Decoupling Multivariate Polynomials Using First-Order Information

TL;DR

This paper introduces a novel method to decompose multivariate polynomials into simpler univariate components using first-order derivative information, enabling efficient analysis and reconstruction.

Contribution

The paper proposes a new tensor-based approach leveraging Jacobian matrices for polynomial decoupling, expanding the toolbox for polynomial analysis.

Findings

Successfully decomposes multivariate polynomials in numerical examples

Provides conditions for the method's applicability

Demonstrates the effectiveness of tensor decomposition in polynomial analysis

Abstract

We present a method to decompose a set of multivariate real polynomials into linear combinations of univariate polynomials in linear forms of the input variables. The method proceeds by collecting the first-order information of the polynomials in a set of operating points, which is captured by the Jacobian matrix evaluated at the operating points. The polyadic canonical decomposition of the three-way tensor of Jacobian matrices directly returns the unknown linear relations, as well as the necessary information to reconstruct the univariate polynomials. The conditions under which this decoupling procedure works are discussed, and the method is illustrated on several numerical examples.