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

doi:10.1007/s10208-014-9239-3

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

Department of Mathematical Sciences

55 Citations (Scopus)

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.

Original language	English
Journal	Foundations of Computational Mathematics
Volume	16
Issue number	1
Pages (from-to)	69-97
Number of pages	29
ISSN	1615-3375
DOIs	https://doi.org/10.1007/s10208-014-9239-3
Publication status	Published - 1 Feb 2016

Access to Document

10.1007/s10208-014-9239-3

Cite this

Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. / Müller, Stefan; Feliu, Elisenda; Regensburger, Georg et al.
In: Foundations of Computational Mathematics, Vol. 16, No. 1, 01.02.2016, p. 69-97.

Research output: Contribution to journal › Journal article › Research › peer-review

@article{48ebda63cb8746f981410eb3f3b537b9,

title = "Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry",

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{\textquoteright} 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 = "math.AG, math.DS",

author = "Stefan M{\"u}ller and Elisenda Feliu and Georg Regensburger and Carsten Conradi and Anne Shiu and Alicia Dickenstein",

year = "2016",

month = feb,

day = "1",

doi = "10.1007/s10208-014-9239-3",

language = "English",

volume = "16",

pages = "69--97",

journal = "Foundations of Computational Mathematics",

issn = "1615-3375",

publisher = "Springer",

number = "1",

}

TY - JOUR

T1 - Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry

AU - Müller, Stefan

AU - Feliu, Elisenda

AU - Regensburger, Georg

AU - Conradi, Carsten

AU - Shiu, Anne

AU - Dickenstein, Alicia

PY - 2016/2/1

Y1 - 2016/2/1

N2 - 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.

AB - 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.

KW - math.AG

KW - math.DS

U2 - 10.1007/s10208-014-9239-3

DO - 10.1007/s10208-014-9239-3

M3 - Journal article

SN - 1615-3375

VL - 16

SP - 69

EP - 97

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

IS - 1

ER -

Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry

Abstract

Access to Document

Fingerprint

Cite this