Foundations of Computational Mathematics

, Volume 16, Issue 1, pp 69–97 | Cite as

Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry

  • Stefan Müller
  • Elisenda Feliu
  • Georg Regensburger
  • Carsten Conradi
  • Anne Shiu
  • Alicia Dickenstein
Article

Abstract

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 Craciun, Garcia-Puente, and Sottile, 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.

Keywords

Sign vector Restricted injectivity Power-law kinetics Descartes’ rule of signs Oriented matroid 

Mathematics Subject Classification

13P15 12D10 70K42 37C10 80A30 52C40 

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Copyright information

© SFoCM 2015

Authors and Affiliations

  • Stefan Müller
    • 1
  • Elisenda Feliu
    • 2
  • Georg Regensburger
    • 1
  • Carsten Conradi
    • 3
  • Anne Shiu
    • 4
  • Alicia Dickenstein
    • 5
  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  3. 3.Max-Planck-Institut Dynamik komplexer technischer SystemeMagdeburgGermany
  4. 4.Department of MathematicsTexas A&M UniversityCollege StationUSA
  5. 5.Dto. de Matemática, FCENUniversidad de Buenos Aires, and IMAS (UBA-CONICET)Buenos AiresArgentina

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