From chemical reaction networks to Descartes' rule of signs

A DICKENSTEIN

Cabo Frío

Workshop; ELGA II; 2015

IMPA

In the context of chemical reactionnetworks with mass-action and other rational kinetics, a major question is topreclude or to guarantee multiple positive steady states. I will explain thismotivation and will present necessary and sufficient conditions in terms ofsign vectors for the injectivity of families of polynomials maps with arbitraryreal exponents defined on the positive orthant. These conditions extendexisting injectivity conditions expressed in terms of Jacobian matrices anddeterminants, obtained by several authors. In the context of real algebraicgeometry, this approach can be seen as the first partial multivariategeneralization of the classical Descartes´ rule, whichbounds the number of positive real roots of a univariate real polynomial interms of the number of sign variations of its coefficients. This is joint workwith Stefan M"uller, Elisenda Feliu, Georg Regensburger, Anne Shiu andCarsten Conradi. Our results can be applied to a general class of biochemicalnetworks involving enzymatic reactions defined in collaboration with MercedesP´erez Mill´an, that I will briefly describe. I will also present somefurther advances in this multivariate generalization obtained in collaborationwith Fr´ed´eric Bihan.