[Paper Review] Bounds on the coefficients of the characteristic and minimal polynomials

Jean‐Guillaume Dumas|arXiv (Cornell University)|Oct 24, 2006

Matrix Theory and Algorithms5 references4 citations

TL;DR

This paper presents tighter bounds on the coefficients of the characteristic and minimal polynomials of integer matrices, using both absolute estimates and adaptive algorithms that refine bounds based on matrix-specific properties. The key contribution is a significant practical speed-up in Chinese remaindering for polynomial reconstruction by reducing the number of required primes, with improvements up to 5% in computational cost for large sparse matrices.

ABSTRACT

This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise input-dependant bounds on these coefficients. Such bounds are e.g. useful to perform deterministic chinese remaindering of the characteristic or minimal polynomial of an integer matrix.

Motivation & Objective

• To provide tighter, input-dependent bounds on the coefficients of the characteristic and minimal polynomials of integer matrices.
• To reduce the number of primes required for deterministic Chinese remaindering of these polynomials in integer matrix computations.
• To develop low-complexity algorithms that compute sharper coefficient bounds during runtime, improving efficiency without significant overhead.
• To improve upon existing bounds—especially Hadamard’s and Mignotte’s—by incorporating matrix-specific properties like spectral radius and minimal polynomial degree.
• To enable more efficient computation of Frobenius normal forms and Smith normal forms via better coefficient estimation.

Proposed method

• Uses Hadamard’s bound on determinants to derive absolute upper bounds on characteristic polynomial coefficients, refined via Stirling’s approximation and Stănică’s bounds on binomial coefficients.
• Introduces a dynamic search procedure that locates the likely position of the largest coefficient by analyzing the behavior of $ F(n,j) = \binom{n}{j} \sqrt{(n-j)B^2}^{n-j} $, reducing the search space to $ \mathcal{O}(\sqrt{n}) $ terms.
• Proposes a recursive scheme to compute bounds on $ \log_2(F(n,j)) $ efficiently, with $ \mathcal{O}(n) $ cost, enabling real-time estimation during polynomial computation.
• Applies eigenvalue-based bounds using Gershgorin disks and Cassini ovals to estimate spectral radius $ \beta $, which informs tighter bounds on minimal polynomial coefficients.
• Uses Mignotte’s bound as a baseline and improves it by distinguishing cases based on whether $ d \leq \beta $ or $ d > \beta $, yielding tighter estimates via binomial coefficient bounds.
• Employs Stănică’s extension of Stirling’s formula to bound binomial coefficients with error terms, improving precision in coefficient size estimation.

Experimental results

Research questions

• RQ1 How can absolute bounds on the coefficients of the characteristic polynomial be improved beyond Hadamard’s bound?
• RQ2 What is the optimal range of indices to search for the largest coefficient in the characteristic polynomial, given matrix size and entry bounds?
• RQ3 Can tighter bounds on minimal polynomial coefficients be derived using spectral radius and degree information rather than general determinant bounds?
• RQ4 To what extent can adaptive, low-cost algorithms reduce the number of primes needed in Chinese remaindering for matrix polynomials?
• RQ5 How do the new bounds compare in practice to existing ones in terms of reducing computational cost for large sparse matrices?

Key findings

• The paper improves the constant in the logarithmic coefficient bound from approximately 3.4427 to 0.21163175, significantly tightening the absolute bound on characteristic polynomial coefficients.
• For the example matrix, the new bound estimates the largest coefficient as 80.66661, compared to Hadamard’s 55.9 and the prior bound of 21793, showing a dramatic improvement in precision.
• The adaptive algorithm reduces the search space for the largest coefficient to at most $ \frac{-1 + \sqrt{1 + 2\delta B^2 n}}{\delta B^2} $ terms, with $ \delta \approx 5.418236 $, enabling efficient on-the-fly estimation.
• For the minimal polynomial, the improved bound reduces the coefficient size estimate by up to 5% compared to prior work, saving 23 modular projections and one hour of computation in a large-scale homology computation.
• The bound $ \|\text{minpoly}_A\|_\infty \leq \min\left\{ \sqrt{\beta d}^d, \sqrt{\frac{2}{d\pi}} 2^d \beta^d \right\} $ when $ d > \beta $ provides a $ \log d $-factor improvement over previous results.
• The proposed algorithms for computing bounds have $ \mathcal{O}(n) $ complexity, making them practical for real-time use during polynomial computation.