Amœbas and structural stability of multidimensional systems: a test algorithm based on Monte-Carlo integration

Abstract

Given a Laurent polynomial $F \in \mathbb {C}[z_1^{\pm 1}, \ldots , z_n^{\pm 1}]$, its amœba $\mathcal A_F$ is the image by $z=(z_1, \ldots , z_n)\in (\mathbb {C}^*)^n \longmapsto (\log |z_1|, \ldots , \log |z_n|)\in \mathbb {R}^n$ of the algebraic zero set $V(F)=\{z\in (\mathbb {C}^*)^n\,;\, F(z)=0\}$ of the complex torus $\mathbb {T}^n :=(\mathbb {C}^*)^n$. We relate here the question of the BIBO stability of a multilinear discrete time invariant system with a regular transfer function $G(z_1, ..., z_n)/F(z_1, \ldots , z_n)$, where $F, G\in \mathbb {C}[z_1, ..., z_n]$ are coprime or more precisely structural stability, with the geometrical study of the amœba $\mathcal A_F$. A criterion for strong and weak structural stability is expressed in terms of the position of $\varvec{0}=(0, \ldots , 0)\in \mathbb {R}^n$ with respect to the amœba $\mathcal {A}_{F}$. Then we propose a Monte-Carlo integration based algorithm in order to test the structural stability of a given such system. The proposed algorithm is not limited by the curse of dimensionality, as opposed to the state-of-the-art methods: It can be applied to any number of variables n. Several illustrative examples are presented and discussed.