Differential Elimination and Algebraic Invariants of Polynomial Dynamical Systems

TL;DR

This paper introduces a novel algebraic approach for deriving invariants of polynomial dynamical systems, enhancing safety verification methods for cyber-physical systems.

Contribution

It adapts the Rosenfeld-Gröbner algorithm to compute invariants without relying on Gr"obner bases or quantifier elimination, focusing on totally real varieties.

Findings

Efficiently computes algebraic invariants for polynomial systems.

Identifies totally real varieties as key for invariance checking.

Provides a systematic elimination-based method for invariants.

Abstract

Invariant sets are a key ingredient for verifying safety and other properties of cyber-physical systems that mix discrete and continuous dynamics. We adapt the elimination-theoretic Rosenfeld-Gr\"{o}bner algorithm to systematically obtain algebraic invariants of polynomial dynamical systems without using Gr\"{o}bner bases or quantifier elimination. We identify totally real varieties as an important class for efficient invariance checking.