Laura Brustenga i Moncusí

University of Copenhagen · Department of Mathematical Sciences

We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric. They are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence...

We give an explicit formula for the Waring rank of every binary binomial form with complex coefficients. We give several examples to illustrate this, and compare the Waring rank and the real Waring rank for binary binomial forms.

We study the reciprocal variety to the linear space of symmetric matrices (LSSM) of catalecticant matrices associated with ternary quartics. With numerical tools, we obtain 85 to be its degree and 36 to be the ML-degree of the LSSM. We provide a geometric explanation to why equality between these two invariants is not reached, as opposed to the cas...

We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric dynamical systems. These systems are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and gl...

We compute the degree of the orbit closure of a generic cubic surface under the action of ${\rm PGL}(\mathbb{C},4)$. The result, 96120, is obtained by using methods from numerical algebraic geometry.

Waring problem for forms is important and classical in mathematics. It has been widely investigated because of its wide applications in several areas. In this paper, we consider the Waring problem for binary forms with complex coefficients. Firstly, we give an explicit formula for the Waring rank of any binary binomial and several examples to illus...

The straight blow up section family of a projection $X\times Y\to X$ along a closed subscheme $Z$ of $X\times Y$ is an object that combines the universal properties of the universal section family of $X\times Y\to X$ and of the blow up of $X\times Y$ along $Z$. It is our attempt to characterise a unique morphism between universal schemes of relativ...

We generalize the concept of $r$-point clusters of a scheme $S$ to $r$-relative clusters of a $B$-scheme $\mathcal{S}$. Define schemes $Cl_r$ that naturally parametrize the $r$-relative clusters which generalize the Kleiman's construction of the iterated blowups by $r$-point clusters. We show that the iterated construction of $Cl_{r+1}$ from $Cl_{r...