TY - BOOK

T1 - Signed Support of Multivariate Polynomials and Applications

AU - Telek, Máté L.

PY - 2024

Y1 - 2024

N2 - This thesis includes six papers that investigate three different areas: chemical reaction network theory, Descartes’ rule of signs, and real tropicalization. A common thread among them is the significant role played by the signed support of multivariate polynomials.Paper I and II focus on chemical reaction networks. In Paper I, we describe a general algorithm for verifying connectivity of the parameter region of multistationarity of a reaction network and apply it to several biologically relevant networks. In Paper II, our focus is on two families of phosphorylation networks, called n-site phosphorylation networks. We provide a proof showing that their parameter region of multistationarity is connected for every n ∈ N≥2.In Paper III and IV, we present combinatorial conditions on the signed support that provide upper bounds on the number of connected components of the set in the positive real orthant where the polynomial takes negative values. We frame this problem as a generalization of Descartes’ rule of signs to multivariate polynomials. The methods developed in Paper III and IV are crucial for the arguments used in Paper I and II.In Paper V, we investigate the real tropicalization of semi-algebraic sets and show its relation to the signed support of the polynomials defining these sets. In Paper VI, we study the signed A-discriminant and show that it has a simple structure if the signed support satisfies some combinatorial conditions. In such cases, Viro’s patchworking becomes applicable for determining all isotopy types of hypersurfaces in the positive real orthant with a prescribed signed support for their defining polynomials.

AB - This thesis includes six papers that investigate three different areas: chemical reaction network theory, Descartes’ rule of signs, and real tropicalization. A common thread among them is the significant role played by the signed support of multivariate polynomials.Paper I and II focus on chemical reaction networks. In Paper I, we describe a general algorithm for verifying connectivity of the parameter region of multistationarity of a reaction network and apply it to several biologically relevant networks. In Paper II, our focus is on two families of phosphorylation networks, called n-site phosphorylation networks. We provide a proof showing that their parameter region of multistationarity is connected for every n ∈ N≥2.In Paper III and IV, we present combinatorial conditions on the signed support that provide upper bounds on the number of connected components of the set in the positive real orthant where the polynomial takes negative values. We frame this problem as a generalization of Descartes’ rule of signs to multivariate polynomials. The methods developed in Paper III and IV are crucial for the arguments used in Paper I and II.In Paper V, we investigate the real tropicalization of semi-algebraic sets and show its relation to the signed support of the polynomials defining these sets. In Paper VI, we study the signed A-discriminant and show that it has a simple structure if the signed support satisfies some combinatorial conditions. In such cases, Viro’s patchworking becomes applicable for determining all isotopy types of hypersurfaces in the positive real orthant with a prescribed signed support for their defining polynomials.

M3 - Ph.D. thesis

BT - Signed Support of Multivariate Polynomials and Applications

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -