Root bounds of vertical systems using tropical geometry

TL;DR

This paper establishes bounds on the number of roots of vertical polynomial systems using tropical geometry, with applications to optimization and chemical networks, and provides Julia implementations.

Contribution

It introduces a novel approach to bounding roots of vertically parametrized systems via tropical intersection theory and mixed volumes, with practical algorithms.

Findings

The generic number of complex zeros equals the tropical intersection number.

Bounds on positive zeros are derived from tropical intersections.

Algorithms for root bounds are implemented in Julia.

Abstract

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.