arXiv subjects

Laura Brustenga i Moncusí

Publications and source records attributed to Laura Brustenga i Moncusí.

6 recordsLinked to original sources

arXiv2022

Disguised toric dynamical systems

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, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. By means of (real) algebraic geometry we show that some reaction networks have an empty toric locus or a toric locus of Lebesgue measure zero in parameter space, while their disguised toric locus is of positive measure. We also propose some algorithms one can use to detect the disguised toric locus.

math.AG↗

arXiv2021

The Waring rank of binary binomial forms

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.

math.AC↗

arXiv2021

Inverting catalecticants of ternary quartics

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 case of binary forms, by describing the intersection of the reciprocal variety and the orthogonal of the LSSM in the rank loci. Moreover, we prove that only the rank-$1$ locus, namely the Veronese surface $ν_4(\mathbb{P}^2)$, contributes to the degree of the reciprocal variety.

math.AG↗

arXiv2019

96120: The degree of the linear orbit of a cubic surface

The projective linear group $\pgl(\comp,4)$ acts on cubic surfaces, considered as points of $\mathbb{P}_{\mathbb{C}}^{19}$. We compute the degree of the $15$-dimensional projective variety given by the Zariski closure of the orbit of a general cubic surface. The result, 96120, is obtained using methods from numerical algebraic geometry.

math.AG↗

arXiv2019

The blow up split sections family

The universal scheme of clusters of sections is an adaption of Kleiman's iterated blow ups (which parametrise clusters of points) to parametrise clusters of sections. They can also be constructed iteratively, but the iterative step is not so clear. Defining the blow up split sections family, we characterise this iterative step. Roughly speaking, it is a morphism that combines the universal properties of blow ups and universal section families. It is a generalisation of blow ups, and as such, we show that it exhibits some sort of birationality. But now, the flattening stratification of a morphism plays also an important role.

math.AG↗

arXiv2019

On the Waring rank of binary forms: The binomial formula and a dihedral cover of rank two forms

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 illustrating it. Secondly, we prove that, up to scalar multiplication, there are exactly $\binom{d-1}{2}$ binary forms of degree $d$ with Waring rank two and multiple of three fixed distinct linear forms.

math.AG↗