Separation bounds for polynomial systems. (November 2020)

• Record Type: Journal Article
• Authors: Emiris, Ioannis; Mourrain, Bernard; Tsigaridas, Elias
• Is Part Of: Journal of symbolic computation. Volume 101(2020)
• Page Start: 128
• Page End: 151
• Publication Date: 2020-11
• Subjects: Separation bound -- Sparse resultant -- DMM -- Arithmetic Nullstellensätze -- Height of the resultant -- Positive polynomial; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.07.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We rely on aggregate separation bounds for univariate polynomials to introduce novel worst-case separation bounds for the isolated roots of zero-dimensional, positive-dimensional, and overdetermined polynomial systems. We exploit the structure of the given system, as well as bounds on the height of the sparse (or toric) resultant, by means of mixed volume, thus establishing adaptive bounds. Our bounds improve upon Canny's Gap theorem (Canny, 1987 ). Moreover, they exploit sparseness and they apply without any assumptions on the input polynomial system. To evaluate the quality of the bounds, we present polynomial systems whose root separation is asymptotically not far from our bounds. We apply our bounds to three problems. First, we use them to estimate the bitsize of the eigenvalues and eigenvectors of an integer matrix; thus we provide a new proof that the problem has polynomial bit complexity. Second, we bound the value of a positive polynomial over the simplex: we improve by at least one order of magnitude upon all existing bounds. Finally, we asymptotically bound the number of steps of any purely subdivision-based algorithm that isolates all real roots of a polynomial system.