# Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics
**Dr. Georg
Regensburger**, Institute for Algebra, JKU Linz,
will give a public talk (followed by a discussion) on **Wed, Nov. 22, 2017**
at **16:15 o'clock** at **HS 13**
on the topic of
"Positive steady states and solutions of polynomial systems with real exponents"
. The organziers of the symposium,

Univ.-Prof. Dr. Gerhard Larcher

A.Univ.-Prof. Dr. Jürgen Maaß, and

die ÖMG (Österreichische Mathematische Gesellschaft),

hereby cordially invite you.

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

### Positive steady states and solutions of polynomial systems with real exponents

Reaction networks with mass action kinetics give rise to polynomial ODE systems. Chemical reaction network theory (CRNT) provides statements about uniqueness, existence, and stability of positive steady states for all rate constants and initial conditions depending on the underlying network structure alone. In terms of polynomial equations, they guarantee existence and uniqueness of positive solutions for all positive parameters.

We first survey several results from CRNT, emphasizing the consequences for polynomial equations with real and symbolic exponents and addressing computational aspects. Then we describe an extension to generalized mass action systems where reaction rates are allowed to be power-laws in the concentrations. In this setting, uniqueness and existence for all parameters additionally depend on sign vectors of related vector subspaces. We also illustrate our results with an implementation in the computer algebra system Maple. This is joint work with Stefan Müller. Finally, we discuss an interpretation of the results as a first partial multivariate generalization of the classical Descartes’ rule for positive roots of univariate polynomials. This is joint work Carsten Conradi, Alicia Dickenstein, Elisenda Feliu, Stefan Müller, and Anne Shiu.