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Numerical algorithm and complexity analysis for diagonalization of multivariate homogeneous polynomials

Lishan Fang, Hua-Lin Huang, Yuechen Li

TL;DR

The paper presents a center-based framework for diagonalizing multivariate homogeneous polynomials via Harrison's center theory. It establishes a polynomial-time criterion (detect nondegeneracy, compute center, and detect diagonalizability) and an iterative diagonalization algorithm that, when feasible, yields a diagonal sum of $d$-th powers; nonlinear steps may be NP-hard, but numerical results show practical efficiency. The work provides explicit complexity costs for each component in terms of $n$, $d$, and the term count $t$, and situates the method relative to randomized factorization, tensor methods, and Milnor-algebra approaches. Its significance lies in delivering a deterministic, algebraic route to diagonalization with documented performance gains and clear avenues for extension to multiple polynomials and to Waring-type decompositions.

Abstract

We study the computational complexity of a diagonalization technique for multivariate homogeneous polynomials, that is, expressing them as sums of powers of independent linear forms. It is based on Harrison's center theory and consists of a criterion and a diagonalization algorithm. Detailed formulations and computational complexity of each component of the technique are given. The complexity analysis focuses on the impacts of the number of variables and the degree of given polynomials. We show that this criterion runs in polynomial time and the diagonalization process performs efficiently in numerical experiments. Other diagonalization techniques are reviewed and compared in terms of complexity.

Numerical algorithm and complexity analysis for diagonalization of multivariate homogeneous polynomials

TL;DR

The paper presents a center-based framework for diagonalizing multivariate homogeneous polynomials via Harrison's center theory. It establishes a polynomial-time criterion (detect nondegeneracy, compute center, and detect diagonalizability) and an iterative diagonalization algorithm that, when feasible, yields a diagonal sum of -th powers; nonlinear steps may be NP-hard, but numerical results show practical efficiency. The work provides explicit complexity costs for each component in terms of , , and the term count , and situates the method relative to randomized factorization, tensor methods, and Milnor-algebra approaches. Its significance lies in delivering a deterministic, algebraic route to diagonalization with documented performance gains and clear avenues for extension to multiple polynomials and to Waring-type decompositions.

Abstract

We study the computational complexity of a diagonalization technique for multivariate homogeneous polynomials, that is, expressing them as sums of powers of independent linear forms. It is based on Harrison's center theory and consists of a criterion and a diagonalization algorithm. Detailed formulations and computational complexity of each component of the technique are given. The complexity analysis focuses on the impacts of the number of variables and the degree of given polynomials. We show that this criterion runs in polynomial time and the diagonalization process performs efficiently in numerical experiments. Other diagonalization techniques are reviewed and compared in terms of complexity.
Paper Structure (23 sections, 8 theorems, 63 equations, 3 tables, 8 algorithms)

This paper contains 23 sections, 8 theorems, 63 equations, 3 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Related work
  4. Methods
  5. Organization of the paper
  6. Centers and diagonalization of homogeneous polynomials
  7. Centers and diagonalization criterion
  8. Diagonalization algorithm
  9. Criterion of diagonalizability
  10. Detect nondegeneracy
  11. Compute center
  12. Detect diagonalizability
  13. Some examples
  14. Diagonalizing homogeneous polynomials
  15. Compute idempotents
  16. ...and 8 more sections

Key Result

proposition 1

Let $f(x_1, x_2, \ldots, x_n)$ be as above and assume $\operatorname{Rank} \left\{\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2},\ldots, \frac{\partial f}{\partial x_n} \right\}=r.$ Then there exists a change of variables $x=Py$ such that Moreover, if we remove the redundant variables from $g(y)$ and view it as an $r$-variate polynomial, then it becomes nondegenerate.

Theorems & Definitions (11)

  • remark 1
  • proposition 1
  • proof
  • proposition 2
  • proof
  • proposition 3
  • proposition 4
  • proposition 5
  • proposition 6
  • proposition 7
  • ...and 1 more