Toward a Salmon Conjecture

References

• Abo , [Abo 09] H. , Ottaviani , G. and Peterson , C. 2009 . “Induction for Secant Varieties of Segre Varieties” . Trans. Amer. Math. Soc. , 361 ( 2 ) : 767 – 792 .
   Web of Science ®Google Scholar
• Allman , [Allman 10] E . 2010 . “ Open Problem: Determine the Ideal Defining ” . Available online (http://www.dms.uaf.edu/~eallman/Papers/salmonPrize.pdf)
   Google Scholar
• [Allman and Rhodes 03] E. Allman and Rhodes , J. 2003 . “Phylogenetic Invariants for the General Markov Model of Sequence Mutation.” . Math. Biosci. , 186 : 113 – 144 .
   PubMed Web of Science ®Google Scholar
• Allman , [Allman and Rhodes 08] E. and Rhodes , J. 2008 . “Phylogenetic Ideals and Varieties for the General Markov Model” . Adv. in Appl. Math. , 40 ( 2 ) : 127 – 148 .
   Web of Science ®Google Scholar
• Bates , [Bates 08] D. J. , Hauenstein , J. D. , Sommese , A. J. and Wampler , C. W. 2008 . “Adaptive Multiprecision Path Tracking.” . SIAM J. Num. Analysis , 46 : 722 – 746 .
   Web of Science ®Google Scholar
• Bates , [Bates 09] D. J. , Hauenstein , J. D. , Sommese , A. J. and Wampler , C. W. 2009 . “Stepsize Control for Adaptive Precision Path Tracking. ” . In Interactions of Classical and Numerical Algebraic Geometry , 21 – 31 . Contemporary Mathematics 496 . Providence: American Mathematical Society
   Google Scholar
• Bates , [Bates 10a] D. J. , Hauenstein , J. D. , Sommese , A. J. and Wampler , C. W. 2010 . “Benchmarking Bertini, Software for Numerical Algebraic Geometry.” . Manuscript
   Google Scholar
• Bates , [Bates 10b] D. J. , Hauenstein , J. D. , Sommese , A. J. and Wampler , C. W. 2010 . “Bertini: Software for Numerical Algebraic Geometry.” . Available online (http://www.nd.edu/sommese/bertini)
   Google Scholar
• Casanellas , [Casanellas and Fernandez-Sanchez 09] M. and Fernandez-Sanchez , J. 2009 . “Relevant Phylogenetic Invariants of Evolutionary Models.” . ArXiv e-prints
   Google Scholar
• [Catalisano 08] , Catalisano , M. V. , Geramita , A. V. and Gimigliano , A. 2008 . “On the Ideals of Secant Varieties to Certain Rational Varieties” . J. Algebra , 319 ( 5 ) : 1913 – 1931 .
   Web of Science ®Google Scholar
• Decker , [Decker 10] W. G.-M. , Greuel, Pfister , G. and H. , Schönemann . 2010 . “Singular 3–1-1: A Computer Algebra System for Polynomial Computations.” . Available online (http://www.singular.uni-kl.de)
   Google Scholar
• Friedland , [Friedland 10] S. 2010 . “On Tensors of Border Rank l in .” . In To appear in Linear Algebra and Its Applications
   Google Scholar
• Friedland , [Friedland and Gross 11] S. and Gross , E. 2011 . “A Proof of the Set-Theoretic Version of the Salmon Conjecture.” . ArXiv e-prints .
   Google Scholar
• Fulton , [Fulton and Harris 91] W. and Harris , J. 1991 . “ Representation Theory: A First Course ” . In Graduate Texts in Mathematics , Vol. 129 , New York : Springer-Verlag .
   Google Scholar
• Hauenstein , [Hauenstein 11] J. D. , Sommese , A. J. and Wampler , C. W. 2011 . “Regeneration Homotopies for Solving Systems of Polynomials.” . Math. Comp. , 80 : 345 – 377 .
   Web of Science ®Google Scholar
• Kredel , [Kredel 09] H. 2009 . “Distributed Parallel Groebner Bases Computation.” . In 2010 International Conference on Complex, Intelligent and Software Intensive Systems , 518 – 524 . Krakow, Poland : IEEE .
   Google Scholar
• [Landsberg 08] and Landsberg , J. M. 2008 . “Geometry and the Complexity of Matrix Multiplication” . Bull. Amer. Math. Soc. (N.S.) , 45 ( 2 ) : 247 – 284 .
   Web of Science ®Google Scholar
• Landsberg , [Landsberg 11] J. M. 2011 . The Geometry of Tensors: Applications to Complexity, Statistics, and Other Areas , In preparation .
   Google Scholar
• [Landsberg and Manivel 04] , Landsberg , J. M. and Manivel , L. 2004 . “On the Ideals of Secant Varieties of Segre Varieties” . Found. Comput. Math. , 4 ( 4 ) : 397 – 422 .
   Web of Science ®Google Scholar
• [Landsberg and Manivel 08] , Landsberg , J. M. and Manivel , Laurent . 2008 . “Generalizations of Strassen’s Equations for Secant Varieties of Segre Varieties” . Comm. Algebra , 36 ( 2 ) : 405 – 422 .
   Web of Science ®Google Scholar
• Landsberg , [Landsberg and Ottaviani 10] J. M. and Ottaviani , G. 2010 . “Equations for Secant Varieties to Veronese Varieties.” , Manuscript .
   Google Scholar
• [Landsberg and Weyman 07] , Landsberg , J. M. and Weyman , J. 2007 . “On the Ideals and Singularities of Secant Varieties of Segre Varieties” . Bull. Lond. Math. Soc. , 39 ( 4 ) : 685 – 697 .
   Google Scholar
• Lee , [Lee 10] T.-L. , Li , T. Y. and Tsai , C.-H. 2010 . “Hom4ps-2.0: A Software Package for Solving Polynomials Systems by the Polyhedral Homotopy Continuation Method.” . Available online (http://www.mth.msu.edu/li/)
   Google Scholar
• Li , [Li 03] T. Y. 2003 . “Numerical Solution of Polynomial Systems by Homotopy Continuation Methods.” . In Handbook of Numerical Analysis, Special Volume: Foundations of Computational Mathematics , Vol. XI , 209 – 304 . Amsterdam: North-Holland .
   Google Scholar
• Oeding. , [Oeding 08] L . 2008 . “Set Theoretic Defining Equations of the Variety of Principal Minors of Symmetric Matrices.” . In To appear in Algebra and Number Theory , ArXiv e-prints .
   Google Scholar
• Oeding. , [Oeding 09] L . 2009 . G-Varieties and the Principal Minors of Symmetric Matrices , Ph.D. thesis, Texas A&M University .
   Google Scholar
• Oeding , [Oeding 11] L. 2011 . “Set-Theoretic Defining Equations of the Tangential Variety of the Segre Variety” . Journal of Pure and Applied Algebra , 215 ( 6 ) : 1516 – 1527 .
   Web of Science ®Google Scholar
• Ottaviani. , [Ottaviani 07] G . 2008 . “Symplectic Bundles on the Plane, Secant Varieties and Lüroth Quartics Revisited.” . In Quaderni di Matematica , Edited by: Casnati , G. F. and Catanese , R. 351 – 352 . Notari . Aracne, vol. 21
   Google Scholar
• Sidman , [Sidman and Sullivant 09] J. and Sullivant , S. 2009 . “Prolongations and Computational Algebra” . Canad. J. Math. , 61 ( 4 ) : 930 – 949 .
   Web of Science ®Google Scholar
• [Sommese and Wampler 05] , Sommese , A. J. and Wampler , C. W. 2005 . Numerical Solution of Polynomial Systems Arising in Engineering and Science , Singapore : World Scientific .
   Google Scholar
• Strassen. , [Strassen 83] V . 1983 . “Rank and Optimal Computation of Generic Tensors.” . Linear Algebra Appl. , 52/53 : 645 – 685 .
   Web of Science ®Google Scholar
• Sturmfels. , [Sturmfels 09] B . 2009 . “Open Problems in Algebraic Statistics.” . In Emerging Applications of Algebraic Geometry , Edited by: Sullivant , S. and Sturmfels , B. 1 – 13 . New York : IMA Volumes in Mathematics and Its Applications 149 . Springer
   Google Scholar
• van Leeuwen , [Van Leeuwen 92] M. A. A. , Coehn , A. M. and Lisser , B. 1992 . “LiE: A Package for Lie Group Computations.” . Computer Algebra Nederland .
   Google Scholar
• Verschelde. , [Verschelde 10] J . 2010 . “PHCpack: A General-Purpose Solver for Polynomial Systems by Homotopy Continuation.” . Available online (http://www.math.uic.edu/jan/PHCpack/phcpack.html
   Google Scholar