arXiv subjects

Oskar Henriksson

Publications and source records attributed to Oskar Henriksson.

11 recordsLinked to original sources

arXiv2026

Root bounds of vertical systems using tropical geometry

Sparse polynomial systems with vertical coefficient dependencies arise naturally when describing the critical points of optimization problems and, when augmented with linear forms, the steady states of chemical reaction networks. Moreover, any polynomial system is the specialization of such a parametrized system. We prove that the generic number of complex zeros of an augmented vertically parametrized system is the tropical intersection number of a tropical linear space and a classical linear space. In the special case when the matroid of the tropical linear space is cotransversal, we express this number as a mixed volume. We also obtain bounds on the maximal number of positive zeros, which is often the significant number in applications. We derive lower bounds from the number of intersections between positive tropicalizations, and when the positive zeros have toric structure, we provide upper bounds that are simpler and in some cases smaller than the generic root count. The resulting algorithms are implemented in Julia.

math.AG↗

arXiv2026

Numerical Elimination: Computing Complements of Real Hypersurfaces Using Pseudo-Witness Sets

Many hypersurfaces in algebraic geometry, such as discriminants, arise as the projection of another variety. The real complement of such a hypersurface decomposes into connected components. In this paper, we propose a new method for computing these components. Existing methods require the explicit equation of the hypersurface as input. However, computing this equation by elimination can be computationally demanding or even infeasible. Our approach instead derives from univariate interpolation by computing the intersection of the hypersurface with a line. Such an intersection may be computed using so-called pseudo-witness sets without computing a defining equation for the hypersurface. We implement our approach in a forthcoming Julia package and demonstrate, on several examples, that the resulting algorithm accurately recovers all components of the real complement of the hypersurface.

math.AG↗

arXiv2025

A flux-based approach for analyzing the disguised toric locus of reaction networks

Dynamical systems with polynomial right-hand sides are very important in various applications, e.g., in biochemistry and population dynamics. The mathematical study of these dynamical systems is challenging due to the possibility of multistability, oscillations, and chaotic dynamics. One important tool for this study is the concept of reaction systems, which are dynamical systems generated by reaction networks for some choices of parameter values. Among these, disguised toric systems are remarkably stable: they have a unique attracting fixed point, and cannot give rise to oscillations or chaotic dynamics. The computation of the set of parameter values for which a network gives rise to disguised toric systems (i.e., the disguised toric locus of the network) is an important but difficult task. We introduce new ideas based on network fluxes for studying the disguised toric locus. We prove that the disguised toric locus of any network $G$ is a contractible manifold with boundary, and introduce an associated graph $G^{\max}$ that characterizes its interior. These theoretical tools allow us, for the first time, to compute the full disguised toric locus for many networks of interest.

math.DS↗

arXiv2025

Maximum likelihood estimation of log-affine models using detailed-balanced reaction networks

A fundamental question in the field of molecular computation is what computational tasks a biochemical system can carry out. In this work, we focus on the problem of finding the maximum likelihood estimate (MLE) for log-affine models. We revisit a construction due to Gopalkrishnan of a mass-action system with the MLE as its unique positive steady state, which is based on choosing a basis for the kernel of the design matrix of the model. We extend this construction to allow for any finite spanning set of the kernel, and explore how the choice of spanning set influences the dynamics of the resulting network, including the existence of boundary steady states, the deficiency of the network, and the rate of convergence. In particular, we prove that using a Markov basis as the spanning set guarantees global stability of the MLE steady state.

q-bio.MN↗

arXiv2025

Moment varieties of the inverse Gaussian and gamma distributions are nondefective

We show that the parameters of a $k$-mixture of inverse Gaussian or gamma distributions are algebraically identifiable from the first $3k-1$ moments, and rationally identifiable from the first $3k+2$ moments. Our proofs are based on Terracini's classification of defective surfaces, careful analysis of the intersection theory of moment varieties, and a recent result on sufficient conditions for rational identifiability of secant varieties by Massarenti--Mella.

math.AG↗

arXiv2024

A tropical method for solving parametrized polynomial systems

We give a framework for constructing generically optimal homotopies for parametrized polynomial systems from tropical data. Here, generically optimal means that the number of paths tracked is equal to the generic number of solutions. We focus on two types of parametrized systems -- vertically parametrized and horizontally parametrized systems -- and discuss techniques for computing the tropical data efficiently. We end the paper with several case studies, where we analyze systems arising from chemical reaction networks, coupled oscillators, and rigid graphs.

math.AG↗

arXiv2024

Generic consistency and nondegeneracy of vertically parametrized systems

We determine the generic consistency, dimension and nondegeneracy of the zero locus over $\mathbb{C}^*$, $\mathbb{R}^*$ and $\mathbb{R}_{>0}$ of vertically parametrized systems: parametric polynomial systems consisting of linear combinations of monomials scaled by free parameters. These systems generalize sparse systems with fixed monomial support and freely varying parametric coefficients. As our main result, we establish the equivalence among three key properties: the existence of nondegenerate zeros, the zero set having generically the expected dimension, and the system being generically consistent. Importantly, we prove that checking whether a vertically parametrized system has these properties amounts to an easily computed matrix rank condition.

math.AG↗

arXiv2024

The generic geometry of steady state varieties

We answer several fundamental geometric questions about reaction networks with power-law kinetics, on topics such as generic finiteness of the number of steady states, robustness, and nondegenerate multistationarity. In particular, we give an ideal-theoretic characterization of generic absolute concentration robustness, as well as conditions under which a network that admits multiple steady states also has the capacity for nondegenerate multistationarity. The key tools underlying our results come from the theory of vertically parametrized systems, and include a linear algebra condition that characterizes when the steady state system has positive nondegenerate zeros.

q-bio.MN↗

arXiv2024

Toric invariance of vertically parametrized systems

We consider the problem of deciding whether the solution sets of a parametrized polynomial system are toric in the sense that they admit a monomial parametrization. We focus on vertically parametrized systems, which are sparse systems where we allow linear dependencies between coefficients in front of the same monomial. We give a matroid-theoretic characterization of the maximal-dimensional torus for which all solution sets are invariant under componentwise multiplication. Building on this, we provide necessary conditions and sufficient conditions for when the solution sets are unions of finitely many or a unique coset of this torus. The motivation of this work comes from the theory of reaction networks, where toric structure of the steady state system substantially simplifies the determination of multistationarity; here, we show that this is also the case for absolute concentration robustness and steady state invariants.

math.AG↗

arXiv2024

Moment varieties from inverse Gaussian and gamma distributions

Motivated by previous work on moment varieties for Gaussian distributions and their mixtures, we study moment varieties for two other statistically important two-parameter distributions: the inverse Gaussian and gamma distributions. In particular, we realize the moment varieties as determinantal varieties and find their degrees and singularities. We also provide computational evidence for algebraic identifiability of mixtures, and study the identifiability degree and Euclidean distance degree.

math.AG↗

arXiv2024

3D genome reconstruction from partially phased Hi-C data

The 3-dimensional (3D) structure of the genome is of significant importance for many cellular processes. In this paper, we study the problem of reconstructing the 3D structure of chromosomes from Hi-C data of diploid organisms, which poses additional challenges compared to the better-studied haploid setting. With the help of techniques from algebraic geometry, we prove that a small amount of phased data is sufficient to ensure finite identifiability, both for noiseless and noisy data. In the light of these results, we propose a new 3D reconstruction method based on semidefinite programming, paired with numerical algebraic geometry and local optimization. The performance of this method is tested on several simulated datasets under different noise levels and with different amounts of phased data. We also apply it to a real dataset from mouse X chromosomes, and we are then able to recover previously known structural features.

q-bio.GN↗