Journal
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
Volume 16, Issue 1, Pages 69-97Publisher
SPRINGER
DOI: 10.1007/s10208-014-9239-3
Keywords
Sign vector; Restricted injectivity; Power-law kinetics; Descartes' rule of signs; Oriented matroid
Funding
- Generalitat de Catalunya
- Spanish research project [MTM2012-38122-C03-01]
- BMBF [FKZ 0315744]
- state Saxony-Anhalt
- NSF [DMS-1004380, DMS-1312473]
- UBACYT [20020130100207BA]
- CONICET [PIP 11220110100580]
- ANPCyT, Argentina [PICT-2013-1110]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1513364] Funding Source: National Science Foundation
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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.
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