Approximating amoebas and coamoebas by sums of squares

TL;DR

This paper introduces a novel method using sums of squares and semidefinite programming to compute and certify membership in amoebas and coamoebas, linking algebraic geometry with optimization techniques.

Contribution

It develops a new approach to approximate amoebas and coamoebas via sums of squares, providing degree bounds and computational methods.

Findings

Polynomial certificates for non-membership are constructed.

Degree bounds for certificates are established.

Computational experiments demonstrate the method's effectiveness.

Abstract

Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the arg-map, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing new connections between (co-)amoebas, semialgebraic and convex algebraic geometry and semidefinite programming. Our approach is based on formulating the membership problem in amoebas (respectively coamoebas) as a suitable real algebraic feasibility problem. Using the real Nullstellensatz, this allows to tackle the problem by sums of squares techniques and semidefinite programming. Our method yields polynomial identities as certificates of non-containment of a point in an amoeba or coamoeba. As the main theoretical result, we establish some degree bounds on the polynomial certificates. Moreover, we provide some actual computations of amoebas based on the sums of squares approach.