arXiv subjects

Yulia Mukhina

Publications and source records attributed to Yulia Mukhina.

4 recordsLinked to original sources

arXiv2026

Engineered Complete Intersections: Algorithmic Aspects

Engineered Complete Intersections (ECI's) are a class of sparse polynomial systems frequently arising in a number of contexts, both in pure mathematics (e.g. enumerative geometry) and applications (e.g. chemical reaction networks). Based on theoretical results given by the first author, we give several contributions. First we give a new effective technique to tropicalize such systems by generalizing the classical notion of mixed subdivisions introduced by Huber and Sturmfels (1995) to ECI's with the particular goal to efficiently count solutions of square systems of equations in ECI form. We further design a tropical homotopy continuation algorithm for computing such mixed subdivisions, inspired by Jensen (2016), Malajovich (2017) and Daisey and Ren (2024). Our techniques can be used to numerically solve such systems by coupling them with the algorithms introduced by Helminck, Henriksson and Ren (2024). Finally, we give an algorithm to compute Newton polytopes of eliminants of ECI's. This gives a new way to compute, for example, Newton polytopes of so-called $A$-discriminants. Coupled with evaluation-interpolation paradigms our algorithm gives an efficient approach to compute such eliminants. We implemented our algorithms in the form of a software package which we use to demonstrate their practical feasibility on a range of examples.

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arXiv2026

Projecting dynamical systems via a support bound

For a polynomial dynamical system, we study the problem of computing the minimal differential equation satisfied by a chosen coordinate (in other words, projecting the system on the coordinate). This problem can be viewed as a special case of the general elimination problem for systems of differential equations and appears in applications to modeling and control. We give a bound for the Newton polytope of such minimal equation. Our bound depends on the dimension of the model and the degrees $d$ and $D$ of the polynomials defining the dynamics of the chosen coordinate and the remaining coordinates, respectively. We show that our bound is sharp if $d \leqslant D$ or the model is planar. We further use this bound to design an algorithm for computing the minimal equation following the evaluation-interpolation paradigm. We demonstrate that our implementation of the algorithm can tackle problems which are out of reach for the state-of-the-art software for differential elimination.

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arXiv2025

Support bound for differential elimination in polynomial dynamical systems

We study an important special case of the differential elimination problem: given a polynomial parametric dynamical system $\mathbf{x}' = \mathbf{g}(\boldsymbolμ, \mathbf{x})$ and a polynomial observation function $y = f(\boldsymbolμ, \mathbf{x})$, find the minimal differential equation satisfied by $y$. In our previous work, for the case $y = x_1$, we established a bound on the support of such a differential equation for the non-parametric case and shown that it can be turned into an algorithm via the evaluation-interpolation approach. The main contribution of the present paper is a generalization of the aforementioned result in two directions: to allow any polynomial function $y = f(\mathbf{x})$, not just a single coordinate, and to allow $\mathbf{g}$ and $f$ depend on unknown symbolic parameters. We conduct computation experiments to evaluate the accuracy of our new bound and show that the approach allows to perform elimination for some cases out of reach for the state of the art software.

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arXiv2025

On the Computation of Newton Polytopes of Eliminants

For systems of polynomial equations, we study the problem of computing the Newton polytope of their eliminants. As was shown by Esterov and Khovanskii, such Newton polytopes are mixed fiber polytopes of the Newton polytopes of the input equations. We use their results in combination with mixed subdivisions to design an algorithm computing these special polytopes. We demonstrate the increase in practical performance of our algorithm compared to existing methods using tropical geometry and discuss the differences that lead to this increase in performance. We also demonstrate an application of our work to differential elimination.

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