The DMM bound: multivariate (aggregate) separation bounds

In this paper we derive aggregate separation bounds, named after Davenport-MahlerMignotte (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 [7] by a factor of O(dn-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 strongly polynomial. We also compare against recent lower bounds on the absolute value of the root coordinates by Brownawell and Yap [5], obtained under the hypothesis there is a 0-dimensional projection. Our bounds are in general comp...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Ioannis Z. Emiris, Bernard Mourrain, Elias P. Tsigaridas
• Keywords: Aggregate Root Bounds, Algorithms, CORR 2010, Education, Polynomial