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Original Articles

On the typical rank of real binary forms

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Pages 657-667 | Received 15 Oct 2009, Accepted 13 Sep 2011, Published online: 25 Nov 2011

References

  • Abo , H , Ottaviani , G and Peterson , C . 2009 . Induction for secant varieties of Segre varieties . Trans. Amer. Math. Soc. , 361 : 767 – 792 .
  • Alexander , J and Hirschowitz , A . 1995 . Polynomial interpolation in several variables . J. Algebraic Geom. , 4 : 201 – 222 .
  • Bader , BW and Kolda , T . 2006 . Algorithm 862: Matlab tensor classes for fast algorithm prototyping . ACM Trans. Math. Soft. , 32 : 635 – 653 .
  • Bernardi , A , Gimigliano , A and Idà , M . 2011 . Computing symmetric rank for symmetric tensors . J. Symbolic Comput. , 46 : 34 – 53 .
  • Brambilla , MC and Ottaviani , G . 2008 . On the Alexander–Hirschowitz theorem . J. Pure Appl. Algebra , 212 : 1229 – 1251 .
  • Buczynski , J and Landsberg , JM . Ranks of tensors and a generalization of secant varieties , arXiv:0909.4262
  • Bürgisser , P , Clausen , M and Shokrollahi , MA . 1997 . Algebraic Complexity Theory , Vol. 315 , Berlin, Heidelberg : Springer .
  • Carroll , JD and Chang , JJ . 1970 . Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition . Psychometrika , 35 : 283 – 319 .
  • Catalisano , MV , Geramita , AV and Gimigliano , A . 2011 . Secant varieties of P 1 × ··· × P 1 (n-times) are not defective for n ≥ 5 . J. Alg. Geom. , 20 : 295 – 327 .
  • Causa , A and Re , R . On the maximum rank of a real binary form , Ann. Mate. Pura Appl. 190 (2011), pp. 55–59
  • Ciliberto , C . 2001 . “ Geometric aspects of polynomial interpolation in more variables and of Waring's problem ” . In European Congress of Mathematics, Vol. I (Barcelona, 2000) , Edited by: Casacuberta , C , Miro-Roig , RM , Verdera , J and Xambo-Descamps , S . 289 – 316 . Basel : 201 of Progress in Mathematics, Birkhäuser .
  • Comas , G and Seiguer , M . On the rank of a binary form, 2001 , Found. Comput. Math., 11 (2011), pp. 65–78
  • Comon , P . 2002 . “ Tensor decompositions, state of the art and applications ” . In Mathematics in Signal Processing V , Edited by: McWhirter , JG and Proudler , IK . 1 – 24 . Oxford , , UK : Clarendon Press .
  • Comon , P and Mourrain , B . 1996 . Decomposition of Quantics in Sums of Powers of linear forms . Signal Process. E. , 53 : 93 – 107 . (special issue on High-Order Statistics)
  • Comon , P , Ten Berge , JMF , De Lathauwer , L and Castaing , J . 2009 . Generic and typical ranks of multi-way arrays . Linear Algebra Appl. , 430 : 2997 – 3007 .
  • De Silva , V and Lim , L-H . 2008 . Tensor rank and the ill-posedness of the best low-rank approximation problem . SIAM J. Matrix Anal. Appl. , 30 : 1084 – 1127 .
  • Ehrenborg , R and Rota , GC . 1993 . Apolarity and canonical forms for homogeneous polynomials . Eur. J. Combin. , 14 : 157 – 181 .
  • Friedland , S . 2011 . On the generic rank of 3-tensors . Linear Algebra Appl. , arXiv:0805.3777 (to appear)
  • Gelfand , IM , Kapranov , MM and Zelevinsky , AV . 1994 . Discriminants, Resultants, and Multidimensional Discriminants , Boston : Birkhäuser .
  • Hackbusch , W , Khoromskij , BN and Tyrtyshnikov , EE . 2005 . Hierarchical Kronecker tensor-product approximation . J. Numer. Math. , 13 : 119 – 156 .
  • Iarrobino , A and Kanev , V . 1999 . Power Sums, Gorenstein Algebras and Determinantal Loci , Berlin, Heidelberg : Springer .
  • Kroonenberg , P . 2008 . Applied Multiway Data Analysis , Hoboken , NJ : Wiley .
  • Landsberg , JM and Teitler , Z . 2010 . On the ranks and border ranks of symmetric tensors . Found. Comput. Math. , 10 : 339 – 366 .
  • Lickteig , T . 1985 . Typical tensorial rank . Linear Algebra Appl. , 69 : 95 – 120 .
  • Morup , M and Hansen , LK . 2006 . Parallel factor analysis as an exploratory tool for wavelet transformed event-related EEG . NeuroImage , 29 : 938 – 947 .
  • Reznick , B . 1992 . Sums of even powers of real linear forms . Mem. AMS , 96 : 1 – 155 .
  • Reznick , B . 2010 . Laws of inertia in higher degree binary forms . Proc. Am. Math. Soc. , 138 : 815 – 826 .
  • Sidiropoulos , ND , Bro , R and Giannakis , GB . 2000 . Parallel factor analysis in sensor array processing . IEEE Trans. Sig. Proc. , 48 : 2377 – 2388 .
  • Smilde , A , Bro , R and Geladi , P . 2004 . Multi-Way Analysis , Chichester , , UK : Wiley .
  • Ten Berge , JMF . 2000 . The typical rank of tall three-way arrays . Psychometrika , 65 : 525 – 532 .

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