Lopsided Approximation of Amoebas

TL;DR

This paper improves the practical efficiency of amoeba approximation for Laurent polynomials by optimizing cyclic resultant computations, enabling faster algorithms and providing open-source implementations.

Contribution

It introduces an optimized method for amoeba approximation using cyclic resultants, overcoming previous computational bottlenecks and offering practical algorithms with open-source code.

Findings

Significant speedup in amoeba approximation algorithms

Effective use of cyclic resultants for practical computation

Open-source implementation demonstrating improved performance

Abstract

The amoeba of a Laurent polynomial is the image of the corresponding hypersurface under the coordinatewise log absolute value map. In this article, we demonstrate that a theoretical amoeba approximation method due to Purbhoo can be used efficiently in practice. To do this, we resolve the main bottleneck in Purbhoo's method by exploiting relations between cyclic resultants. We use the same approach to give an approximation of the Log preimage of the amoeba of a Laurent polynomial using semi-algebraic sets. We also provide a SINGULAR/SAGE implementation of these algorithms, which shows a significant speedup when our specialized cyclic resultant computation is used, versus a general purpose resultant algorithm.