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A family of explicit Waring decompositions of a polynomial

Kangjin Han, Hyunsuk Moon

TL;DR

The paper addresses explicit Waring decompositions for monomials and homogeneous forms, introducing a parametric combinatorial identity that writes $D_\\mathbf{a} X_0^{a_0}\\cdots X_n^{a_n}$ as a sum of $d$-th powers of linear forms with computable coefficients. This yields a completely explicit decomposition and an upper bound for the monomial Waring rank $\\mathrm{rank}_\\Bbbk(M) \le \tfrac12(\\prod_i (a_i+1) - \\prod_i (a_i-1))$, and extends to general homogeneous forms via (de)homogenization. The authors compare with existing apolarity-based and rational decompositions, showing asymptotically fewer summands through a refined count $K(n,D)$, and provide a Macaulay2 implementation with a concrete case $X_0^4X_1^3X_2^2$ yielding a one-parameter family of 27-summand decompositions. These results have practical impact for polynomial computation and symmetric tensor problems by delivering explicit, low-sum representations and a usable computational toolkit.

Abstract

In this paper we settle some polynomial identity which provides a family of explicit Waring decompositions of any monomial $X_0^{a_0}X_1^{a_1}\cdots X_n^{a_n}$ over a field $\Bbbk$. This gives an upper bound for the Waring rank of a given monomial and naturally leads to an explicit Waring decomposition of any homogeneous form and, eventually, of any polynomial via (de)homogenization. Note that such decomposition is very useful in many applications dealing with polynomial computations, symmetric tensor problems and so on. We discuss some computational aspect of our result as comparing with other known methods and also present a computer implementation for potential use in the end.

A family of explicit Waring decompositions of a polynomial

TL;DR

The paper addresses explicit Waring decompositions for monomials and homogeneous forms, introducing a parametric combinatorial identity that writes as a sum of -th powers of linear forms with computable coefficients. This yields a completely explicit decomposition and an upper bound for the monomial Waring rank , and extends to general homogeneous forms via (de)homogenization. The authors compare with existing apolarity-based and rational decompositions, showing asymptotically fewer summands through a refined count , and provide a Macaulay2 implementation with a concrete case yielding a one-parameter family of 27-summand decompositions. These results have practical impact for polynomial computation and symmetric tensor problems by delivering explicit, low-sum representations and a usable computational toolkit.

Abstract

In this paper we settle some polynomial identity which provides a family of explicit Waring decompositions of any monomial over a field . This gives an upper bound for the Waring rank of a given monomial and naturally leads to an explicit Waring decomposition of any homogeneous form and, eventually, of any polynomial via (de)homogenization. Note that such decomposition is very useful in many applications dealing with polynomial computations, symmetric tensor problems and so on. We discuss some computational aspect of our result as comparing with other known methods and also present a computer implementation for potential use in the end.
Paper Structure (4 sections, 6 theorems, 58 equations, 1 table)

This paper contains 4 sections, 6 theorems, 58 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main Result
  3. Case of arbitrary homogeneous form
  4. Macaulay2 code for the decompositions

Key Result

Theorem 1.1

For a monomial $X_0^{a_0}X_1^{a_1}\ldots X_n^{a_n}$ with each $a_i>0$, it holds that Further, the same result is true for $\mathop{\mathrm{rank}}\nolimits_\mathbb{Q}(X_0^{a_0}X_1^{a_1}\ldots X_n^{a_n})$.

Theorems & Definitions (15)

  • Theorem 1.1: HM
  • Lemma 2.2
  • proof
  • Example 2.3
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • proof
  • Remark 2.6: Waring rank and decomposition of a monomial over any infinite field ${\Bbbk}$
  • Remark 2.7: Linear forms $\ell_{A,\mathbf{k},\mathbf{s}}$'s via Apolarity
  • ...and 5 more