[Paper Review] Classification of hyperbolicity and stability preservers: the multivariate Weyl algebra case

Julius Borcea, Petter Brändén+1 more|arXiv (Cornell University)|Jun 15, 2006

Mathematical Dynamics and Fractals36 references10 citations

TL;DR

This paper characterizes finite-order linear differential operators that preserve stability in multivariate polynomials, introducing a multivariate generalization of multiplier sequences and proving a duality between a differential operator and its Fischer-Fock adjoint. The key contribution is a multivariate analog of the Lax conjecture for real stable polynomials in two variables, enabling classification of stability-preserving operators via determinants and homogenized symbols.

ABSTRACT

Abstract. A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We characterize all finite order linear differential operators that preserve stability. An important technical tool that we develop in the process is the multivariate generalization of the classical notion of multiplier sequence. We give a complete description of all multivariate multiplier sequences as well as those of finite order. Next we formulate and prove a natural analog of the Lax conjecture for real stable polynomials in two variables and use it to classify all finite order linear differential operators that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols. As a further consequence of our methods we establish a duality theorem showing that a differential operator preserves stability if and only if its Fischer-Fock adjoint has the same property. This is a vast generalization of the Hermite-Poulain-Jensen theorem in the univariate case and a natural multivariate extension of the latter. We also discuss several other applications of our results as well as further directions and open problems. Contents

Motivation & Objective

• To extend the classical theory of multiplier sequences to the multivariate setting.
• To classify finite-order linear differential operators that preserve stability in multivariate polynomials.
• To formulate and prove a multivariate analog of the Lax conjecture for real stable polynomials in two variables.
• To establish a duality theorem showing that a differential operator preserves stability if and only if its Fischer-Fock adjoint does.
• To provide a determinant-based classification of operators preserving univariate hyperbolic polynomials via homogenized symbols.

Proposed method

• Develops a multivariate generalization of the classical notion of multiplier sequences, characterizing all such sequences and those of finite order.
• Applies the theory of real stable polynomials and their homogenized symbols to derive determinant-based characterizations of stability-preserving operators.
• Introduces and utilizes the Fischer-Fock adjoint of a differential operator to establish a duality between stability preservation and adjoint preservation.
• Employs tools from multivariate complex analysis and polynomial theory to analyze the zero sets of multivariate polynomials under differential operators.
• Leverages the structure of the multivariate Weyl algebra to analyze finite-order differential operators acting on polynomials.
• Uses the Lax-type characterization for two-variable real stable polynomials as a key technical tool to derive broader classification results.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a finite-order linear differential operator to preserve stability in multivariate polynomials?
• RQ2 How can the classical notion of multiplier sequences be generalized to the multivariate setting, and what are the complete characterizations of such sequences?
• RQ3 What is the multivariate analog of the Lax conjecture, and how does it enable the classification of stability-preserving operators?
• RQ4 How is the Fischer-Fock adjoint related to stability preservation, and does a duality principle hold in the multivariate case?
• RQ5 Can univariate hyperbolicity-preserving operators be fully characterized using determinants and homogenized symbols in the multivariate framework?

Key findings

• All multivariate multiplier sequences are completely characterized, including those of finite order.
• A multivariate analog of the Lax conjecture is formulated and proven for real stable polynomials in two variables.
• Finite-order linear differential operators preserving stability are fully classified using determinants and homogenized symbols.
• A duality theorem is established: a differential operator preserves stability if and only if its Fischer-Fock adjoint does.
• The Hermite-Poulain-Jensen theorem is generalized to the multivariate setting as a natural consequence of the duality result.
• The results provide a complete framework for analyzing stability-preserving operators in the multivariate Weyl algebra setting.