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Parametric "Non-nested" Discriminants for Multiplicities of Univariate Polynomials

Hoon Hong, Jing Yang

TL;DR

The paper tackles the problem of classifying complex root multiplicities of univariate polynomials by deriving coefficient-conditions that distinguish all possible multiplicity vectors. It introduces a new family of non-nested determinant discriminants $D(\boldsymbol{\gamma})$ built from derivatives and divided differences, and proves a main result that identifies the multiplicity vector $\operatorname{mult}(F)$ via a lexicographic ordering of conjugates and vanishing/non-vanishing of $D(\overline{\boldsymbol{\mu}})$. The approach connects divided differences, generic-root determinants, and higher-order derivatives to embed multiplicity information directly into determinant polynomials, yielding linear-in-$n$ maximum degrees and simpler non-nested forms. Compared to prior methods (YHZ, HY21, HY22), this framework achieves non-nested determinants with smaller maximum degrees and a clear, scalable criterion for all multiplicity structures, enhancing computational tractability for root-multiplicity classification. This work thus provides a principled, generalizable extension of subdiscriminant concepts to higher-order multiplicities with practical implications for symbolic computation and algebraic analysis of polynomials.

Abstract

We consider the problem of complex root classification, i.e., finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients. Those polynomials in the coefficients are called discriminants for multiplicities. It is well known that discriminants can be obtained by using repeated parametric gcd's. The resulting discriminants are usually nested determinants, that is, determinants of matrices whose entries are determinants, and so son. In this paper, we give a new type of discriminants which are not based on repeated gcd's. The new discriminants are simpler in that they are non-nested determinants and have smaller maximum degrees.

Parametric "Non-nested" Discriminants for Multiplicities of Univariate Polynomials

TL;DR

The paper tackles the problem of classifying complex root multiplicities of univariate polynomials by deriving coefficient-conditions that distinguish all possible multiplicity vectors. It introduces a new family of non-nested determinant discriminants built from derivatives and divided differences, and proves a main result that identifies the multiplicity vector via a lexicographic ordering of conjugates and vanishing/non-vanishing of . The approach connects divided differences, generic-root determinants, and higher-order derivatives to embed multiplicity information directly into determinant polynomials, yielding linear-in- maximum degrees and simpler non-nested forms. Compared to prior methods (YHZ, HY21, HY22), this framework achieves non-nested determinants with smaller maximum degrees and a clear, scalable criterion for all multiplicity structures, enhancing computational tractability for root-multiplicity classification. This work thus provides a principled, generalizable extension of subdiscriminant concepts to higher-order multiplicities with practical implications for symbolic computation and algebraic analysis of polynomials.

Abstract

We consider the problem of complex root classification, i.e., finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients. Those polynomials in the coefficients are called discriminants for multiplicities. It is well known that discriminants can be obtained by using repeated parametric gcd's. The resulting discriminants are usually nested determinants, that is, determinants of matrices whose entries are determinants, and so son. In this paper, we give a new type of discriminants which are not based on repeated gcd's. The new discriminants are simpler in that they are non-nested determinants and have smaller maximum degrees.
Paper Structure (10 sections, 7 theorems, 17 equations)

This paper contains 10 sections, 7 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Problem
  3. Main Result
  4. Proof of the Main Theorem
  5. Multiplicity discriminant in terms of roots
  6. Connection between the multiplicity discriminants and multiplicity vectors
  7. Proof of Theorem \ref{['thm:main_result']}
  8. Comparison
  9. Form of discriminants
  10. Maximum degree of discriminants

Key Result

Theorem 3.5

Let $F=\sum_{i=0}^{n}a_{i}x^{i}\ \ $where$\ \ a_{n}\neq0$. Let $\mathcal{M}(n)=\left\{ \boldsymbol{\mu}_{0},\boldsymbol{\mu}_{1},\ldots,\boldsymbol{\mu}_{p}\right\}$ where the entries are ordered in the lexicographically decreasing order in their conjugates $\overline{\boldsymbol{\mu}_{i}}$'s. Then Equivalently,

Theorems & Definitions (19)

  • Definition 2.1: Multiplicity vector
  • Example 2.2
  • Definition 2.3: Potential multiplicity vectors
  • Example 2.4
  • Definition 3.1: Determinant polynomial
  • Definition 3.2: Multiplicity Discriminant
  • Example 3.3
  • Definition 3.4: Conjugate
  • Theorem 3.5: Main Result
  • Example 3.6
  • ...and 9 more