[Paper Review] The DMM bound: multivariate (aggregate) separation bounds

Ioannis Z. Emiris, Bernard Mourrain+1 more|arXiv (Cornell University)|May 31, 2010

Polynomial and algebraic computation22 references19 citations

TL;DR

This paper introduces the DMM bound, a novel multivariate aggregate separation bound for isolated roots of polynomial systems, leveraging mixed volume and sparse resultant height to improve upon Canny’s Gap theorem by a factor of $\mathcal{O}(d^{n-1})$. It provides the first complexity bound for Milne’s subdivision algorithm in $\mathbb{R}^2$ and offers tighter bounds for eigenvalue computation and positive polynomials over the simplex.

ABSTRACT

In this paper we derive aggregate separation bounds, named after Davenport-Mahler-Mignotte (\dmm), on the isolated roots of polynomial systems, specifically on the minimum distance between any two such roots. The bounds exploit the structure of the system and the height of the sparse (or toric) resultant by means of mixed volume, as well as recent advances on aggregate root bounds for univariate polynomials, and are applicable to arbitrary positive dimensional systems. We improve upon Canny's gap theorem \cite{c-crmp-87} by a factor of $\OO(d^{n-1})$, where $d$ bounds the degree of the polynomials, and $n$ is the number of variables. One application is to the bitsize of the eigenvalues and eigenvectors of an integer matrix, which also yields a new proof that the problem is polynomial. We also compare against recent lower bounds on the absolute value of the root coordinates by Brownawell and Yap \cite{by-issac-2009}, obtained under the hypothesis there is a 0-dimensional projection. Our bounds are in general comparable, but exploit sparseness; they are also tighter when bounding the value of a positive polynomial over the simplex. For this problem, we also improve upon the bounds in \cite{bsr-arxix-2009,jp-arxiv-2009}. Our analysis provides a precise asymptotic upper bound on the number of steps that subdivision-based algorithms perform in order to isolate all real roots of a polynomial system. This leads to the first complexity bound of Milne's algorithm \cite{Miln92} in 2D.

Motivation & Objective

• To derive aggregate separation bounds for isolated roots of multivariate polynomial systems, including positive-dimensional cases.
• To improve upon Canny’s Gap theorem by exploiting sparsity via mixed volume and resultant height.
• To provide tighter bounds for the minimum value of positive polynomials over the simplex compared to prior work.
• To establish the first complexity bound for Milne’s subdivision algorithm in $\mathbb{R}^2$ using real-root counting.
• To offer a new proof of polynomial bit complexity for integer matrix eigenvalue/eigenvector computation.

Proposed method

• The DMM bound is derived using the sparse resultant and its height, combined with mixed volume to exploit sparsity in polynomial systems.
• The method extends univariate DMM bounds to multivariate systems via resultants and polynomial remainder sequences.
• It employs iterated resultants to eliminate variables and compute a univariate polynomial in a volume parameter $u$, enabling root isolation.
• The algorithm uses Sturm sequences and sign variation counting over box vertices to isolate real roots in a domain.
• The complexity analysis relies on bit-size bounds for resultants and multivariate polynomial evaluation, with careful tracking of arithmetic and bit complexity.
• The approach is applied to eigenvalue problems and positive polynomial minimization, using bounds on critical points via derivatives.

Experimental results

Research questions

• RQ1 Can aggregate separation bounds for multivariate polynomial systems be improved by exploiting sparsity and mixed volume?
• RQ2 How does the new DMM bound compare to Canny’s Gap theorem and Brownawell–Yap bounds in terms of tightness and applicability?
• RQ3 Can the DMM bound be extended to positive-dimensional systems, where roots are not isolated?
• RQ4 What is the complexity of Milne’s subdivision algorithm in $\mathbb{R}^2$, and can it be bounded using the new separation bound?
• RQ5 Can the DMM bound lead to a new proof of polynomial bit complexity for integer matrix eigenvalue problems?

Key findings

• The DMM bound improves upon Canny’s Gap theorem by a factor of $\mathcal{O}(d^{n-1})$ for $n$-variable systems with degree $d$ polynomials.
• The bound is within $\mathcal{O}(2^n)$ of optimal for small $n$, and tighter than [5] when expressed using mixed volumes.
• For the minimum value of a positive polynomial over the simplex, the DMM bound improves upon [2], [5], and [16] when the total degree exceeds the number of variables.
• The first complexity bound for Milne’s algorithm in $\mathbb{R}^2$ is established, with runtime $\widetilde{\mathcal{O}}_B(d^{27}\tau + d^{26}\tau^2)$.
• The DMM bound yields a new proof that integer matrix eigenvalue/eigenvector computation is of polynomial bit complexity.
• In numerical comparisons, the DMM bound achieves significantly lower bitsize estimates than prior bounds—e.g., $324,506$ bits for $d=32$, $\tau=85$, compared to $1.6$ billion for [2].