Math. Soc. Ser. A 34 (1983), no. 3, 306–315. MR 695915
[AV08] M., Agrawal and V., Vinay, Arithmetic circuits: A chasm at depth four, Proc. 49th IEEE Symposium on Foundations of Computer Science (2008), 67–75.
[AW07] Kaan, Akin and Jerzy, Weyman, Primary ideals associated to the linear strands of Lascoux's resolution and syzygies of the corresponding irreducible representations of the Lie superalgebra gl(mn), J. Algebra 310 (2007), no. 2, 461–490. MR 2308168 (2009c:17007)
[Bar77] W., Barth, Moduli of vector bundles on the projective plane, Invent. Math. 42 (1977), 63–91. MR MR0460330 (57 #324)
[Bas15] Saugata, Basu, A complexity theory of constructible functions and sheaves, Found. Comput. Math. 15 (2015), no. 1, 199–279. MR 3303696
[BB] Austin R., Benson and Grey, Ballard, A framework for practical parallel fast matrix multiplication, arXiv:1409.2908 (2014).
[BB14] Weronika, Buczynska and Jaroslaw, Buczynski, Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes, J. Algebraic Geom. 23 (2014), no. 1, 63–90. MR 3121848
[BCRL79] Dario, Bini, Milvio, Capovani, Francesco, Romani, and Grazia, Lotti, O(n2.7799) complexity for n × n approximate matrix multiplication, Inform. Process. Lett. 8 (1979), no. 5, 234–235. MR MR534068 (80h:68024)
[BCS97] Peter, Bürgisser, Michael, Clausen, and M., Amin Shokrollahi, with Thomas Lickteig, Algebraic complexity theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 315, Springer, Berlin, 1997. MR 99c:68002
[BDHM15] A., Bernardi, N.S., Daleo, J.D., Hauenstein, and B., Mourrain, Tensor decomposition and homotopy continuation, ArXiv e-prints (2015).
[Bea00] Arnaud, Beauville, Determinantal hypersurfaces, Mich.Math. J. 48 (2000), 39–64. MR 1786479 (2002b:14060)
[BGL13] Jaroslaw, Buczynski, Adam, Ginensky, and J.M., Landsberg, Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture, J. Lond. Math. Soc. (2) 88 (2013), no. 1, 1–24. MR 3092255
[BGL14] H., Bermudez, S., Garibaldi, and V., Larsen, Linear preservers and representations with a 1-dimensional ring of invariants, Trans. Am.Math. Soc. 366 (2014), no. 9, 4755–4780. MR 3217699
[BILR] Grey, Ballard, Christian, Ikenmeyer, J.M., Landsberg, and Nick, Ryder, The geometry of rank decompositions of matrix multiplication ii: 3x3 matrices, preprint.
[Bin80] D., Bini, Relations between exact and approximate bilinear algorithms. Applications, Calcolo 17 (1980), no. 1, 87–97. MR 605920 (83f:68043b)
[BIP16] Peter, Bürgisser, Christian, Ikenmeyer, and Greta, Panova, No occurrence obstructions in geometric complexity theory, CoRR abs/1604.06431 (2016).
[BKs07] Anita, Buckley and Tomaž, Košir, Determinantal representations of smooth cubic surfaces, Geom. Dedicata 125 (2007), 115–140. MR 2322544
[BL89] S.C., Black and R.J., List, A note on plethysm, Eur. J. Combin. 10 (1989), no. 1, 111–112. MR 977186 (89m:20011)
[BL14] Jaroslaw, Buczynski and J.M., Landsberg, On the third secant variety, J. Algebraic Combin. 40 (2014), no. 2, 475–502. MR 3239293
[BL16] M., Bläser and V., Lysikov, On degeneration of tensors and algebras, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016), LIPIcs (2016), 58: 19:1–19:11.
[Blä00] Markus, Bläser, Lower bounds for the bilinear complexity of associative algebras, Comput. Complexity 9 (2000), no. 2, 73–112. MR 1809686
[Bla01a] Markus, Bläser, Improvements of the Alder-Strassen bound: algebras with nonzero radical, Automata, Languages and Programming, Lecture Notes in Comput. Sci., vol. 2076, Springer, Berlin, 2001, pp. 79–91. MR 2065853
[Blä01b] Markus, Bläser, Complete problems for Valiant's class of qp-computable families of polynomials, Computing and Combinatorics (Guilin, 2001), Lecture Notes in Comput. Sci., vol. 2108, Springer, Berlin, 2001, pp. 1–10. MR 1935355 (2003j:68051)
[Blä03], On the complexity of the multiplication of matrices of small formats, J. Complexity 19 (2003), no. 1, 43–60. MR MR1951322 (2003k:68040)
[Blä13] Markus, Bläser, Fast matrix multiplication, Graduate Surveys, no. 5, Theory of Computing Library, 2013.
[Blä14] Markus, Bläser, Explicit tensors, Perspectives in Computational Complexity, Springer, New York, 2014, pp. 117–130.
[BLMW11] Peter, Bürgisser, J.M., Landsberg, Laurent, Manivel, and Jerzy, Weyman, An overview of mathematical issues arising in the geometric complexity theory approach to VP _= VNP, SIAM J. Comput. 40 (2011), no. 4, 1179–1209. MR 2861717
[BLR80] Dario, Bini, Grazia, Lotti, and Francesco, Romani, Approximate solutions for the bilinear form computational problem, SIAM J. Comput. 9 (1980), no. 4, 692–697. MR MR592760 (82a:68065)
[BO08] Maria Chiara, Brambilla and Giorgio, Ottaviani, On the Alexander-Hirschowitz theorem, J. Pure Appl. Algebra 212 (2008), no. 5, 1229–1251. MR 2387598 (2008m:14104)
[BOC92] Micheal, Ben Or and Richard, Cleve, Computing algebraic formulas using a constant number of registers, SIAM J. Comput. 21 (1992), no. 21, 54–58.
[BR13] Alessandra, Bernardi and Kristian, Ranestad, On the cactus rank of cubics forms, J. Symbolic Comput. 50 (2013), 291–297. MR 2996880
[Bre70] R.P., Brent, Algorithms for matrix multiplication, Technical Report TR-CS-70-157 DCS, Stanford (1970), 1–52.
[Bre74] Richard P., Brent, The parallel evaluation of general arithmetic expressions, J. Assoc. Comput. Mach. 21 (1974), 201–206. MR 0660280 (58 #31996)
[Bri93] A., Brill, Uber symmetrische functionen von variabelnpaaren, Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen 20 (1893), 757–762.
[Bri93] Michel, Brion, Stable properties of plethysm: on two conjectures of Foulkes, Manuscripta Math. 80 (1993), no. 4, 347–371. MR MR1243152 (95c:20056)
[Bri97] Michel, Brion, Sur certains modules gradues associes aux produits symetriques, Algebre non commutative, groupes quantiques et invariants (Reims, 1995), Sémin. Congr., vol. 2, Soc. Math. France, Paris, 1997, pp. 157–183. MR 1601139 (99e:20054)
[Bri02] Emmanuel, Briand, Polynomes multisymetriques, Ph.D. thesis, Université de Rennes 1 et Universidad de Cantabria, 2002.
[Bri10], Covariants vanishing on totally decomposable forms, Liaison, Schottky problem and invariant theory, Progr. Math., vol. 280, Birkhäuser Verlag, Basel, 2010, pp. 237–256. MR 2664658
[Bsh98] Nader H., Bshouty, On the direct sum conjecture in the straight line model, J. Complexity 14 (1998), no. 1, 49–62. MR 1617757 (99c:13056)
[BT15] Grigoriy, Blekherman and Zach, Teitler, On maximum, typical and generic ranks, Math. Ann. 362 (2015), no. 3-4, 1021–1031. MR 3368091
[Bur14] Vladimir P., Burichenko, On symmetries of the strassen algorithm, CoRR abs/1408.6273 (2014).
[Bur15] Vladimir P., Burichenko, Symmetries of matrix multiplication algorithms. I, CoRRabs/1508.01110 (2015).
[Cai90] Jin-Yi, Cai, A note on the determinant and permanent problem, Inform. and Comput. 84 (1990), no. 1, 119–127. MR MR1032157 (91d:68028)
[CCC+15a] E., Carlini, M.V., Catalisano, L., Chiantini, A.V., Geramita, and Y., Woo, Symmetric tensors: rank, Strassen's conjecture and e-computability, ArXiv e-prints (2015).
[CCC15b] Enrico, Carlini, Maria, Virginia Catalisano, and Luca, Chiantini, Progress on the symmetric Strassen conjecture, J. Pure Appl. Algebra 219 (2015), no. 8, 3149– 3157. MR 3320211
[CEVV09] Dustin A., Cartwright, Daniel, Erman, Mauricio, Velasco, and Bianca, Viray, Hilbert schemes of 8 points, Algebra Number Theory 3 (2009), no. 7, 763–795. MR 2579394
[CHI+] Luca, Chiantini, Jon, Hauenstein, Christian, Ikenmeyer, J.M., Landsberg, and Giorgio, Ottaviani, Polynomials and the exponent of matrix multiplication, in preparation.
[CILO16] Luca, Chiantini, Christian, Ikenmeyer, J.M., Landsberg, and Giorgio, Ottaviani, The geometry of rank decompositions of matrix multiplication I: 2x2 matrices, CoRR abs/1610.08364 (2016).
[CIM17] Man-Wai, Cheung, Christian, Ikenmeyer, and Sevak, Mkrtchyan, Symmetrizing tableaux and the 5th case of the Foulkes conjecture, J. Symbolic Comput. 80 (2017), part 3, 833–843. MR 3574536
[CKSU05] H., Cohn, R., Kleinberg, B., Szegedy, and C., Umans, Group-theoretic algorithms for matrix multiplication, Proceedings of the 46th Annual Symposium on Foundations of Computer Science (2005), 379–388.
[CKSV16] S., Chillara, M., Kumar, R., Saptharishi, and V., Vinay, The chasm at depth four, and tensor rank: old results, new insights, ArXiv e-prints (2016).
[CKW10] Xi, Chen, Neeraj, Kayal, and Avi, Wigderson, Partial derivatives in arithmetic complexity and beyond, Found. Trends Theor. Comput. Sci. 6 (2010), no. 1–2, front matter, 1–138 (2011). MR 2901512
[Com02] P., Comon, Tensor decompositions, state of the art and applications, Mathematics in Signal Processing V (J.G. McWhirter and I.K., Proudler, eds.), Clarendon Press, Oxford, 2002, arXiv:0905.0454v1, pp. 1–24.
[Csa76] L., Csanky, Fast parallel matrix inversion algorithms, SIAM J. Comput. 5 (1976), no. 4, 618–623. MR 0455310 (56 #13549)
[CU03] H., Cohn and C., Umans, A group theoretic approach to fast matrix multiplication, Proceedings of the 44th Annual Symposium on Foundations of Computer Science (2003), no. 2, 438–449.
[CU13] H., Cohn and C., Umans, Fast matrix multiplication using coherent configurations, Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (2013), no. 2, 1074–1087.
[CW82] D., Coppersmith and S., Winograd, On the asymptotic complexity of matrix multiplication, SIAM J. Comput. 11 (1982), no. 3, 472–492. MR 664715 (83j:68047b)
[CW90] Don, Coppersmith and Shmuel, Winograd, Matrix multiplication via arithmetic progressions, J. Symbolic Comput. 9 (1990), no. 3, 251–280. MR 91i:68058
[CW16] Nicholas J., Cavenagh and Ian M, Wanless, There are asymptotically the same number of Latin squares of each parity, Bull. Austr. Math. Soc. First View (2016), 1–8.
[dG78] Hans F., de Groote, On varieties of optimal algorithms for the computation of bilinear mappings. I. The isotropy group of a bilinear mapping, Theoret. Comput. Sci. 7 (1978), no. 1, 1–24. MR 0506377 (58 #22132)
[Die49] Jean, Dieudonné, Sur une generalisation du groupe orthogonal a quatre variables, Arch. Math. 1 (1949), 282–287. MR 0029360 (10,586l)
[Dri97] Arthur A., Drisko, On the number of even and odd Latin squares of order p + 1, Adv. Math. 128 (1997), no. 1, 20–35. MR 1451417 (98e:05018)
[DS13] A. M., Davie and A. J., Stothers, Improved bound for complexity of matrix multiplication, Proc. R. Soc. Edinburgh Sect. A 143 (2013), no. 2, 351–369. MR 3039815
[dSP16] Clément de Seguins Pazzis, Large spaces of bounded rank matrices revisited, Linear Algebra Appl. 504 (2016), 124–189. MR 3502533
[Dyn52] E. B., Dynkin, Maximal subgroups of the classical groups, Trudy Moskov. Mat. Obšc. 1 (1952), 39–166. MR 0049903 (14,244d)
[EH88] David, Eisenbud and Joe, Harris, Vector spaces of matrices of low rank, Adv. Math. 70 (1988), no. 2, 135–155. MR 954659
[Eis95] David, Eisenbud, Commutative Algebra, Graduate Texts in Mathematics, vol. 150, Springer, New York, 1995. MR MR1322960 (97a:13001)
[Eis05] David, Eisenbud, The Geometry of Syzygies, Graduate Texts in Mathematics, vol. 229, Springer, New York, 2005. MR 2103875 (2005h:13021)
[ELSW15] K., Efremenko, J. M., Landsberg, H., Schenck, and J., Weyman, On minimal free resolutions of sub-permanents and other ideals arising in complexity theory, ArXiv e-prints (2015).
[ELSW16], The method of shifted partial derivatives cannot separate the permanent from the determinant, ArXiv e-prints, to appear in MCOM (2016).
[FH91] William, Fulton and Joe, Harris, Representation Theory, Graduate Texts in Mathematics, vol. 129, Springer, New York, 1991. MR 1153249 (93a:20069)
[Fis94] Ismor, Fischer, Sums of like powers of multivariate linear forms, Math. Mag. 67 (1994), no. 1, 59–61. MR 1573008
[FLR85] P., Fillmore, C., Laurie, and H., Radjavi, On matrix spaces with zero determinant, Linear Multilinear Algebra 18 (1985), no. 3, 255–266. MR 828407
[Fou50] H. O., Foulkes, Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form, J. London Math. Soc. 25 (1950), 205–209. MR MR0037276 (12,236e)
[Fro97] G., Frobenius, Uber die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber Deutsch. Akad. Wiss. Berlin (1897), 994–1015.
[Frö85] Ralf, Fröberg, An inequality for Hilbert series of graded algebras, Math. Scand. 56 (1985), no. 2, 117–144. MR 813632 (87f:13022)
[FS13a] Michael A., Forbes and Amir, Shpilka, Explicit Noether normalization for simultaneous conjugation via polynomial identity testing, Approximation, Randomization, and Combinatorial Optimization, Lecture Notes in Comput. Sci., vol. 8096, Springer, Heidelberg, 2013, pp. 527–542. MR 3126552
[FS13b] Michael A., Forbes and Amir, Shpilka, Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs, 54th annual symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 2013. MR 3246226
[FSS13] Michael A., Forbes, Ramprasad, Saptharishi, and Amir, Shpilka, Pseudorandomness for multilinear read-once algebraic branching programs, in any order, CoRR abs/1309.5668 (2013).
[FW84] Ephraim, Feig and Shmuel, Winograd, On the direct sum conjecture, Linear Algebra Appl. 63 (1984), 193–219. MR 766508 (86h:15022)
[Gal17] Maciej, Galcazka, Vector bundles give equations of cactus varieties, Linear Algebra Appl. 521 (2017), 254–262. MR 3611482
[Gat87] Joachim von zur, Gathen, Feasible arithmetic computations: Valiant's hypothesis, J. Symbolic Comput. 4 (1987), no. 2, 137–172. MR MR922386 (89f:68021)
[Gay76] David A., Gay, Characters of the Weyl group of SU(n) on zero weight spaces and centralizers of permutation representations, Rocky Mountain J. Math. 6 (1976), no. 3, 449–455. MR MR0414794 (54 #2886)
[Ger61] Murray, Gerstenhaber, On dominance and varieties of commuting matrices, Ann. Math. 73 (1961), 324–348. MR 0132079 (24 #A1926)
[Ger89] Anthony V., Geramita (ed.), The Curves Seminar at Queen's. Vol. VI, Queen's Papers in Pure and Applied Mathematics, vol. 83, Queen's University, Kingston, ON, 1989, Papers from the seminar held at Queen's University, Kingston, Ontario, 1989. MR 1036030
[Ges16] Fulvio, Gesmundo, Geometric aspects of iterated matrix multiplication, J. Algebra 461 (2016), 42–64. MR 3513064
[GGOW15] A., Garg, L., Gurvits, R., Oliveira, and A., Wigderson, A deterministic polynomial time algorithm for non-commutative rational identity testing with applications, ArXiv e-prints (2015).
[GH79] Phillip, Griffiths and Joseph, Harris, Algebraic geometry and local differential geometry, Ann. Sci. École Norm. Sup. 12 (1979), no. 3, 355–452. MR 81k:53004
[GHIL16] Fulvio, Gesmundo, Jonathan D., Hauenstein, Christian, Ikenmeyer, and J.M., Landsberg, Complexity of linear circuits and geometry, Found. Comput.Math. 16 (2016), no. 3, 599–635. MR 3494506
[GHPS14] Zachary A., Griffin, Jonathan D., Hauenstein, Chris, Peterson, and Andrew J., Sommese, Numerical computation of the Hilbert function and regularity of a zero dimensional scheme, Connections between algebra, combinatorics, and geometry, Springer Proc. Math. Stat., vol. 76, Springer, New York, 2014, pp. 235–250. MR 3213522
[GKKS13a] Ankit, Gupta, Pritish, Kamath, Neeraj, Kayal, and Ramprasad, Saptharishi, Approaching the chasm at depth four, Proceedings of the Conference on Computational Complexity (CCC) (2013).
[GKKS13b] Ankit, Gupta, Pritish, Kamath, Neeraj, Kayal, and Ramprasad, Saptharishi, Arithmetic circuits: A chasm at depth three, Electronic Colloquium on Computational Complexity (ECCC) 20 (2013), 26.
[GKKS17] Ankit, Gupta, Pritish, Kamath, Neeraj, Kayal, and Ramprasad, Saptharishi, Unexpected power of low-depth arithmetic circuits, preprint, to appear in CACM (2017).
[GKZ94] I. M., Gel_fand, M. M., Kapranov, and A. V., Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser, Boston, MA, 1994. MR 95e:14045
[GL17] Gesmundo, Fulvio and Landsberg, J.M., Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K.Mulmuley arXiv:1705.03866.
[Gly10] David G., Glynn, The conjectures of Alon-Tarsi and Rota in dimension prime minus one, SIAM J. Discrete Math. 24 (2010), no. 2, 394–399. MR 2646093 (2011i:05034)
[Gly13] David G., Glynn, Permanent formulae from the Veronesean, Des. Codes Cryptogr. 68 (2013), no. 1–3, 39–47. MR 3046335
[GM16] J. A., Grochow and C., Moore, Matrix multiplication algorithms from group orbits, ArXiv e-prints (2016).
[Gor94] P., Gordan, Das Zerfallen der Curven in gerade Linien, Math. Ann. 45 (1894), no. 3, 410–427. MR 1510871
[Got78] Gerd, Gotzmann, Eine Bedingung fur die Flachheit und das Hilbertpolynom eines graduierten Ringes, Math. Z. 158 (1978), no. 1, 61–70. MR 0480478 (58 #641)
[Gra55] H., Grassmann, Die stereometrischen Gleichungen dritten Grades, und die dadurch erzeugten Oberflachen, J. Reine Angew. Math. 49 (1855), 47–65. MR 1578905
[Gre78] Edward L., Green, Complete intersections and Gorenstein ideals, J. Algebra 52 (1978), no. 1, 264–273. MR 0480472
[Gre98] Mark L., Green, Generic initial ideals, Six lectures on commutative algebra (Bellaterra, 1996), Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 119–186. MR 1648665 (99m:13040)
[Gre11] Bruno, Grenet, An Upper Bound for the Permanent versus Determinant Problem, manuscript (submitted), 2011.
[Gri86] B., Griesser, A lower bound for the border rank of a bilinearmap, Calcolo 23 (1986), no. 2, 105–114 (1987). MR 88g:15021
[Gua15a] Y., Guan, Brill's equations as a GL(V)-module, ArXiv e-prints (2015).
[Gua15b] Y., Guan, Flattenings and Koszul Young flattenings arising in complexity theory, ArXiv e-prints (2015).
[Gun86] S., Gundelfinger, Zur theorie der binaren formen, J. Reine Angew. Math. 100 (1886), 413–424.
[GW09] Roe, Goodman and Nolan R., Wallach, Symmetry, representations, and invariants, Graduate Texts inMathematics, vol. 255, Springer, Dordrecht, 2009. MR 2522486
[Had97] J., Hadamard, Memoire sur l'elimination, Acta Math. 20 (1897), no. 1, 201–238. MR 1554881
[Had99] J., Hadamard, Sur les conditions de decomposition des formes, Bull. Soc. Math. France 27 (1899), 34–47. MR 1504330
[Har77] Robin, Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52 Springer, New York, 1977, MR MR0463157 (57 #3116)
[Har95] Joe, Harris, Algebraic geometry, Graduate Texts inMathematics, vol. 133, Springer, New York, 1995. MR 1416564 (97e:14001)
[HIL13] Jonathan D., Hauenstein, Christian, Ikenmeyer, and J. M., Landsberg, Equations for lower bounds on border rank, Exp. Math. 22 (2013), no. 4, 372–383. MR 3171099
[HL16] Jesko, Hüttenhain and Pierre, Lairez, The boundary of the orbit of the 3-by-3 determinant polynomial, C. R., Math. Acad. Sci. Paris 354 (2016), no. 9, 931–935. MR 3535348
[How87] Roger, Howe, (GLn, GLm)-duality and symmetric plethysm, Proc. Ind. Acad. Sci. Math. Sci. 97 (1987), no. 1–3, 85–109 (1988). MR MR983608 (90b:22020)
[HR94] Rosa, Huang and Gian-Carlo, Rota, On the relations of various conjectures on Latin squares and straightening coefficients, Discrete Math. 128 (1994), no. 1–3, 225– 236. MR 1271866 (95i:05036)
[HS82] J., Heintz and C.-P., Schnorr, Testing polynomials which are easy to compute, Logic and algorithmic (Zurich, 1980), Monograph. Enseign. Math., vol. 30, Univ. Geneve, Geneva, 1982, pp. 237–254. MR 648305
[HS13] Jonathan D., Hauenstein and Andrew J., Sommese, Membership tests for images of algebraic sets by linear projections, Appl. Math. Comput. 219 (2013), no. 12, 6809–6818. MR 3027848
[Hü17] Jesko, Hüttenhain, A Note on normalizations of orbit closures, Commun. Algebra 45 (2017), no. 9, 3716–3723. MR 3627624
[HWY10] Pavel, Hrubes, Avi, Wigderson, and Amir, Yehudayoff, Relationless completeness and separations, 25th annual Conference on Computational Complexity—CCC 2010, IEEE Computer Soc., Los Alamitos, CA, 2010, pp. 280–290. MR 2932363
[Iar95] A., Iarrobino, Inverse system of a symbolic power. II. TheWaring problem for forms, J. Algebra 174 (1995), no. 3, 1091–1110. MR 1337187
[Iar97] A., Iarrobino, Inverse system of a symbolic power. III. Thin algebras and fat points, Compositio Math. 108 (1997), no. 3, 319–356. MR 1473851 (98k:13017)
[IE78] A., Iarrobino and J., Emsalem, Some zero-dimensional generic singularities; finite algebras having small tangent space, Compositio Math. 36 (1978), no. 2, 145–188. MR 515043
[IK99] Anthony, Iarrobino and Vassil, Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721, Springer, Berlin, 1999, Appendix C by Iarrobino and Steven L. Kleiman. MRMR 1735271 (2001d:14056)
[Ike15] C., Ikenmeyer, On McKay's propagation theorem for the Foulkes conjecture, ArXiv e-prints (2015).
[IL99] Bo, Ilic and J. M., Landsberg, On symmetric degeneracy loci, spaces of symmetric matrices of constant rank and dual varieties, Math. Ann. 314 (1999), no. 1, 159– 174. MR MR1689267 (2000e:14091)
[IL16a] C., Ikenmeyer and J. M., Landsberg, On the complexity of the permanent in various computational models, ArXiv e-prints, to appear in JPAA (2016).
[IL16b] Thomas A., Ivey and J. M., Landsberg, Cartan for beginners: differential geometry via moving frames and exterior differential systems, second edition, Graduate Studies in Mathematics, vol. 175, American Mathematical Society, Providence, RI, 2016.
[IM05] Atanas, Iliev and Laurent, Manivel, Varieties of reductions for gln , Projective varieties with unexpected properties, Walter de Gruyter, Berlin, 2005, pp. 287–316. MR MR2202260 (2006j:14056)
[IP15] C., Ikenmeyer and G., Panova, Rectangular Kronecker coefficients and plethysms in geometric complexity theory, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (2015).
[JM86] Rodney W., Johnson and Aileen M., McLoughlin, Noncommutative bilinear algorithms for 3 × 3 matrix multiplication, SIAM J. Comput. 15 (1986), no. 2, 595–603. MR 837607
[JT86] Joseph, Ja'Ja' and Jean, Takche, On the validity of the direct sum conjecture, SIAM J. Comput. 15 (1986), no. 4, 1004–1020. MR MR861366 (88b:68084)
[Kem78] George R., Kempf, Instability in invariant theory, Ann. Math. 108 (1978), no. 2, 299–316. MR MR506989 (80c:20057)
[KL14] Harlan, Kadish and J. M., Landsberg, Padded polynomials, their cousins, and geometric complexity theory, Comm. Algebra 42 (2014), no. 5, 2171–2180. MR 3169697
[KL15] Shrawan, Kumar and J. M., Landsberg, Connections between conjectures of Alon- Tarsi, Hadamard-Howe, and integrals over the special unitary group, Discrete Math. 338 (2015), no. 7, 1232–1238. MR 3322811
[KLPSMN09] Abhinav, Kumar, Satyanarayana V., Lokam, Vijay M., Patankar, and Jayalal Sarma, M. N., Using elimination theory to construct rigid matrices, Foundations of software technology and theoretical computer science—FSTTCS 2009, LIPIcs. Leibniz Int. Proc. Inform., vol. 4, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2009, pp. 299–310. MR 2870721
[Koi12] Pascal, Koiran, Arithmetic circuits: the chasm at depth four gets wider, Theoret. Comput. Sci. 448 (2012), 56–65. MR 2943969
[Kum13] Shrawan, Kumar, Geometry of orbits of permanents and determinants, Comment. Math. Helv. 88 (2013), no. 3, 759–788. MR 3093509
[Kum15] Shrawan, Kumar, A study of the representations supported by the orbit closure of the determinant, Compos. Math. 151 (2015), no. 2, 292–312. MR 3314828
[Lad76] Julian D., Laderman, A noncommutative algorithm for multiplying 3 × 3 matrices using 23 muliplications, Bull. Am. Math. Soc. 82 (1976), no. 1, 126–128. MR MR0395320 (52 #16117)
[Lan06] J. M., Landsberg, The border rank of the multiplication of 2 × 2 matrices is seven, J. Am. Math. Soc. 19 (2006), no. 2, 447–459. MR 2188132 (2006j:68034)
[Lan10] J. M., Landsberg, P versus NP and geometry, J. Symbolic Comput. 45 (2010), no. 12, 1369– 1377. MR 2733384 (2012c:68065)
[Lan12] J. M., Landsberg, Tensors: geometry and applications, Graduate Studies in Mathematics, vol. 128, American Mathematical Society, Providence, RI, 2012. MR 2865915
[Lan14] J. M., Landsberg, New lower bounds for the rank of matrix multiplication, SIAM J. Comput. 43 (2014), no. 1, 144–149. MR 3162411
[Lan15a] J. M., Landsberg, Geometric complexity theory: an introduction for geometers, Ann. Univ. Ferrara Sez. VII Sci. Mat. 61 (2015), no. 1, 65–117. MR 3343444
[Lan15b] J. M., Landsberg, Nontriviality of equations and explicit tensors in Cm ⊗ Cm ⊗ Cm of border rank at least 2m− 2, J. Pure Appl. Algebra 219 (2015), no. 8, 3677–3684. MR 3320240
[Las78] Alain, Lascoux, Syzygies des varietes determinantales, Adv. in Math. 30 (1978), no. 3, 202–237. MR 520233 (80j:14043)
[Lee16] Hwangrae, Lee, Power sum decompositions of elementary symmetric polynomials, Linear Algebra Appl. 492 (2016), 89–97. MR 3440150
[Lei16] Arielle, Leitner, Limits under conjugacy of the diagonal subgroup in SLn(R), Proc. Am. Math. Soc. 144 (2016), no. 8, 3243–3254. MR 3503693
[LG14] François, Le Gall, Powers of tensors and fast matrix multiplication, ISSAC 2014— Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2014, pp. 296–303. MR 3239939
[Lic84] Thomas, Lickteig, A note on border rank, Inform. Process. Lett. 18 (1984), no. 3, 173–178. MR 86c:68040
[Lic85] Thomas, Lickteig, Typical tensorial rank, Linear Algebra Appl. 69 (1985), 95–120. MR 87f:15017
[Lit06] Dudley E., Littlewood, The theory of group characters and matrix representations of groups, AMS Chelsea, Providence, RI, 2006, reprint of the second (1950) edition. MR MR2213154 (2006m:20013)
[LM04] J. M., Landsberg and Laurent, Manivel, On the ideals of secant varieties of Segre varieties, Found. Comput. Math. 4 (2004), no. 4, 397–422. MR MR2097214 (2005m:14101)
[LM08a] J. M., Landsberg and L., Manivel, Generalizations of Strassen's equations for secant varieties of Segre varieties, Comm. Algebra 36 (2008), no. 2, 405–422. MR 2387532 (2009f:14109)
[LM08b] J. M., Landsberg and Laurent, Manivel, Generalizations of Strassen's equations for secant varieties of Segre varieties, Comm. Algebra 36 (2008), no. 2, 405–422. MR MR2387532
[LM15] J. M., Landsberg and M., Michalek, Abelian Tensors, ArXiv e-prints, to appear in JMPA (2015).
[LM17a] J. M., Landsberg, A2n 2 − log(n) − 1 lower bound for the border rank of matrix multiplication, Int Math Res Notices (2017).
[LM17b] J. M., Landsberg and Mateusz, Michalek, On the Geometry of Border Rank Decompositions for Matrix Multiplication and Other Tensors with Symmetry, SIAM J. Appl. Algebra Geom. 1 (2017), no. 1, 2–19. MR 3633766
[LMR13] Joseph M., Landsberg, Laurent, Manivel, and Nicolas, Ressayre, Hypersurfaces with degenerate duals and the geometric complexity theory program, Comment. Math. Helv. 88 (2013), no. 2, 469–484. MR 3048194
[LMS16] Nutan, Limaye, Guillaume, Malod, and Srikanth, Srinivasan, Lower bounds for noncommutative skew circuits, Theory Comput. 12 (2016), Paper No. 12, 38. MR 3546728
[LO13] J. M., Landsberg and Giorgio, Ottaviani, Equations for secant varieties of Veronese and other varieties, Ann. Mat. Pura Appl. (4) 192 (2013), no. 4, 569–606. MR 3081636
[LO15] Joseph M., Landsberg and Giorgio, Ottaviani, New lower bounds for the border rank of matrix multiplication, Theory Comput. 11 (2015), 285–298. MR 3376667
[Lok08] Satyanarayana V., Lokam, Complexity lower bounds using linear algebra, Found. Trends Theor. Comput. Sci. 4 (2008), no. 1–2, front matter, 1–155 (2009). MR 2539154 (2011c:68060)
[LR06] Hong, Liu and Kenneth W., Regan, Improved construction for universality of determinant and permanent, Inform. Process. Lett. 100 (2006), no. 6, 233–237. MR 2270826 (2007f:68084)
[LR15] J. M., Landsberg and N., Ressayre, Permanent v. determinant: an exponential lower bound assumingsymmetry and a potential path towards Valiant's conjecture, ArXiv e-prints, to appear in DGA special issue on Geometry and complexity (2015).
[LR17] J. M., Landsberg and Nicholas, Ryder, On the Geometry of Border Rank Algorithms for n × 2 by 2 × 2 Matrix Multiplication, Exp. Math. 26 (2017), no. 3, 275–286. MR 3642105
[Mac95] I. G., Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs, Clarendon Press, New York, 1995, with contributions by A. Zelevinsky, Oxford Science Publications. MR 1354144 (96h:05207)
[Man97] Laurent, Manivel, Applications de Gauss et plethysme, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 3, 715–773. MR MR1465785 (98h:20078)
[Man98] Laurent, Manivel, Gaussian maps and plethysm, Algebraic geometry (Catania, 1993/ Barcelona, 1994), Lecture Notes in Pure and Appl. Math., vol. 200, Dekker, New York, 1998, pp. 91–117. MR MR1651092 (99h:20070)
[Man15a] Laurent, Manivel, On the asymptotics of Kronecker coefficients, J. Algebraic Combin. 42 (2015), no. 4, 999–1025. MR 3417256
[Man15b] Laurent, Manivel, On the asymptotics of Kronecker coefficients, 2, Sém. Lothar.Combin. 75 (2015), Art. B75d, 13. MR 3461556
[Mat60] Yozô, Matsushima, Espaces homogenes de Stein des groupes de Lie complexes, Nagoya Math. J 16 (1960), 205–218. MR MR0109854 (22 #739)
[McK08] Tom, McKay, On plethysm conjectures of Stanley and Foulkes, J. Algebra 319 (2008), no. 5, 2050–2071. MR 2394689 (2008m:20023)
[MM61] Marvin, Marcus and Henryk, Minc, On the relation between the determinant and the permanent, Illinois J. Math. 5 (1961), 376–381. MR 0147488 (26 #5004)
[MM62] Marvin, Marcus and F. C., May, The permanent function, Can. J. Math. 14 (1962), 177–189. MR MR0137729 (25 #1178)
[MN05] Jurgen, Müller and Max, Neunhöffer, Some computations regarding Foulkes' conjecture, Exp. Math. 14 (2005), no. 3, 277–283. MR MR2172706 (2006e:05186)
[MO34] S., Mazur and W., Orlicz, Grundlegende eigenschaften der polynomischen operationen, Studia Math. 5 (1934), 50–68.
[MP08] G., Malod and N., Portier, Characterizing Valiant's algebraic complexity classes, J. Complexity 24 (2008), 16–38.
[MP13] Davesh, Maulik and Rahul, Pandharipande, Gromov-Witten theory and Noether- Lefschetz theory, A celebration of algebraic geometry, Clay Math. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 2013, pp. 469–507. MR 3114953
[MR04] Thierry, Mignon and Nicolas, Ressayre, A quadratic bound for the determinant and permanent problem, Int. Math. Res. Not. (2004), no. 79, 4241–4253. MR MR2126826 (2006b:15015)
[MR13] Alex, Massarenti and Emanuele, Raviolo, The rank of n × n matrix multiplication is at least 3n 2 – 2-√2n3/2 , Linear Algebra Appl. 438 (2013), no. 11, 4500–4509. MR 3034546
[MS01] Ketan D., Mulmuley and Milind, Sohoni, Geometric complexity theory. I. An approach to the P vs. NP and related problems, SIAM J. Comput. 31 (2001), no. 2, 496–526 (electronic). MR MR1861288 (2003a:68047)
[MS08] Ketan D., Mulmuley, Geometric complexity theory. II. Towards explicit obstructions for embeddings among class varieties, SIAM J. Comput. 38 (2008), no. 3, 1175–1206. MR MR2421083
[Mul99] Ketan, Mulmuley, Lower bounds in a parallel model without bit operations, SIAM J. Comput. 28 (1999), no. 4, 1460–1509 (electronic). MR 1681069
[Mul12] Ketan, Mulmuley, Geometric complexity theory V: equivalence between blackbox derandomization of polynomial identity testing and derandomization of noether's normalization lemma, CoRR abs/1209.5993 (2012).
[Mul14] K., Mulmuley, The GCT chasm, lecture (2014).
[Mum66] David, Mumford, Lectures on curves on an algebraic surface, with a section by G. M., Bergman. Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N. J., 1966. MR 0209285
[Mum95] David, Mumford, Algebraic geometry. I, Classics in Mathematics, Springer, Berlin, 1995, Complex projective varieties, reprint of the 1976 edition. MR 1344216 (96d:14001)
[MV97] Meena, Mahajan and V., Vinay, Determinant: combinatorics, algorithms, and complexity, Chicago J. Theoret. Comput. Sci. (1997), Article 5, 26 pp. (electronic).MR 1484546 (98m:15016)
[Nis91] Noam, Nisan, Lower bounds for non-commutative computation, Proceedings of the 23rd annual Symposium on Theory of Computing, ACM, New York, 1991, pp. 410–418.
[NR16] J. F., Nash and M. T., Rassias, Open problems in mathematics, Springer International, Berlin, 2016.
[NW97] Noam, Nisan and Avi, Wigderson, Lower bounds on arithmetic circuits via partial derivatives, Comput. Complexity 6 (1996/97), no. 3, 217–234. MR 1486927 (99f:68107)
[Oed16] L., Oeding, Border ranks of monomials, ArXiv e-prints (2016).
[Ott07] Giorgio, Ottaviani, Symplectic bundles on the plane, secant varieties and Luroth quartics revisited, Vector bundles and low codimensional subvarieties: state of the art and recent developments, Quad. Mat., vol. 21, Dept. Math., Seconda Univ. Napoli, Caserta, 2007, pp. 315–352. MR 2554725
[Pan66] V. Ja., Pan, On means of calculating values of polynomials, Uspehi Mat. Nauk 21 (1966), no. 1 (127), 103–134. MR 0207178
[Pan78] V. Ya., Pan, Strassen's algorithm is not optimal. Trilinear technique of aggregating, uniting and canceling for constructing fast algorithms for matrix operations, 19th annual Symposium on Foundations of Computer Science, IEEE, Long Beach, CA, 1978, pp. 166–176. MR 539838
[Pra94] V. V., Prasolov, Problems and theorems in linear algebra, Translations of Mathematical Monographs, vol. 134, American Mathematical Society, Providence, RI, 1994, translated from the Russian manuscript by D. A.|Leıtes. MR 1277174
[Pro76] C., Procesi, The invariant theory of n × n matrices, Adv. Math. 19 (1976), no. 3, 306–381. MR 0419491
[Pro07] Claudio, Procesi, Lie groups, Universitext, Springer, New York, 2007. MR MR2265844 (2007j:22016)
[Raz74] Ju. P., Razmyslov, Identities with trace in full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 723–756. MR 0506414
[Raz09] Ran, Raz, Multi-linear formulas for permanent and determinant are of superpolynomial size, J. ACM 56 (2009), no. 2, Art. 8, 17. MR 2535881
[Raz10a] Ran, Raz, Elusive functions and lower bounds for arithmetic circuits, Theory Comput. 6 (2010), 135–177. MR 2719753
[Raz10b] Ran, Raz, Tensor-rank and lower bounds for arithmetic formulas, Proceedings of the 2010 International Symposium on Theory of Computing, ACM, New York, 2010, pp. 659–666. MR 2743315 (2011i:68044)
[Rob17] P., Roberts, A minimal free complex associated to the minors of a matrix, JCA, to appear (2017).
[RS00] Kristian, Ranestad and Frank-Olaf, Schreyer, Varieties of sums of powers, J. Reine Angew. Math. 525 (2000), 147–181. MR MR1780430 (2001m:14009)
[RS11] Kristian, Ranestad, On the rank of a symmetric form, J. Algebra 346 (2011), 340–342. MR 2842085 (2012j:13037)
[Sap] R., Saptharishi, A survey of lower bounds in arithmetic circuit complexity,
[Sax08] Nitin, Saxena, Diagonal circuit identity testing and lower bounds, Automata, languages and programming. Part I, Lecture Notes in Comput. Sci., vol. 5125, Springer, Berlin, 2008, pp. 60–71. MR 2500261
[Sch81] A., Schönhage, Partial and total matrix multiplication, SIAM J. Comput. 10 (1981), no. 3, 434–455. MR MR623057 (82h:68070)
[Seg10] C., Segre, Preliminari di una teoria delle varieta luoghi di spazi, Rend. Circ. Mat. Palermo XXX (1910), 87–121.
[Sha07] Igor R., Shafarevich, Basic algebraic geometry.1 3rd ed. Springer, Heidelberg, 2013, translated from the 2007 third Russian edition by Miles Reid.
[Sha13] Igor R., Shafarevich, Basic algebraic geometry. 2, 3rd ed., Springer, Heidelberg, 2013, Schemes and complex manifolds, translated from the 2007 third Russian edition by Miles Reid. MR 3100288
[Shp02] Amir, Shpilka, Affine projections of symmetric polynomials, J. Comput. System Sci. 65 (2002), no. 4, 639–659, special issue on complexity, 2001 (Chicago, IL). MR 1964647
[Sip92] Michael, Sipser, The history and status of the p versus np question, Proceedings of the 24th annual Symposium on Theory of Computing, ACM, 1992, pp. 603–618.
[Smi13] A. V., Smirnov, The bilinear complexity and practical algorithms for matrix multiplication, Comput.Math.Math. Phys. 53 (2013), no. 12, 1781–1795. MR 3146566
[Spi79] Michael, Spivak, A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish Inc., Wilmington, DE, 1979. MR MR532830 (82g:53003a)
[SS42] R., Salem and D. C., Spencer, On sets of integers which contain no three terms in arithmetical progression, Proc. Natl. Acad. Sci. U.S.A. 28 (1942), 561–563. MR 0007405
[SS09] Jessica, Sidman and Seth, Sullivant, Prolongations and computational algebra, Can. J. Math. 61 (2009), no. 4, 930–949. MR 2541390
[Sto] A., Stothers, On the complexity of matrix multiplication, PhD thesis, University of Edinburgh, 2010.
[Str69] Volker, Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969), 354–356. MR 40 #2223
[Str73] Volker, Strassen, Vermeidung von Divisionen, J. Reine Angew. Math. 264 (1973), 184–202. MR MR0521168 (58 #25128)
[Str83] V., Strassen, Rank and optimal computation of generic tensors, Linear Algebra Appl. 52/53 (1983), 645–685. MR 85b:15039
[Str87] V., Strassen, Relative bilinear complexity and matrix multiplication, J. Reine Angew. Math. 375/376 (1987), 406–443. MR MR882307 (88h:11026)
[Str91] V., Strassen, Degeneration and complexity of bilinear maps: some asymptotic spectra, J. Reine Angew. Math. 413 (1991), 127–180. MR 92m:11038
[SVW01] A. J., Sommese, J., Verschelde, and C. W., Wampler, Using monodromy to decompose solution sets of polynomial systems into irreducible components, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Academic, Dordrecht, 2001, pp. 297–315. MR 1866906
[SVW02] Andrew J., Sommese, Jan, Verschelde, and Charles W., Wampler, Symmetric functions applied to decomposing solution sets of polynomial systems, SIAMJ. Numer. Anal. 40 (2002), no. 6, 2026–2046 (2003). MR 1974173
[SW01] Amir, Shpilka and Avi, Wigderson, Depth-3 arithmetic circuits over fields of characteristic zero, Comput. Complexity 10 (2001), no. 1, 1–27. MR 1867306 (2003a:68048)
[SY09] Amir, Shpilka and Amir, Yehudayoff, Arithmetic circuits: a survey of recent results and open questions, Found. Trends Theor. Comput. Sci. 5 (2009), no. 3–4, 207–388 (2010). MR 2756166
[Syl52] J. J., Sylvester, On the principles of the calculus of forms, Cambridge and Dublin Math. J. (1852), 52–97.
[Tav15] Sébastien, Tavenas, Improved bounds for reduction to depth 4 and depth 3, Inf. Comput. 240 (2015), 2–11. MR 3303254
[Ter11] A., Terracini, Sulla vk per cui la varieta degli sh(h + 1)-seganti ha dimensione minore dell'ordinario, Rend. Circ. Mat. Palermo 31 (1911), 392–396.
[Tod92] S., Toda, Classes of arithmetic circuits capturing the complexity of computing the determinant, IEICE Trans. Inf. Syst. E75-D (1992), 116–124.
[Toe77] Emil, Toeplitz, Ueber ein Flachennetz zweiter Ordnung, Math. Ann. 11 (1877), no. 3, 434–463. MR 1509924
[Tra84] B. A., Trakhtenbrot, A survey of Russian approaches to perebor (brute-force search) algorithms, Ann. Hist. Comput. 6 (1984), no. 4, 384–400. MR 763733
[Val77] Leslie G., Valiant, Graph-theoretic arguments in low-level complexity, Mathematical foundations of computer science (Proc. Sixth Sympos., Tatranská Lomnica, 1977), Springer, Berlin, 1977, pp. 162–176. Lecture Notes in Comput. Sci., Vol. 53. MR 0660702 (58 #32067)
[Val79] Leslie G., Valiant, Completeness classes in algebra, Proceedings of the 11th STOC, ACM, 1979, pp. 249–261.
[VSBR83] L. G., Valiant, S., Skyum, S., Berkowitz, and C., Rackoff, Fast parallel computation of polynomials using few processors, SIAM J. Comput. 12 (1983), no. 4, 641–644. MR 721003 (86a:68044)
[vzG87] Joachim von, zur Gathen, Permanent and determinant, Linear Algebra Appl. 96 (1987), 87–100. MR MR910987 (89a:15005)
[Wah91] Jonathan, Wahl, Gaussian maps and tensor products of irreducible representations, Manuscripta Math. 73 (1991), no. 3, 229–259. MR 1132139 (92m:14066a)
[Wey97] Hermann, Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997, Their invariants and representations, 15th printing, Princeton Paperbacks. MR 1488158 (98k:01049)
[Wey03] Jerzy, Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003. MR MR1988690 (2004d:13020)
[Wil] Virginia, Williams, Breaking the coppersimith-winograd barrier, preprint.
[Win71] S., Winograd, On multiplication of 2 × 2 matrices, Linear Algebra and Appl. 4 (1971), 381–388. MR 45 #6173
[Ye11] Ke, Ye, The stabilizer of immanants, Linear Algebra Appl. 435 (2011), no. 5, 1085– 1098. MR 2807220 (2012e:15017)
[Zui15] J., Zuiddam, A note on the gap between rank and border rank, ArXiv e-prints (2015).


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 673 *
Loading metrics...

Book summary page views

Total views: 1836 *
Loading metrics...

* Views captured on Cambridge Core between 3rd October 2017 - 22nd April 2018. This data will be updated every 24 hours.