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Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. (English) Zbl 1382.92272
Summary: We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry, our work recognizes a prior result of G. Craciun et al. [Lect. Notes Comput. Sci. 5862, 111–135 (2010; Zbl 1274.65033)], together with work of two of the authors, as the first partial multivariate generalization of the classical Descartes’ rule, which bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients.

MSC:
92E20 Classical flows, reactions, etc. in chemistry
34A34 Nonlinear ordinary differential equations and systems
37C10 Dynamics induced by flows and semiflows
52C40 Oriented matroids in discrete geometry
80A30 Chemical kinetics in thermodynamics and heat transfer
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References:
[1] Althaus, E; Dumitriu, D, Certifying feasibility and objective value of linear programs, Oper. Res. Lett., 40, 292-297, (2012) · Zbl 1247.90187
[2] R. M. Anderson and R. M. May, Infectious diseases of humans: Dynamics and control, Oxford University Press, Oxford, 1991.
[3] D. L. Applegate, W. Cook, S. Dash, and D. G. Espinoza, Exact solutions to linear programming problems, Oper. Res. Lett. 35 (2007), 693-699. · Zbl 1177.90282
[4] D. L. Applegate, W. Cook, S. Dash, and D. G. Espinoza, QSopt_ex (2009). Available online at http://www.math.uwaterloo.ca/ bico//qsopt/ex/
[5] E. Babson, L. Finschi, and K. Fukuda, Cocircuit graphs and efficient orientation reconstruction in oriented matroids, European J. Combin. 22 (2001), 587-600. · Zbl 0988.52032
[6] M. Banaji and G. Craciun, Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements, Commun. Math. Sci. 7 (2009), 867-900. · Zbl 1195.05038
[7] M. Banaji and G. Craciun, Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems, Adv. Appl. Math. 44 (2010), 168-184. · Zbl 1228.05204
[8] M. Banaji, P. Donnell, and S. Baigent, \(P\)matrix properties, injectivity, and stability in chemical reaction systems, SIAM J. Appl. Math. 67 (2007), 1523-1547. · Zbl 1132.80004
[9] M. Banaji and C. Pantea, Some results on injectivity and multistationarity in chemical reaction networks, Available online at arXiv:1309.6771, 2013. · Zbl 1342.80011
[10] S. Basu, R. Pollack, and M. F. Roy, Algorithms in real algebraic geometry, second ed., Algorithms and Computation in Mathematics, vol. 10, Springer-Verlag, Berlin, 2006, Updated online version available at http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html. · Zbl 1102.14041
[11] F. Bihan and A. Dickenstein, Descartes’ rule of signs for polynomial systems supported on circuits, Preprint, 2014.
[12] Birch, MW, Maximum likelihood in three-way contingency tables, J. Roy. Stat. Soc. B Met., 25, 220-233, (1963) · Zbl 0121.14001
[13] A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler, Oriented matroids, second ed., Encyclopedia Math. Appl., vol. 46, Cambridge University Press, Cambridge, 1999.
[14] S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, A tutorial on geometric programming, Optim. Eng. 8 (2007), 67-127. · Zbl 1178.90270
[15] R. Brualdi and B. Shader, Matrices of sign-solvable linear systems, Cambridge University Press, 1995. · Zbl 0833.15002
[16] S. Chaiken, Oriented matroid pairs, theory and an electric application, Matroid theory (Seattle, WA, 1995), Contemp. Math., vol. 197, Amer. Math. Soc., Providence, RI, 1996, pp. 313-331. · Zbl 0866.05016
[17] C. Conradi and D. Flockerzi, Multistationarity in mass action networks with applications to ERK activation, J. Math. Biol. 65 (2012), 107-156. · Zbl 1278.37058
[18] C. Conradi and D. Flockerzi, Switching in mass action networks based on linear inequalities, SIAM J. Appl. Dyn. Syst. 11 (2012), 110-134. · Zbl 1235.37034
[19] C. Conradi, D. Flockerzi, and J. Raisch, Multistationarity in the activation of a MAPK: parametrizing the relevant region in parameter space, Math. Biosci. 211 (2008), 105-131. · Zbl 1130.92024
[20] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks. I. The injectivity property, SIAM J. Appl. Math. 65 (2005), 1526-1546. · Zbl 1094.80005
[21] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: extensions to entrapped species models, Systems Biology, IEE Proceedings 153 (2006), 179-186. · Zbl 1417.37298
[22] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks. II. The species-reaction graph, SIAM J. Appl. Math. 66 (2006), 1321-1338. · Zbl 1136.80306
[23] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: semiopen mass action systems, SIAM J. Appl. Math. 70 (2010), 1859-1877. · Zbl 1255.80020
[24] G. Craciun, L. Garcia-Puente, and F. Sottile, Some geometrical aspects of control points for toric patches, Mathematical Methods for Curves and Surfaces (Heidelberg) (M. Dæhlen, M. S. Floater, T. Lyche, J. L. Merrien, K. Morken, and L. L. Schumaker, eds.), Lecture Notes in Computer Science, vol. 5862, Springer, 2010, pp. 111-135. · Zbl 1274.65033
[25] G. Craciun, J. W. Helton, and R. J. Williams, Homotopy methods for counting reaction network equilibria, Math. Biosci. 216 (2008), 140-149. · Zbl 1153.92015
[26] M. Feinberg, Complex balancing in general kinetic systems, Arch. Rational Mech. Anal. 49 (1972/73), 187-194.
[27] M. Feinberg, Lectures on chemical reaction networks, Available online at http://www.crnt.osu.edu/LecturesOnReactionNetworks, (1980).
[28] M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors-I. The deficiency zero and deficiency one theorems, Chem. Eng. Sci. 42 (1987), 2229-2268. · Zbl 1283.92030
[29] M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors-II. Multiple steady states for networks of deficiency one, Chem. Eng. Sci. 43 (1988), 1-25.
[30] M. Feinberg, The existence and uniqueness of steady states for a class of chemical reaction networks, Arch. Rational Mech. Anal. 132 (1995), 311-370. · Zbl 0853.92024
[31] M. Feinberg, Multiple steady states for chemical reaction networks of deficiency one, Arch. Rational Mech. Anal. 132 (1995), 371-406. · Zbl 0853.92025
[32] Feliu, E; Wiuf, C, Preclusion of switch behavior in reaction networks with mass-action kinetics, Appl. Math. Comput., 219, 1449-1467, (2012) · Zbl 1417.37298
[33] Feliu, E; Wiuf, C, A computational method to preclude multistationarity in networks of interacting species, Bioinformatics, 29, 2327-2334, (2013) · Zbl 1381.82025
[34] Gale, D; Nikaidô, H, The Jacobian matrix and global univalence of mappings, Math. Ann., 159, 81-93, (1965) · Zbl 0158.04903
[35] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, reprint of the 1994 ed., Boston, MA: Birkhäuser, 2008. · Zbl 1138.14001
[36] A. M. Gleixner, D. E. Steffy, and K. Wolter, Improving the accuracy of linear programming solvers with iterative refinement, Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation (ISSAC ’12) (New York, NY, USA), ACM, 2012, pp. 187-194. · Zbl 1308.65091
[37] Gnacadja, G, A Jacobian criterion for the simultaneous injectivity on positive variables of linearly parameterized polynomials maps, Linear Algebra Appl., 437, 612-622, (2012) · Zbl 1244.26021
[38] C. M. Guldberg and P. Waage, Studies Concerning Affinity, C. M. Forhandlinger: Videnskabs-Selskabet i Christiana 35 (1864).
[39] J. Gunawardena, Chemical reaction network theory for in-silico biologists, Available online at http://vcp.med.harvard.edu/papers/crnt.pdf, 2003.
[40] J. W. Helton, V. Katsnelson, and I. Klep, Sign patterns for chemical reaction networks, J. Math. Chem. 47 (2010), 403-429. · Zbl 1197.92061
[41] J. W. Helton, I. Klep, and R. Gomez, Determinant expansions of signed matrices and of certain Jacobians, SIAM J. Matrix Anal. Appl. 31 (2009), 732-754. · Zbl 1191.15023
[42] Holstein, K; Flockerzi, D; Conradi, C, Multistationarity in sequential distributed multisite phosphorylation networks, Bull. Math. Biol., 75, 2028-2058, (2013) · Zbl 1283.92030
[43] Horn, F, Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Rational Mech. Anal., 49, 172-186, (1972)
[44] F. Horn and R. Jackson, General mass action kinetics, Arch. Ration. Mech. Anal. 47 (1972), 81-116.
[45] I. Itenberg and M. F. Roy, Multivariate Descartes’ rule, Beiträge zur Algebra und Geometrie 37 (1996), 337-346. · Zbl 0870.14038
[46] B. Joshi and A. Shiu, Simplifying the Jacobian criterion for precluding multistationarity in chemical reaction networks, SIAM J. Appl. Math. 72 (2012), 857-876. · Zbl 1253.65070
[47] M. Joswig and T. Theobald, Polyhedral and algebraic methods in computational geometry, Universitext, Springer, London, 2013. · Zbl 1267.52001
[48] A. G. Khovanskiĭ, Fewnomials, Translations of Mathematical Monographs, vol. 88, American Mathematical Society, Providence, RI, 1991, Translated from the Russian by Smilka Zdravkovska.
[49] V. Klee, R. Ladner, and R. Manber, Signsolvability revisited, Linear Algebra Appl. 59 (1984), 131-157.
[50] F. Kubler and K. Schmedders, Tackling multiplicity of equilibria with Gröbner bases, Oper. Res. 58 (2010), 1037-1050. · Zbl 1228.90125
[51] O. L. Mangasarian, Nonlinear programming, Classics in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994, Corrected reprint of the 1969 original. · Zbl 0833.90108
[52] M. Mincheva and G. Craciun, Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks, Proceedings of the IEEE 96 (2008), 1281-1291. · Zbl 1251.92016
[53] Müller, S; Regensburger, G, Generalized mass action systems: complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM J. Appl. Math., 72, 1926-1947, (2012) · Zbl 1261.92063
[54] S. Müller and G. Regensburger, Generalized mass-action systems and positive solutions of polynomial equations with real and symbolic exponents, Computer Algebra in Scientific Computing (V. P. Gerdt, W. Koepf, W. M. Seiler, and E. V. Vorozhtsov, eds.), Lecture Notes in Computer Science, vol. 8660, Springer International Publishing, 2014, pp. 302-323. · Zbl 1421.92043
[55] J. D. Murray, Mathematical biology: I. An introduction, third ed., Interdisciplinary Applied Mathematics, vol. 17, Springer-Verlag, New York, 2002. · Zbl 1006.92001
[56] C. Pantea, H. Koeppl, and G. Craciun, Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), 2153-2170. · Zbl 1253.80023
[57] Pérez Millán, M; Dickenstein, A; Shiu, A; Conradi, C, Chemical reaction systems with toric steady states, Bull. Math. Biol., 74, 1027-1065, (2012) · Zbl 1251.92016
[58] J. Rambau, TOPCOM: Triangulations of point configurations and oriented matroids, Mathematical software (Beijing, 2002), World Sci. Publ., River Edge, NJ, 2002, pp. 330-340. · Zbl 1057.68150
[59] J. Richter-Gebert and G. M. Ziegler, Oriented matroids, Handbook of discrete and computational geometry, CRC, Boca Raton, FL, 1997, pp. 111-132. · Zbl 0902.05015
[60] M. Safey El Din, RAGLib, Available online at http://www-polsys.lip6.fr/ safey/RAGLib/, 2013.
[61] I. W. Sandberg and A. N. Willson, Existence and uniqueness of solutions for the equations of nonlinear DC networks, SIAM J. Appl. Math. 22 (1972), 173-186. · Zbl 0241.94023
[62] Savageau, MA, Biochemical systems analysis. I. some mathematical properties of the rate law for the component enzymatic reactions, J. Theor. Biol., 25, 365-369, (1969)
[63] M. A. Savageau and E. O. Voit, Recasting nonlinear differential equations as S-systems: a canonical nonlinear form, Math. Biosci. 87 (1987), 83-115. · Zbl 0631.34015
[64] A. Schrijver, Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons Ltd., Chichester, 1986, A Wiley-Interscience Publication.
[65] G. Shinar and M. Feinberg, Concordant chemical reaction networks, Math. Biosci. 240 (2012), 92-113. · Zbl 1316.92100
[66] G. Shinar and M. Feinberg, Concordant chemical reaction networks and the species-reaction graph, Math. Biosci. 241 (2013), 1-23. · Zbl 1309.92094
[67] A. J. Sommese and C. W. Wampler, II, The numerical solution of systems of polynomials arising in engineering and science, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. · Zbl 1091.65049
[68] F. Sottile and C. Zhu, Injectivity of 2D toric Bézier patches, Proceedings of 12th International Conference on Computer-Aided Design and Computer Graphics (Jinan, China) (R. Martin, H. Suzuki, and C. Tu, eds.), IEEE CPS, 2011, pp. 235-238.
[69] F. Sottile, Real solutions to equations from geometry, University Lecture Series, vol. 57, American Mathematical Society, Providence, RI, 2011. · Zbl 1233.14001
[70] W. A. Stein et al., Sage Mathematics Software, Available online at http://www.sagemath.org, 2013.
[71] D. J. Struik (ed.), A source book in mathematics, 1200-1800, Source Books in the History of the Sciences. Cambridge, Mass.: Harvard University Press, XIV, 427 p., 1969. · Zbl 0205.29202
[72] M. Uhr, Structural analysis of inference problems arising in systems biology, Ph.D. thesis, ETH Zurich, 2012.
[73] Wiuf, C; Feliu, E, 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
[74] G. M. Ziegler, Lectures on polytopes, Springer-Verlag, New York, 1995. · Zbl 0823.52002
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