Periodic Reporting for period 1 - POSALG (Signs, polynomials, and reaction networks)
Période du rapport: 2023-01-01 au 2025-06-30
The main goal of this project is to advance in selected topics within algebra in order to contribute to the analysis of biochemical reaction networks. These networks are commonly modeled using systems of differential equations that contain several unknown parameters. To study their behavior, researchers often apply dynamical systems theory, which helps identify main properties on how the species change in time as a consequence of the interactions within the network.
In many cases, these differential equations take the form of polynomial systems, making algebraic geometry and computational algebra valuable tools for analysis. A critical aspect of this analysis is understanding the equilibria of the network, which correspond to the solutions of a system of polynomial equations. While classical algebraic methods focus on complex solutions, this project specifically examines positive solutions, as the variables represent concentrations of chemical species and must remain nonnegative.
This project focuses on developing mathematical theory to study the positive solutions of parametrized polynomial systems. Key research questions include determining the number of such solutions and analyzing the (real) dimension of the solution set. In particular, a weaker version of the solution-counting problem can often be reframed as determining whether a given multivariate polynomial attains negative values and identifying the number of connected components in the complement of its zero set within the positive orthant—a question rooted in semi-algebraic geometry.
In many cases, these differential equations take the form of polynomial systems, making algebraic geometry and computational algebra valuable tools for analysis. A critical aspect of this analysis is understanding the equilibria of the network, which correspond to the solutions of a system of polynomial equations. While classical algebraic methods focus on complex solutions, this project specifically examines positive solutions, as the variables represent concentrations of chemical species and must remain nonnegative.
This project focuses on developing mathematical theory to study the positive solutions of parametrized polynomial systems. Key research questions include determining the number of such solutions and analyzing the (real) dimension of the solution set. In particular, a weaker version of the solution-counting problem can often be reframed as determining whether a given multivariate polynomial attains negative values and identifying the number of connected components in the complement of its zero set within the positive orthant—a question rooted in semi-algebraic geometry.
We have established a general framework, called “vertically parametrized systems”, for studying the systems arising from reaction networks. Within this framework, we have conducted a detailed analysis of the fundamental properties of these systems. We have particularly understood the generic (with respect to the parameters) dimension of the zero sets, and properties about nondegeneracy of equilibria. With this in place we have been able to characterize relevant properties of vertically parametrized systems (and hence of reaction networks) such as the existence of absolute concentration robustness or monomial parametrizations.
In addition, we have explored the nonnegativity of polynomials and the topology of the complement of the zero set of a multivariate polynomial. For the latter, we have established conditions, based on the sign of the coefficients and the geometry of the exponents of the multivariate polynomial, to determine whether a given polynomial has at most one connected component in its complement where it takes negative values. With respect to nonnegativity, we have identified supports that allow for a simple algorithm to decide whether the polynomial can never attain negative values. The algorithm is based on vertically parametrized systems and homotopy continuation.
Finally, our ongoing work explores tropical methods and Gale duality to derive bounds on the total number of positive solutions for verticallly parametrized systems, hence of positive equilibria.
In addition, we have explored the nonnegativity of polynomials and the topology of the complement of the zero set of a multivariate polynomial. For the latter, we have established conditions, based on the sign of the coefficients and the geometry of the exponents of the multivariate polynomial, to determine whether a given polynomial has at most one connected component in its complement where it takes negative values. With respect to nonnegativity, we have identified supports that allow for a simple algorithm to decide whether the polynomial can never attain negative values. The algorithm is based on vertically parametrized systems and homotopy continuation.
Finally, our ongoing work explores tropical methods and Gale duality to derive bounds on the total number of positive solutions for verticallly parametrized systems, hence of positive equilibria.
The project has revealed two particularly promising research directions. The first is the study of nonnegative polynomials and the second is the study of vertically parametrized systems. Both lines of research have led to original findings that have openend new avenues for future study. Moreover, these two areas are interconnected and hence can progress in parallel.
The theory of vertically parametrized systems has already resolved fundamental questions in reaction network theory that were previously poorly understood. Meanwhile, our work on nonnegative polynomials is at an earlier stage; so far, we have focused on polynomials expressible as sums of nonnegative circuits when nonnegative. However, our methods show promise for extending to broader classes of polynomials in future research.
The theory of vertically parametrized systems has already resolved fundamental questions in reaction network theory that were previously poorly understood. Meanwhile, our work on nonnegative polynomials is at an earlier stage; so far, we have focused on polynomials expressible as sums of nonnegative circuits when nonnegative. However, our methods show promise for extending to broader classes of polynomials in future research.