Separation bounds for polynomial systems - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Journal of Symbolic Computation Year : 2020

Separation bounds for polynomial systems


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 overde- termined 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 [9]. 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.
Fichier principal
Vignette du fichier
emt-dmm-j.pdf (260.91 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-01105276 , version 1 (21-01-2015)
hal-01105276 , version 2 (07-02-2015)
hal-01105276 , version 3 (09-12-2015)
hal-01105276 , version 4 (08-02-2017)
hal-01105276 , version 5 (28-06-2019)



Ioannis Z. Emiris, Bernard Mourrain, Elias Tsigaridas. Separation bounds for polynomial systems. Journal of Symbolic Computation, 2020, 101, pp.128-151. ⟨10.1016/j.jsc.2019.07.001⟩. ⟨hal-01105276v5⟩
611 View
1544 Download



Gmail Facebook X LinkedIn More