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The multivariate Hermite method for counting real and complex solutions to polynomial systems

Volodymyr Oleksiyuk

TL;DR

The paper introduces a multivariate Hermite criterion to count distinct real and complex roots of zero-dimensional polynomial systems by working in the quotient algebra $A=\mathbb{R}[X_1,\dots,X_n]/I$ with $I=(f_1,\dots,f_k)$. It defines a bilinear form $H(f,g)=\operatorname{tr}(\varphi_{fg})$ via multiplication operators $\varphi_f$, showing $\operatorname{Rank}(H)$ equals the number of distinct complex solutions and $\operatorname{Tr}(H)$ the number of distinct real solutions, extending the classical univariate Hermite theory. The authors provide a rigorous reduction from the general case to a special setting, leveraging Hilbert's Nullstellensatz and the nilradical to justify finite-dimensional Gram matrices, and they give explicit proofs (including an alternative proof) and a practical Macaulay2 implementation. Their algorithm hinges on constructing a monomial basis of the quotient, forming the corresponding Hermite matrix from traces of products, and extracting counts from its rank and trace. This work offers a robust, exact-root-counting tool for polynomial systems with potential impact in computational algebraic geometry and real algebraic geometry, complementing numerical solvers.

Abstract

This note presents the multivariate Hermite criterion: a practical and powerful algorithm for determining the number of distinct real and complex roots of a zero-dimensional system of polynomials in any finite number of variables. The final section includes an implementation in Macaulay2, a free and open-source computer algebra system.

The multivariate Hermite method for counting real and complex solutions to polynomial systems

TL;DR

The paper introduces a multivariate Hermite criterion to count distinct real and complex roots of zero-dimensional polynomial systems by working in the quotient algebra with . It defines a bilinear form via multiplication operators , showing equals the number of distinct complex solutions and the number of distinct real solutions, extending the classical univariate Hermite theory. The authors provide a rigorous reduction from the general case to a special setting, leveraging Hilbert's Nullstellensatz and the nilradical to justify finite-dimensional Gram matrices, and they give explicit proofs (including an alternative proof) and a practical Macaulay2 implementation. Their algorithm hinges on constructing a monomial basis of the quotient, forming the corresponding Hermite matrix from traces of products, and extracting counts from its rank and trace. This work offers a robust, exact-root-counting tool for polynomial systems with potential impact in computational algebraic geometry and real algebraic geometry, complementing numerical solvers.

Abstract

This note presents the multivariate Hermite criterion: a practical and powerful algorithm for determining the number of distinct real and complex roots of a zero-dimensional system of polynomials in any finite number of variables. The final section includes an implementation in Macaulay2, a free and open-source computer algebra system.

Paper Structure

This paper contains 9 sections, 4 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. The Hermite Criterion and Its Generalization
  3. Classic Hermite Method
  4. Multivariate Hermite criterion
  5. Proof under an assumption
  6. Alternative proof under the assumption
  7. Reduction of the general case to the assumption
  8. Implementation
  9. Speed

Key Result

Theorem 1

Let $f = t^n + a_{n-1} t^{n-1} + \ldots + a_1 t + a_0 \in \mathbb{R}[t]$. Let $\alpha_1, \ldots, \alpha_n \in \mathbb{C}$ be the (not necessarily distinct) roots of $f$. Define the Newton sums as $p_k = \alpha_1^k + \alpha_2^k + \ldots + \alpha_n^k$. Then, for the Hermite matrix $H(f) = (p_{i+j-1})_

Theorems & Definitions (5)

  • Theorem : Classic Hermite Criterion
  • Theorem : Multivariate Hermite criterion, plaumann2023seminar
  • Theorem : Hilbert's Nullstellensatz
  • Lemma 1
  • proof