Nonlinear algebra and applications. (English) Zbl 1522.13040
In this article, some overview of explicit applications of Non-linear Algebra in different areas of sciences and engineering is presented.
Reviewer: Hanieh Keneshlou (Leipzig)
MSC:
| 13P25 | Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) |
| 14Q20 | Effectivity, complexity and computational aspects of algebraic geometry |
| 62R01 | Algebraic statistics |
Keywords:
nonlinear algebra; polynomial optimization; partial differential equations; algebraic statistics; integrable systems; configuration spaces; reaction networks; computer vision; tensor decompositionsSoftware:
Julia; Bertini; hgm; gfun; DiscreteStatisticalModelsWithRationalMLE; NumericalAlgebraicGeometry; Theta.jl; GraphicalModels; NAG4M2; ore_algebra; hgm R; PHCpack; SageMath; SumsOfSquares; AlgebraicOptimization; Magma; R; crntwin; Maple; Macaulay2; SINGULAR; CoNtRol; dmod.lib; SE-Sync; Hom4PS-3References:
| [1] | S. P. G. Abenda Grinevich, Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline soliton, Selecta Math., 25, 1-64 (2019) · Zbl 1426.37052 · doi:10.1007/s00029-019-0488-5 |
| [2] | M. J. Ablowitz, Line soliton interactions, Available at: https://sites.google.com/site/ablowitz/line-solitons. |
| [3] | M. J. D. E. Ablowitz Baldwin, Nonlinear shallow ocean-wave soliton interactions on flat beaches, Phys. Rev. E (3), 86, 036305 (2012) |
| [4] | M. F. A. L. A.-L. B. Adamer Lőrincz Sattelberger Sturmfels, Algebraic analysis of rotation data, Algebraic Statistics, 11, 189-211 (2020) · Zbl 1461.62220 · doi:10.2140/astat.2020.11.189 |
| [5] | M. J. Adler Moser, On a class of polynomials connected with the Korteweg-de Vries equation, Comm. Math. Phys., 61, 1-30 (1978) · Zbl 0428.35067 |
| [6] | S. Y. N. I. B. S. M. R. Agarwal Furukawa Snavely Simon Curless Seitz Szeliski, Building Rome in a day, Communications of the ACM, 54, 105-112 (2011) |
| [7] | S. Agarwal, A. Pryhuber, R. Sinn and R. R. Thomas, The chiral domain of a camera arrangement, arXiv: 2003.09265, 2020. · Zbl 07632073 |
| [8] | D. C. Agostini Améndola, Discrete Gaussian distributions via theta functions, SIAGA, 2, 1-30 (2019) · Zbl 1425.60020 · doi:10.1137/18M1164937 |
| [9] | D. Agostini, T. Ö. Çelik and D. Eken, Numerical reconstruction of curves from their Jacobians, arXiv: 2103.03138, 2021. · Zbl 1506.14115 |
| [10] | D. Agostini, T. Ö. Çelik, J. Struwe and B. Sturmfels, Theta surfaces, Vietnam J. Math., 2020. |
| [11] | D. Agostini, T. Ö. Çelik and B. Sturmfels, The Dubrovin threefold of an algebraic curve, arXiv: 2005.08244, 2020. To appear in Nonlinearity. · Zbl 1465.14036 |
| [12] | D. L. Agostini Chua, Computing theta functions with Julia, J. Softw. Algebra Geom., 11, 41-51 (2021) · Zbl 1483.14002 · doi:10.2140/jsag.2021.11.41 |
| [13] | D. Agostini, C. Fevola, Y. Mandelshtam and B. Sturmfels, KP solitons from tropical limits, arXiv: 2101.10392, 2021. · Zbl 1495.14046 |
| [14] | C. L. Aholt Oeding, The ideal of the trifocal variety, Math. Comp., 83, 2553-2574 (2014) · Zbl 1347.13011 · doi:10.1090/S0025-5718-2014-02842-1 |
| [15] | C. B. R. Aholt Sturmfels Thomas, A Hilbert scheme in computer vision, Canad. J. Math., 65, 961-988 (2013) · Zbl 1284.13035 · doi:10.4153/CJM-2012-023-2 |
| [16] | R. Ait El Manssour and A.-L. Sattelberger, Combinatorial differential algebra of \(x^p\), arXiv: 2102.03182, 2021. |
| [17] | E. S. S. J. A. S. Allman Petrović Rhodes Sullivant, Identifiability of two-tree mixtures for group-based models, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 8, 710-722 (2010) |
| [18] | C. Améndola, L. D. García-Puente, R. Homs, O. Kuznetsova and H. J. Motwani, Computing maximum likelihood estimates for Gaussian graphical models with Macaulay2, arXiv: 2012.11572, 2020. · Zbl 1514.62040 |
| [19] | C. K. P. A. Améndola Kohn Reichenbach Seigal, Invariant theory and scaling algorithms for maximum likelihood estimation, SIAM Journal on Applied Algebra and Geometry, 5, 304-337 (2021) · Zbl 1518.14065 · doi:10.1137/20M1328932 |
| [20] | E. Angelini, On complex and real identifiability of tensors, Rivista di Matematica della Universita di Parma, 8, 367-377 (2017) · Zbl 1391.14104 |
| [21] | S. Aoki, H. Hara and A. Takemura, Markov Bases in Algebraic Statistics, Volume 199, Springer Science & Business Media, 2012. · Zbl 1304.62015 |
| [22] | F. B. A. Arrigoni Rossi Fusiello, Spectral synchronization of multiple views in SE (3), SIAM Journal on Imaging Sciences, 9, 1963-1990 (2016) · Zbl 1409.68299 · doi:10.1137/16M1060248 |
| [23] | D. J. Bates, J. D. Hauenstein, A. J. Sommese and C. W. Wampler, Bertini: Software for Numerical Algebraic Geometry, Available at bertini.nd.edu with permanent doi: dx.doi.org/10.7274/R0H41PB5. |
| [24] | E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its and V. B. Matveev, Algebro-Geometric Approach to Non- Linear Integrable Equations, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1994. · Zbl 0809.35001 |
| [25] | C. P. N. Beltrán Breiding Vannieuwenhoven, Pencil-based algorithms for tensor rank decomposition are not stable, SIAM J. Matrix Anal. Appl., 40, 02 (2019) · Zbl 1451.14170 · doi:10.1137/18M1200531 |
| [26] | A. Bernardi, C. De Lazzari and F. Gesmundo, Dimension of tensor network varieties, arXiv: 2101.03148, 2021. |
| [27] | D. I. S. Bernstein Sullivant, Unimodular binary hierarchical models, J. Combin. Theory Ser. A, 123, 97-125 (2017) · Zbl 1354.05150 · doi:10.1016/j.jctb.2016.11.003 |
| [28] | J. A. S. V. B. Bezanson Edelman Karpinski Shah, Julia: A fresh approach to numerical computing, SIAM Review, 59, 65-98 (2017) · Zbl 1356.68030 · doi:10.1137/141000671 |
| [29] | F. A. M. Bihan Dickenstein Giaroli, Lower bounds for positive roots and regions of multistationarity in chemical reaction networks, J. Algebra, 542, 367-411 (2020) · Zbl 1453.92120 · doi:10.1016/j.jalgebra.2019.10.002 |
| [30] | J.-E. Björk, Analytic \({\mathcal{D}} \)-Modules and Applications, Volume 247 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1993. · Zbl 0805.32001 |
| [31] | G. Blekherman, P. Parrilo and R. Thomas, Semidefinite Optimization and Convex Algebraic Geometry, Society for Industrial and Applied Mathematics, 2012. |
| [32] | T. Boege, J. I. Coons, C. Eur, A. Maraj and F. Röttger, Reciprocal maximum likelihood degrees of Brownian motion tree models, arXiv: 2009.11849, 2020. |
| [33] | W. J. C. Bosma Cannon Playoust, The Magma algebra system, I. The user language, J. Symbolic Comput., 24, 235-265 (1997) · Zbl 0898.68039 · doi:10.1006/jsco.1996.0125 |
| [34] | P. Breiding and S. Timme, Homotopy continuation, jl: A package for homotopy continuation in Julia, In Mathematical Software - ICMS 2018, Springer International Publishing, (2018), 458-465. · Zbl 1396.14003 |
| [35] | N. Bruin, J. Sijsling and A. Zotine, Numerical computation of endomorphism rings of Jacobians, In 13th Algorithmic Number Theory Symposium, volume 2 of Open Book Series, (2019), 155-171. · Zbl 1516.14060 |
| [36] | C. H. J. Bruschek Mourtada Schepers, Arc spaces and Rogers-Ramanujan identities, Ramanujan J., 30, 9-38 (2013) · Zbl 1281.14003 · doi:10.1007/s11139-012-9401-y |
| [37] | V. V. D. Buchstaber Enolski Leykin, Rational analogs of abelian functions, Funct. Anal. Appl., 33, 83-94 (1999) · Zbl 1056.14049 · doi:10.1007/BF02465189 |
| [38] | N. Budur, Bernstein-Sato ideals and local systems, Ann. Inst. Fourier (Grenoble), 65, 549-603 (2015) · Zbl 1332.32038 |
| [39] | N. M. Z. Budur Mustaţǎ Teitler, The monodromy conjecture for hyperplane arrangements, Geom. Dedicata, 153, 131-137 (2011) · Zbl 1227.32035 · doi:10.1007/s10711-010-9560-1 |
| [40] | N. Budur, R. van der Veer, L. Wu and P. Zhou, Zero loci of Bernstein-Sato ideals, arXiv: 1907.04010, 2019. To appear in Invent. Math. |
| [41] | B. F. Caviness and J. R. Johnson, Quantifier Elimination And Cylindrical Algebraic Decomposition, Springer Science & Business Media, 2012. |
| [42] | T. Ö. Çelik, Thomae-Weber formula: Algebraic computations of theta constants, Int. Math. Res. Not. IMRN, rnz302. |
| [43] | T. Ö. A. G. B. L. Çelik Jamneshan Montúfar Sturmfels Venturello, Wasserstein distance to independence models, J. Symbolic Comput., 104, 855-873 (2021) · Zbl 1460.62205 · doi:10.1016/j.jsc.2020.10.005 |
| [44] | T. Ö. A. G. B. L. Çelik Jamneshan Montúfar Sturmfels Venturello, Optimal transport to a variety, Mathematical Aspects of Computer and Information Sciences, Springer Lecture Notes in Computer Science, 11989, 364-381 (2020) · Zbl 07441083 · doi:10.1007/s00454-021-00322-3 |
| [45] | C. Chen, J. H. Davenport, M. Moreno Maza, B. Xia and R. Xiao, Computing with semi-algebraic sets represented by triangular decomposition, In Proceedings of the 2011 International Symposium on Symbolic and Algebraic Computation (ISSAC 2011), ACM Press, (2011), 75-82. · Zbl 1323.14029 |
| [46] | T. Chen, T.-L. Lee and T.-Y. Li, Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods, In Mathematical Software - ICMS 2014 (eds. H. Hong and C. Yap), Springer Berlin Heidelberg, (2014), 183-190. · Zbl 1434.65065 |
| [47] | L. G. N. Chiantini Ottaviani Vannieuwenhoven, On generic identifiability of symmetric tensors of subgeneric rank, Trans. Amer. Math. Soc., 369, 4021-4042 (2017) · Zbl 1360.14021 · doi:10.1090/tran/6762 |
| [48] | Y. Cid-Ruiz and B. Sturmfels, Primary decomposition with differential operators, arXiv: 2101.03643, 2021. · Zbl 1524.13078 |
| [49] | D. T. P. Cifuentes Kahle Parrilo, Sums of squares in Macaulay2, J. Softw. Algebra Geom., 10, 17-24 (2020) · Zbl 1439.13068 · doi:10.2140/jsag.2020.10.17 |
| [50] | G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decompostion, in Automata Theory and Formal Languages, Springer, (1975), 134-183. · Zbl 0318.02051 |
| [51] | R. Connelly, Generic global rigidity, Discrete Comput. Geom., 33, 549-563 (2005) · Zbl 1072.52016 · doi:10.1007/s00454-004-1124-4 |
| [52] | R. H. Connelly Servatius, Higher-order rigidity—what is the proper definition?, Discrete Comput. Geom., 11, 193-200 (1994) · Zbl 0793.52005 · doi:10.1007/BF02574003 |
| [53] | C. Conradi, E. Feliu, M. Mincheva and C. Wiuf, Identifying parameter regions for multistationarity, PLoS Comput. Biol., 13 (2017), e1005751. |
| [54] | C. D. J. J. Conradi Flockerzi Raisch Stelling, Subnetwork analysis reveals dynamic features of complex (bio)chemical networks, Proc. Nat. Acad. Sci. USA, 104, 19175-19180 (2007) |
| [55] | C. M. A. Conradi Mincheva Shiu, Emergence of oscillations in a mixed-mechanism phosphorylation system, Bull. Math. Biol., 81, 1829-1852 (2019) · Zbl 1415.92092 · doi:10.1007/s11538-019-00580-6 |
| [56] | J. I. Coons, J. Cummings, B. Hollering and A. Maraj, Generalized cut polytopes for binary hierarchical models, arXiv: 2008.00043, 2020. |
| [57] | J. I. S. Coons Sullivant, Quasi-independence models with rational maximum likelihood estimator, J. Symbolic Comput., 104, 917-941 (2021) · Zbl 1461.62226 · doi:10.1016/j.jsc.2020.10.006 |
| [58] | J. I. S. Coons Sullivant, Toric geometry of the Cavender-Farris-Neyman model with a molecular clock, Adv. in Appl. Math, 123, 102119 (2021) · Zbl 1457.92125 · doi:10.1016/j.aam.2020.102119 |
| [59] | S. C. Coutinho, A Primer of Algebraic \(D\)-Modules, Volume 33 of Student Texts, London Mathematical Society, 1995. · Zbl 0848.16019 |
| [60] | D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics. Springer New York, 4th edition, 2007. · Zbl 1118.13001 |
| [61] | G. J. W. R. J. Craciun Helton Williams, Homotopy methods for counting reaction network equilibria, Math. Biosci, 216, 140-149 (2008) · Zbl 1153.92015 · doi:10.1016/j.mbs.2008.09.001 |
| [62] | L. De Lathauwer, Blind separation of exponential polynomials and the decomposition of a tensor in rank-\((L_r, L_r, 1)\) terms, SIAM J. Matrix Anal. Appl., 32, 1451-1474 (2011) · Zbl 1239.15016 · doi:10.1137/100805510 |
| [63] | J. A. De Loera, J. Rambau and F. Santos, Triangulations, Volume 25 of Algorithms and Computations in Mathematics, Springer-Verlag Berlin Heidelberg, 2010. · Zbl 1207.52002 |
| [64] | W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-1-3 —A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, 2020. |
| [65] | B. M. A. M. M. Deconinck Heil Bobenko van Hoeij Schmies, Computing Riemann theta functions, Math. Comp., 73, 1417-1442 (2004) · Zbl 1092.33018 · doi:10.1090/S0025-5718-03-01609-0 |
| [66] | M. Demazure, Sur deux problemes de reconstruction, Technical Report RR-0882, INRIA, July 1988, Available at: https://hal.inria.fr/inria-00075672. |
| [67] | H. Derksen, V. Makam and M. Walter, Maximum likelihood estimation for tensor normal models via castling transforms, arXiv: 2011.03849, 2020. · Zbl 1505.13012 |
| [68] | P. B. et Diaconis Sturmfels al, Algebraic algorithms for sampling from conditional distributions, Ann. Statist., 26, 363-397 (1998) · Zbl 0952.62088 · doi:10.1214/aos/1030563990 |
| [69] | A. M. P. A. X. Dickenstein Millan Shiu Tang, Multistationarity in structured reaction networks, Bull. Math. Biol., 81, 1527-1581 (2019) · Zbl 1415.92083 · doi:10.1007/s11538-019-00572-6 |
| [70] | V. Dolotin and A. Morozov, Introduction to Non-linear Algebra, World Scientific Publishing, Hackensack, NJ, 2007. · Zbl 1134.15001 |
| [71] | I. L. D. Domanov Lathauwer, On Uniqueness and computation of the decomposition of a tensor into multilinear rank-\((1, L_r, L_r)\) terms, SIAM J. Matrix Anal. Appl., 41, 747-803 (2020) · Zbl 1442.15042 · doi:10.1137/18M1206849 |
| [72] | P. M. A. C. Donnell Banaji Marginean Pantea, Control: an open source framework for the analysis of chemical reaction networks, Bioinformatics, 30, 1633-1634 (2014) |
| [73] | J. R. R. A. Draisma Eggermont Krone Leykin, Noetherianity for infinite-dimensional toric varieties, Algebra Number Theory, 9, 1857-1880 (2015) · Zbl 1354.13025 · doi:10.2140/ant.2015.9.1857 |
| [74] | J. E. G. B. R. R. Draisma Horobeţ Ottaviani Sturmfels Thomas, The Euclidean distance degree of an algebraic variety, Found. Comput. Math., 16, 99-149 (2016) · Zbl 1370.51020 · doi:10.1007/s10208-014-9240-x |
| [75] | J. S. P. et Draisma Kuhnt Zwiernik al, Groups acting on Gaussian graphical models, Ann. Statist., 41, 1944-1969 (2013) · Zbl 1292.62098 · doi:10.1214/13-AOS1130 |
| [76] | J. Draisma, G. Ottaviani and A. Tocino, Best rank-k approximations for tensors: generalizing Eckart-Young, Res. Math. Sci., (2018), 5-27. · Zbl 1417.15034 |
| [77] | E. C. Duarte Görgen, Equations defining probability tree models, J. Symbolic Comput., 99, 127-146 (2020) · Zbl 1451.13086 · doi:10.1016/j.jsc.2019.04.001 |
| [78] | E. O. B. Duarte Marigliano Sturmfels, Discrete statistical models with rational maximum likelihood etimates, Bernoulli, 27, 135-154 (2021) · Zbl 1465.62184 · doi:10.3150/20-BEJ1231 |
| [79] | B. Dubrovin, Theta functions and non-linear equations, Russian Math. Surveys, 36, 11-92 (1981) · Zbl 0549.58038 |
| [80] | T. Duff, K. Kohn, A. Leykin and T. Pajdla, PLMP-point-line minimal problems in complete multi-view visibility, In Proceedings of the IEEE International Conference on Computer Vision, (2019), 1675-1684. |
| [81] | S. Dye, K. Kohn, F. Rydell and R. Sinn, Maximum likelihood estimation for nets of conics, arXiv: 2011.08989, 2020. · Zbl 1481.14015 |
| [82] | L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Volume XVII of Pure and Applied Mathematics, Wiley-Interscience Publishers, New York-London-Sydney, 1970. · Zbl 0195.10401 |
| [83] | M. Eichler and D. Zagier, The Theory of Jacobi Forms, Volume 55 of Progress in Mathematics, Birkhäuser Boston, 1985. · Zbl 0554.10018 |
| [84] | D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Graduate Texts in Mathematics. Springer New York, 2013. |
| [85] | P. Ellison, M. Feinberg, H. Ji, and D. Knight, Chemical reaction network toolbox, version 2.2, Available online at http://www.crnt.osu.edu/CRNTWin, 2012. |
| [86] | R. Fabbri, T. Duff, H. Fan, M. H. Regan, D. d. C. d. Pinho, E. Tsigaridas, C. W. Wampler, J. D. Hauenstein, P. J. Giblin, B. Kimia, A. Leykin and T. Pajdla, TRPLP - Trifocal Relative Pose from Lines at Points, In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020. |
| [87] | M. Feinberg, The existence and uniqueness of steady states for a class of chemical reaction networks, Arch. Rational Mech. Anal., 132, 311-370 (1995) · Zbl 0853.92024 · doi:10.1007/BF00375614 |
| [88] | M. Feinberg, Foundations of Chemical Reaction Network Theory, Volume 202 of Applied Mathematical Sciences, Springer International Publishing, 2019. · Zbl 1420.92001 |
| [89] | E. Feliu, Injectivity, multiple zeros, and multistationarity in reaction networks, Proc. Roy. Soc. Edinburgh Sect. A, doi: 10.1098/rspa.2014.0530,2014. MR3298836 · Zbl 1371.92147 |
| [90] | E. Feliu, N. Kaihnsa, T. de Wolff and O. Yürük, The kinetic space of multistationarity in dual phosphorylation, J. Dynam. Differential Equations, 2020. · Zbl 1496.92028 |
| [91] | C. Fevola, Y. Mandelshtam and B. Sturmfels, Pencils of quadrics: Old and new, arXiv: 2009.04334, 2020. · Zbl 1480.14037 |
| [92] | M. A. R. C. Fischler Bolles, Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography, Communications of the ACM, 24, 381-395 (1981) · doi:10.1145/358669.358692 |
| [93] | G. J. G. Fløystad Kileel Ottaviani, The Chow form of the essential variety in computer vision, J. Symbolic Comput., 86, 97-119 (2018) · Zbl 1380.68369 · doi:10.1016/j.jsc.2017.03.010 |
| [94] | J. C. C. Frauendiener Jaber Klein, Efficient computation of multidimensional theta functions, J. Geom. Phys., 127, 147-158 (2019) · Zbl 1444.33011 · doi:10.1016/j.geomphys.2019.03.011 |
| [95] | L. D. M. B. Garcia Stillman Sturmfels, Algebraic geometry of Bayesian networks, J. Symbolic Comput., 39, 331-355 (2005) · Zbl 1126.68102 · doi:10.1016/j.jsc.2004.11.007 |
| [96] | P. Gaudry, Fast genus 2 arithmetic based on Theta functions, J. Math. Cryptol. J., 3, 243-265 (2007) · Zbl 1145.11048 · doi:10.1515/JMC.2007.012 |
| [97] | D. C. B. et Geiger Meek Sturmfels al, On the toric algebra of graphical models, Ann. Statist., 34, 1463-1492 (2006) · Zbl 1104.60007 · doi:10.1214/009053606000000263 |
| [98] | I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1994. · Zbl 0827.14036 |
| [99] | C. Görgen, A. Maraj and L. Nicklasson, Staged tree models with toric structure, 2107.04516. · Zbl 1491.14074 |
| [100] | P. C. and A.-L. Görlach Lehn Sattelberger, Algebraic analysis of the hypergeometric function \(_1F_1\) of a matrix argument, Beiträge Algebra Geom., 62, 397-427 (2021) · Zbl 1478.33007 · doi:10.1007/s13366-020-00546-z |
| [101] | S. J. A. D. D. P. Gortler Healy Thurston, Characterizing generic global rigidity, Amer. J. Math., 132, 897-939 (2010) · Zbl 1202.52020 · doi:10.1353/ajm.0.0132 |
| [102] | H.-C. K. Graf von Bothmer Ranestad, A general formula for the algebraic degree in semidefinite programming, Bull. Math. Biol., 41, 193-197 (2009) · Zbl 1185.14047 · doi:10.1112/blms/bdn114 |
| [103] | J. Graver, B. Servatius and H. Servatius, Combinatorial Rigidity, Volume 2 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993. · Zbl 0788.05001 |
| [104] | D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/, 2020. |
| [105] | G. Greuel, G. Pfister, O. Bachmann, C. Lossen and H. Schönemann, A Singular Introduction to Commutative Algebra, Springer Nature Book Archives Millennium. Springer, 2002. · Zbl 1023.13001 |
| [106] | C. L. G. H. F. P. T. M. Häne Heng Lee Fraundorfer Furgale Sattler Pollefeys, 3D visual perception for self-driving cars using a multi-camera system: Calibration, mapping, localization, and obstacle detection, Image and Vision Computing, 68, 14-27 (2017) |
| [107] | M. B. F. T. J. I. Härkönen Hollering Kashani Rodriguez, Algebraic optimization degree, ACM Commun. Comput. Algebra, 54, 44-48 (2020) · Zbl 1499.14003 |
| [108] | R. F. Hartley Kahl, Critical configurations for projective reconstruction from multiple views, Int. J. Comput. Vis., 71, 5-47 (2007) · Zbl 1477.68366 |
| [109] | R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, Cambridge, 2nd edition, 2003. · Zbl 0956.68149 |
| [110] | R. I. Hartley, Chirality, Int. J. Comput. Vis., 26, 41-61 (1998) |
| [111] | R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer New York, 2013. |
| [112] | H. Y. N. A. Hashiguchi Numata Takayama Takemura, The holonomic gradient method for the distribution function of the largest root of a Wishart matrix, J. Multivariate Anal., 117, 296-312 (2013) · Zbl 1282.33010 · doi:10.1016/j.jmva.2013.03.011 |
| [113] | J. D. Hauenstein and A. C. Liddell, A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems, In Proceedings of the 2015 International Workshop on Parallel Symbolic Computation, PASCO ’15, Association for Computing Machinery, (2015), 53-60. |
| [114] | K. Heal, J. Wang, S. J. Gortler and T. Zickler, A lighting-invariant point processor for shading, (2020), 94-102. |
| [115] | A. Heaton and S. Timme, Catastrophe in elastic tensegrity frameworks, arXiv: 2009.13408, 2020. · Zbl 1508.14066 |
| [116] | D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann., 32, 342-350 (1888) · JFM 20.0198.02 · doi:10.1007/BF01443605 |
| [117] | C. J. S. Hillar Sullivant, Finite Gröbner bases in infinite dimensional polynomial rings and applications, Adv. Math., 229, 1-25 (2012) · Zbl 1233.13012 · doi:10.1016/j.aim.2011.08.009 |
| [118] | R. Hirota, The Direct Method in Soliton Theory (2004) · doi:10.1017/CBO9780511543043 |
| [119] | N. G. R. S. Hitchin Segal Ward, Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces (1999) |
| [120] | B. S. Hollering Sullivant, Identifiability in phylogenetics using algebraic matroids, J. Symbolic Comput., 104, 142-158 (2021) · Zbl 1455.92103 · doi:10.1016/j.jsc.2020.04.012 |
| [121] | S. S. Hoşten Sullivant, Gröbner bases and polyhedral geometry of reducible and cyclic models, J. Combin. Theory Ser. A, 100, 277-301 (2002) · Zbl 1044.62065 · doi:10.1006/jcta.2002.3301 |
| [122] | R. Hotta, K. Takeuchi, and T. Tanisaki, \(D\)-Modules, Perverse Sheaves, and Representation Theory, Volume 236 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. · Zbl 1136.14009 |
| [123] | J. Huh, Varieties with maximum likelihood degree one, J. Algebr. Stat., 5, 1-17 (2014) · Zbl 1346.14118 · doi:10.18409/jas.v5i1.22 |
| [124] | J. Huh and B. Sturmfels, Likelihood geometry, In Combinatorial Algebraic Geometry, volume 2108 of Lecture notes in mathematics, Springer, New York, (2014), 63-117. · Zbl 1328.14004 |
| [125] | B. Hunyadi, D. Camps, L. Sorber, W. V. Paesschen, M. D. Vos, S. V. Huffel and L. D. Lathauwer, Block term decomposition for modelling epileptic seizures, EURASIP J. Adv. Signal Process., 139 (2014). |
| [126] | A. Hurwitz, Ueber den Vergleich des arithmetischen und des geometrischen Mittels, J. Reine Angew. Math., 108, 266-268 (1891) · JFM 23.0263.01 · doi:10.1515/crll.1891.108.266 |
| [127] | S. Iliman and T. de Wolff, Amoebas, nonnegative polynomials and sums of squares supported on circuits, Res. Math. Sci., 3 (2016). · Zbl 1415.11071 |
| [128] | B. A. Joshi Shiu, A survey of methods for deciding whether a reaction network is multistationary, Math. Model. Nat. Phenom., 10, 47-67 (2015) · Zbl 1371.92148 · doi:10.1051/mmnp/201510504 |
| [129] | F. D. Kahl Henrion, Globally optimal estimates for geometric reconstruction problems, Int. J. Comput. Vis., 74, 3-15 (2007) · Zbl 1477.68382 |
| [130] | M. Kashiwara, \(B\)-functions and holonomic systems, rationality of roots of \(B\)-functions, Invent. Math., 38, 33-53 (1976) · Zbl 0354.35082 · doi:10.1007/BF01390168 |
| [131] | M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., 20, 319-365 (1984) · Zbl 0566.32023 · doi:10.2977/prims/1195181610 |
| [132] | M. Kauers, M. Jaroschek and F. Johansson, Ore polynomials in Sage, arXiv: 1306.4263, 2013. · Zbl 1439.16049 |
| [133] | J. Kileel, Minimal problems for the calibrated trifocal variety, SIAM J. Appl. Algebra Geom., 1, 575-598 (2017) · Zbl 1386.14185 · doi:10.1137/16M1104482 |
| [134] | Y. Kodama, KP Solitons and the Grassmannians. Combinatorics and Geometry and Two-dimensional Wave Patterns, Volume 22 of Briefs in Mathematical Physics, Springer, 2017. · Zbl 1372.35266 |
| [135] | Y. L. Kodama Williams, KP solitons and total positivity for the Grassmannian, Invent. Math., 198, 637-699 (2014) · Zbl 1306.35109 · doi:10.1007/s00222-014-0506-3 |
| [136] | Y. Y. Kodama Xie, Space curves and solitons of the KP hierarchy: I. The l-th generalized KdV hierarchy, SIGMA, 17, 024 (2021) · Zbl 1468.37052 · doi:10.3842/SIGMA.2021.024 |
| [137] | E. R. Kolchin, Differential Algebra and Algebraic Groups (1973) · Zbl 0264.12102 |
| [138] | C. Koutschan, Holonomic functions (user’s guide), Technical Report 10-01, RISC Report Series, Johannes Kepler University, Linz, Austria, 2010. |
| [139] | T. Koyama, The annihilating ideal of the Fisher integral, Kyushu J. Math., 74, 415-427 (2020) · Zbl 1470.16048 |
| [140] | I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Russian Math. Surveys, 32, 44-48 (1977) |
| [141] | R. A. A. Krone Leykin Snowden, Hilbert series of symmetric ideals in infinite polynomial rings via formal languages, J. Algebra, 485, 353-362 (2017) · Zbl 1401.13048 · doi:10.1016/j.jalgebra.2017.05.014 |
| [142] | J. B. Kruskal, Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics, Linear Algebra Appl., 18, 95-138 (1977) · Zbl 0364.15021 · doi:10.1016/0024-3795(77)90069-6 |
| [143] | Z. Kukelova, M. Bujnak and T. Pajdla, Automatic generator of minimal problem solvers, In European Conference on Computer Vision, Springer, (2008), 302-315. |
| [144] | Z. Kukelova and T. Pajdla, A minimal solution to the autocalibration of radial distortion, In 2007 IEEE Conference on Computer Vision and Pattern Recognition, (2007), 1-7. |
| [145] | P. Lairez, Computing periods of rational integrals, Math. Comp., 85, 1719-1752 (2016) · Zbl 1337.68301 · doi:10.1090/mcom/3054 |
| [146] | P. Lairez, M. Mezzarobba and M. Safey El Din, Computing the volume of compact semi-algebraic sets, In ISSAC 2019 - International Symposium on Symbolic and Algebraic Computation, Beijing, China, July 2019, ACM. · Zbl 1467.14139 |
| [147] | G. Laman, On graphs and rigidity of plane skeletal structures, J. Engrg. Math., 4, 331-340 (1970) · Zbl 0213.51903 · doi:10.1007/BF01534980 |
| [148] | J. M. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics, AMS, Providence, Rhode Island, 2012. · Zbl 1238.15013 |
| [149] | V. Larsson, K. Astrom and M. Oskarsson, Efficient solvers for minimal problems by syzygy-based reduction, In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2017), 820-829. |
| [150] | J. B. Lasserre, Moments, Positive Polynomials and Their Applications, Volume 1 of Imperial College Press Optimization, World Scientific, 2010. · Zbl 1211.90007 |
| [151] | L. D. Lathauwer, Decompositions of a higher-order tensor in block terms - Part II: Definitions and uniqueness, SIAM J. Matrix Anal. Appl., 30, 1033-1066 (2008) · Zbl 1177.15032 · doi:10.1137/070690729 |
| [152] | M. N. Laurent Kellershohn, Multistability: a major means of differentiation and evolution in biological systems, Trends Biochem. Sciences, 24, 418-422 (1999) |
| [153] | S. C. P. et Lauritzen Uhler Zwiernik al., Maximum likelihood estimation in Gaussian models under total positivity, Ann. Statist., 47, 1835-1863 (2019) · Zbl 1467.62092 · doi:10.1214/17-AOS1668 |
| [154] | V. Levandovskyy and J. Martín-Morales, dmod_lib: A \(\mathtt{Singular: Plural}\) library for algorithms for algebraic \(D\)-modules, https://www.singular.uni-kl.de/Manual/4-2-0/sing_537. |
| [155] | A. Leykin, Numerical algebraic geometry for macaulay2, J. Softw. Algebra Geom., 3, 5-10 (2011) · Zbl 1311.14057 · doi:10.2140/jsag.2011.3.5 |
| [156] | M. L. Lieblich Van Meter, Two Hilbert schemes in computer vision, SIAM J. Appl. Algebra Geom., 4, 297-321 (2020) · Zbl 1440.14056 · doi:10.1137/18M1200117 |
| [157] | J. Little, Translation manifolds and the converse to Abel’s theorem, Compos. Math., 49, 147-171 (1983) · Zbl 0521.14017 |
| [158] | Q. Liu, Algebraic Geometry and Arithmetic Curves (2006) · Zbl 1103.14001 |
| [159] | M. Z. Luxton Qu, Some results on tropical compactifications, Trans. Amer. Math. Soc., 363, 4853-4876 (2011) · Zbl 1230.14014 · doi:10.1090/S0002-9947-2011-05254-2 |
| [160] | D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Volume 161 of Graduate Studies in Mathematics, American Mathematical Society, Providence, R.I., 2015. · Zbl 1321.14048 |
| [161] | P. Maisonobe, Filtration relative, l’idéal de Bernstein et ses pentes, hal-01285562v2, 2016. |
| [162] | Maplesoft, A division of Waterloo Maple Inc., Maple, 2019. |
| [163] | A. Maraj, Algebraic and Geometric Properties of Hierarchical Models, PhD Thesis, 2020. |
| [164] | A. U. Maraj Nagel, Equivariant Hilbert series for hierarchical models, Algebraic Statistics, 12, 21-42 (2021) · Zbl 1465.13025 · doi:10.2140/astat.2021.12.21 |
| [165] | M. Marshall, Positive Polynomials and Sums Of Squares, Volume 146 of Mathematical Surveys and Monographs, American Mathematical Society, 2008. · Zbl 1169.13001 |
| [166] | Max Planck Institute for Mathematics in the Sciences, Mathrepo. Mathematical data and software, Repository website of the MPI MiS, https://mathrepo.mis.mpg.de/. |
| [167] | L. G. J. I. B. Maxim Rodriguez Wang, Euclidean distance degree of the multiview variety, SIAM J. Appl. Algebra Geom., 4, 28-48 (2020) · Zbl 1437.13049 · doi:10.1137/18M1233406 |
| [168] | Z. Mebkhout, Une équivalence de catégories, Compos. Math., 51, 51-62 (1984) · Zbl 0566.32021 |
| [169] | M. Michałek and B. Sturmfels, Lecture: Introduction to Nonlinear Algebra, Available at https://youtu.be/1EryuvBLY80, 2018. |
| [170] | M. Michałek and B. Sturmfels, Invitation to Nonlinear Algebra, Volume 211 of Graduate Studies in Mathematics, American Mathematical Society, 2021. Available at https://personal-homepages.mis.mpg.de/michalek/NonLinearAlgebra.pdf. · Zbl 1477.14001 |
| [171] | E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Volume 227 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2005. · Zbl 1090.13001 |
| [172] | P. Misra and S. Sullivant, Gaussian graphical models with toric vanishing ideals, Ann. Inst. Statist. Math., (2020), 1-29. · Zbl 1469.62420 |
| [173] | R. J. Muirhead, Systems of partial differential equations for hypergeometric functions of matrix argument, Ann. Math. Statist., 41, 991-1001 (1970) · Zbl 0225.62078 · doi:10.1214/aoms/1177696975 |
| [174] | R. J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1982. · Zbl 0556.62028 |
| [175] | D. Mumford, Tata Lectures on Theta I, Modern Birkhäuser Classics. Birkhäuser Boston, 2007. |
| [176] | U. T. Nagel Römer, Equivariant Hilbert series in non-Noetherian polynomial rings, J. Algebra, 486, 204-245 (2017) · Zbl 1372.13016 · doi:10.1016/j.jalgebra.2017.05.011 |
| [177] | R. Nagpal and A. Snowden, Symmetric ideals of the infinite polynomial ring, arXiv: 2107.13027, 2021. · Zbl 1436.20096 |
| [178] | H. K. M. K. T. N. A. Nakayama Nishiyama Noro Ohara Sei Takayama Takemura, Holonomic gradient descent and its application to the Fisher-Bingham integral, Adv. in Appl. Math., 47, 639-658 (2011) · Zbl 1226.90137 · doi:10.1016/j.aam.2011.03.001 |
| [179] | A. Nakayashiki, On Algebraic Expansions of Sigma Functions for \((n, s)\) curves, Asian J. Math., 14, 175-212 (2010) · Zbl 1214.14028 · doi:10.4310/AJM.2010.v14.n2.a2 |
| [180] | A. Nakayashiki, Degeneration of trigonal curves and solutions of the KP-hierarchy, Nonlinearity, 31, 3567-3590 (2018) · Zbl 1398.37081 · doi:10.1088/1361-6544/aabf00 |
| [181] | J. K. Nie Ranestad, Algebraic degree of polynomial optimization, SIAM J. Optim., 20, 485-502 (2009) · Zbl 1190.14051 · doi:10.1137/080716670 |
| [182] | J. K. B. Nie Ranestad Sturmfels, The algebraic degree of semidefinite programming, Math. Program., 122, 379-405 (2010) · Zbl 1184.90119 · doi:10.1007/s10107-008-0253-6 |
| [183] | D. Nistér, An efficient solution to the five-point relative pose problem, IEEE Transactions on Pattern Analysis And Machine Intelligence, 26, 756-770 (2004) |
| [184] | M. Noro, System of partial differential equations for the hypergeometric function \(_1F_1\) of a matrix argument on diagonal regions, ISSAC’16: Proceedings of the ACM on International Symposium of Symbolic and Algebraic Computation, July 2016,381-388. · Zbl 1429.35052 |
| [185] | S. Novikov, S. Manakov, L. Pitaevskii and V. Zakharov, Theory of Solitons: The Inverse Scattering Method, Monographs in Contemporary Mathematics, Springer, 1984. · Zbl 0598.35002 |
| [186] | F. Ollivier, Standard bases of differential ideals, Lecture Notes in Comput. Sci., 508, 304-321 (1990) · Zbl 0737.13009 · doi:10.1007/3-540-54195-0_60 |
| [187] | G. Ottaviani and P. Reichenbach, Tensor rank and complexity, arXiv: 2004.01492, 2020. |
| [188] | G. L. Ottaviani Sodomaco, The distance function from a real algebraic variety, Comput. Aided Geom. Design, 82, 101927 (2020) · Zbl 1453.65046 · doi:10.1016/j.cagd.2020.101927 |
| [189] | E. M. M. H. N. B. I. A. Ozbudak Thattai Lim Shraiman Van Oudenaarden, Multistability in the lactose utilization network of escherichia coli, Nature, 427, 737-740 (2004) |
| [190] | V. P. Palamodov, Linear Differential Qperators with Constant Coefficients, Volume 168 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York-Berlin, 1970. · Zbl 0191.43401 |
| [191] | E. E. Papalexakis, C. Faloutsos and N. D. Sidiropoulos, Tensors for data mining and data fusion: Models, applications, and scalable algorithms, ACM Trans. Intell. Syst. Technol., 8 (2016). |
| [192] | M. A. A. and C. Pérez Millán Dickenstein Shiu Conradi, Chemical reaction systems with toric steady states, Bull. Math. Biol., 74, 1027-1065 (2012) · Zbl 1251.92016 · doi:10.1007/s11538-011-9685-x |
| [193] | S. Petrović, What is... a markov basis?, Notices of the American Mathematical Society, 66, 1088-1092 (2019) · Zbl 1426.62195 |
| [194] | G. H. Pistone Wynn, Generalised confounding with grobner bases, Biometrika, 83, 653-666 (1996) · Zbl 0928.62060 · doi:10.1093/biomet/83.3.653 |
| [195] | H. Pollaczek-Geiringer, Über die Gliederung ebener Fachwerke, J. Appl. Math. Mech., 7, 58-72 (1927) · JFM 53.0743.02 |
| [196] | A. Pryhuber, R. Sinn and R. R. Thomas, Existence of two view chiral reconstructions, arXiv: 2011.07197, 2020. |
| [197] | Y. P. L.-H. Qi Comon Lim, Semialgebraic geometry of nonnegative tensor rank, SIAM J. Matrix Anal. Appl., 37, 1556-1580 (2016) · Zbl 1349.14183 · doi:10.1137/16M1063708 |
| [198] | R Core Team, \( \mathtt{R} \): A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2013. |
| [199] | O. N. Regev Stephens-Davidowitz, An inequality for Gaussians on lattices, SIAM J. Discrete Math., 31, 749-757 (2017) · Zbl 1395.11105 · doi:10.1137/15M1052226 |
| [200] | T. Reichelt, M. Schulze, C. Sevenheck and U. Walther, Algebraic aspects of hypergeometric differential equations, Beiträge Algebra Geom., 2021. · Zbl 1468.14042 |
| [201] | J. F. Ritt, Differential Algebra, Volume 14 of American Mathematical Society: Colloquium Publications, American Mathematical Society, 1950. · Zbl 0037.18402 |
| [202] | A. A. Rontogiannis, E. Kofidis and P. Giampouras, Block-term tensor decomposition: Model selection and computation, arXiv: 2002.09759, 2020. |
| [203] | D. M. L. A. S. and J. J. Rosen Carlone Bandeira Leonard, SE-Sync: A certifiably correct algorithm for synchronization over the special Euclidean group, The International Journal of Robotics Research, 38, 95-125 (2019) |
| [204] | C. Sabbah, Proximité évanescente. I. La structure polaire dun \(D\)-module, Compositio Math., 62, 283-328 (1987) · Zbl 0622.32012 |
| [205] | M. Saito, B. Sturmfels and N. Takayama, Gröbner deformations of hypergeometric differential equations, volume 6 of Algorithms and Computation in Mathematics, Springer-Verlag, Berlin, 2000. · Zbl 0946.13021 |
| [206] | B. P. Salvy Zimmermann, \( \mathtt{gfun} \): A maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software, 20, 163-177 (1994) · Zbl 0888.65010 |
| [207] | M. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, In Nonlinear Partial Differential Equations in Applied Science; Proceedings of The U.S.-Japan Seminar, Tokyo, 1982, North-Holland Mathematics Studies, North-Holland, (1983), 259-271. · Zbl 0528.58020 |
| [208] | A.-L. Sattelberger and B. Sturmfels, \(D\)-modules and holonomic functions, arXiv: 1910.01395, 2019. |
| [209] | A.-L. Sattelberger and R. van der Veer, Maximum likelihood estimation from a tropical and a Bernstein-Sato perspective, arXiv: 2101.03570, 2021. |
| [210] | G. G. Segal Wilson, Loop groups and equations of KdV type, Publ. Math. Inst. Hautes Etudes Sci., 61, 5-65 (1985) · Zbl 0592.35112 |
| [211] | H. S. R. S. Segal Grushevsky Manni, An explicit solution to the weak Schottky problem, Algebr. Geom., 8, 358-373 (2021) · Zbl 1454.14087 · doi:10.14231/ag-2021-009 |
| [212] | T. H. A. K. N. Sei Shibata Takemura Ohara Takayama, Properties and applications of Fisher distribution on the rotation group, J. Multivariate Anal., 116, 440-455 (2013) · Zbl 1283.60011 · doi:10.1016/j.jmva.2013.01.010 |
| [213] | Y. Shitov, A counterexample to Comon’s conjecture, SIAM Journal on Applied Algebra and Geometry, 2, 428-443 (2018) · Zbl 1401.15004 · doi:10.1137/17M1131970 |
| [214] | N. L. X. K. E. C. Sidiropoulos De Lathauwer Fu Huang Papalexakis Faloutsos, Tensor decomposition for signal processing and machine learning, IEEE Transactions on Signal Processing, 65, 3551-3582 (2017) · Zbl 1415.94232 · doi:10.1109/TSP.2017.2690524 |
| [215] | R. Silhol, The Schottky problem for real genus 3 M-curves, Math. Z., 236, 841-881 (2001) · Zbl 0988.32014 · doi:10.1007/PL00004854 |
| [216] | M. A. St. J. ed Sitharam John Sidman itors, Handbook of Geometric Constraint Systems Principles (2019) · Zbl 1397.05005 |
| [217] | A. J. Sommese and C. W. Wampler, The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific, 2005. · Zbl 1091.65049 |
| [218] | M. L. Sorensen De Lathauwer, Coupled canonical polyadic decompositions and (coupled) decompositions in multilinear rank-\((L_r, n, L_r, n, 1)\) terms—Part I: Uniqueness, SIAM J. Matrix Anal. Appl., 36, 496-522 (2015) · Zbl 1320.15010 · doi:10.1137/140956853 |
| [219] | M. L. Sorensen De Lathauwer, Multidimensional harmonic retrieval via coupled canonical polyadic decomposition Part I: Model and identifiability, IEEE Trans. Signal Process., 65, 517-527 (2017) · Zbl 1414.94579 · doi:10.1109/TSP.2016.2614796 |
| [220] | M. L. Sorensen De Lathauwer, Multidimensional harmonic retrieval via coupled canonical polyadic decomposition Part II: Algorithm and multirate sampling, IEEE Trans. Signal Process., 65, 528-539 (2017) · Zbl 1414.94580 · doi:10.1109/TSP.2016.2614797 |
| [221] | D. L. L. Do Sorensen M. manov I., Coupled canonical polyadic decompositions and (coupled) decompositions in multilinear rank-\((L_r, n, L_r, n, 1)\) terms — Part II: Algorithms, SIAM J. Matrix Anal. Appl., 36, 1015-1045 (2015) · Zbl 1321.65068 · doi:10.1137/140956865 |
| [222] | G. Strang, Computational Science and Engineering (2007) · Zbl 1194.65001 |
| [223] | B. Sturmfels, Gröbner Bases and Convex Polytopes, Volume 8 of University Lecture Series, American Mathematical Soc., 1996. · Zbl 0856.13020 |
| [224] | B. Sturmfels, Solving Systems of Polynomial Equations, Number 97 in CBMS Regional Conferences Series. American Mathematical Society, 2002. |
| [225] | B. Sturmfels, What is … a Gröbner basis?, Notices Amer. Math. Soc., 52, 1199-1200 (2005) · Zbl 1093.13512 |
| [226] | B. S. Sturmfels Sullivant, Toric ideals of phylogenetic invariants, J. Comput. Biol., 12, 204-228 (2005) · Zbl 1391.13058 |
| [227] | B. Sturmfels and S. Telen, Likelihood equations and scattering amplitudes, arXiv: 2012.05041, 2020. · Zbl 1515.62137 |
| [228] | B. C. Sturmfels Uhler, Multivariate Gaussian, semidefinite matrix completion, and convex algebraic geometry, Ann. Inst. Statist. Math., 62, 603-638 (2010) · Zbl 1440.62255 · doi:10.1007/s10463-010-0295-4 |
| [229] | B. C. P. Sturmfels Uhler Zwiernik, Brownian motion tree models are toric, Kybernetika, 56, 1154-1175 (2020) · Zbl 1538.62361 · doi:10.14736/kyb-2020-6-1154 |
| [230] | S. Sullivant, Gaussian conditional independence relations have no finite complete characterization, J. Pure Appl. Algebra, 213, 1502-1506 (2009) · Zbl 1162.13014 · doi:10.1016/j.jpaa.2008.11.026 |
| [231] | S. Sullivant, Algebraic Statistics, Volume 194 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2018. · Zbl 1408.62004 |
| [232] | C. B. Swierczewski Deconinck, Computing Riemann theta functions in Sage with applications, Math. Comput. Simulation, 127, 263-272 (2016) · Zbl 1520.33013 · doi:10.1016/j.matcom.2013.04.018 |
| [233] | R. Szeliski, Computer Vision - Algorithms and Applications, Texts in Computer Science, Springer, 2011. · Zbl 1219.68009 |
| [234] | N. Takayama, T. Koyama, T. Sei, H. Nakayama, and K. Nishiyama, \( \mathtt{hgm} \): Holonomic Gradient Method and Gradient Descent, \( \mathtt{R}\) package version 1.17, 2017. |
| [235] | The Sage Developers, SageMath, the Sage Mathematics Software System, https://www.sagemath.org. |
| [236] | A. Torres and E. Feliu, Detecting parameter regions for bistability in reaction networks, arXiv: 1909.13608, 2019. |
| [237] | R. Vakil, The rising sea: Foundations of algebraic geometry, Available at http://math.stanford.edu/vakil/216blog/index.html. |
| [238] | J. Verschelde, Algorithm 795: PHCpack: A General-Purpose Solver for Polynomial Systems by Homotopy Continuation, ACM Trans. Math. Softw., 25, 251C276 (1999) · Zbl 0961.65047 |
| [239] | R. Y. S. Vidal Ma Sastry, Generalized principal component analysis (GPCA), IEEE Transactions on Pattern Analysis and Machine Intelligence, 27, 1945-1959 (2005) · Zbl 1349.62006 · doi:10.1007/978-0-387-87811-9 |
| [240] | C. E. Wiuf Feliu, Power-law kinetics and determinant criteria for the preclusion of multistationarity in networks of interacting species, SIAM J. Appl. Dyn. Syst., 12, 1685-1721 (2013) · Zbl 1278.92012 · doi:10.1137/120873388 |
| [241] | R. J. Woodham, Photometric method for determining surface orientation from multiple images, In Shape From Shading, (1989), 513-531. |
| [242] | W. J. E. Xiong Ferrell Jr, A positive-feedback-based bistable ’memory module’ that governs a cell fate decision, Nature, 426, 460-465 (2003) |
| [243] | M. Yang, On partial and generic uniqueness of block term tensor decompositions, Annali dell’universita’ di Ferrara, 60, 465-493 (2014) · Zbl 1315.15026 · doi:10.1007/s11565-013-0190-z |
| [244] | D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math., 32, 321-368 (1990) · Zbl 0738.33001 · doi:10.1016/0377-0427(90)90042-X |
| [245] | A. I. Zobnin, One-element differential standard bases with respect to inverse lexicographical orderings, J. Math. Sci. (N. Y.), 163, 523-533 (2009) · Zbl 1288.13021 · doi:10.1007/s10958-009-9690-x |
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