Emma Boniface | Searcharxiv

arXiv2024

Tropical toric maximum likelihood estimation

We consider toric maximum likelihood estimation over the field of Puiseux series and study critical points of the likelihood function using tropical methods. This problem translates to finding the intersection points of a tropical affine space with a classical linear subspace. We derive new structural results on tropical affine spaces and use these to give a complete and explicit description of the tropical critical points in certain cases. In these cases, we associate tropical critical points to the simplices in a regular triangulation of the polytope giving rise to the toric model.

math.AG↗

arXiv2022

Trinomials and Deterministic Complexity Limits for Real Solving

We detail an algorithm that -- for all but a $\frac{1}{\Omega(\log(dH))}$ fraction of $f\in\mathbb{Z}[x]$ with exactly $3$ monomial terms, degree $d$, and all coefficients in $\{-H,\ldots, H\}$ -- produces an approximate root (in the sense of Smale) for each real root of $f$ in deterministic time $\log^{4+o(1)}(dH)$ in the classical Turing model. (Each approximate root is a rational with logarithmic height $O(\log(dH))$.) The best previous deterministic bit complexity bounds were exponential in $\log d$. We then relate this to Koiran's Trinomial Sign Problem (2017): Decide the sign of a degree $d$ trinomial $f\in\mathbb{Z}[x]$ with coefficients in $\{-H,\ldots,H\}$, at a point $r\!\in\!\mathbb{Q}$ of logarithmic height $\log H$, in (deterministic) time $\log^{O(1)}(dH)$. We show that Koiran's Trinomial Sign Problem admits a positive solution, at least for a fraction $1-\frac{1}{\Omega(\log(dH))}$ of the inputs $(f,r)$.

math.AG↗