From chemical reaction networks to Descartes' rule of signs

A DICKENSTEIN

Perugia

Jornada; Giornata INdAM 2016; 2016

Istituto Nazionale di Alta Matematica ?Francesco Severi? - INdAM

In the context of chemical reaction networks with mass-action andother rational kinetics, a major question is to preclude or to guarantee multiple positive steady states.  I will explain this motivation and I will present necessary and sufficient conditions interms of sign vectors for the injectivity of families of polynomialsmaps with arbitrary real exponents defined on the positive orthant. These conditions extend existing injectivity conditions expressed in terms of Jacobian matrices and determinants, obtained by several authors. In the context of real algebraic geometry, this approach can be seen 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. This is joint work with Stefan Müller, Elisenda Feliu, Georg Regensburger, Anne Shiu and Carsten Conradi. I will also present some further advances in this multivariate generalization obtained in collaboration with Frédéric Bihan, together with applications to biochemical MESSI systems obtained in collaboration with Mercedes Pérez Millán.