Absolute Root Separation

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute value of the coefficients) of the polynomial. We improve the known bounds for this problem and related ones. Then we report on extensive experiments in low degrees, suggesting that the current bounds are still very pessimistic.

[1]  Chee-Keng Yap,et al.  Fundamental problems of algorithmic algebra , 1999 .

[2]  Stephen Melczer,et al.  Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables , 2016, ISSAC.

[3]  B. Salvy,et al.  Absolute Real Root Separation , 2016, Am. Math. Mon..

[4]  Min Sha,et al.  Counting and Testing Dominant Polynomials , 2014, Exp. Math..

[5]  Maurice Mignotte,et al.  Mathematics for computer algebra , 1991 .

[6]  Stephen Melczer,et al.  Effective Coefficient Asymptotics of Multivariate Rational Functions via Semi-Numerical Algorithms for Polynomial Systems , 2019, J. Symb. Comput..

[7]  Yann Bugeaud,et al.  Root separation for reducible integer polynomials , 2013, 1306.2128.

[8]  K. Mahler.  An inequality for the discriminant of a polynomial. , 1964 .

[9]  Bruno Salvy,et al.  Effective asymptotics of linear recurrences with rational coefficients , 1996, Discret. Math..

[10]  Arnold Schönhage,et al.  Polynomial root separation examples , 2006, J. Symb. Comput..

[11]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[12]  Jean-Charles Faugère,et al.  FGb: A Library for Computing Gröbner Bases , 2010, ICMS.

[13]  B. Brindza.  Distances between the conjugates of an algebraic number , 2005 .

[14]  Pascal Koiran,et al.  Root Separation for Trinomials , 2017, J. Symb. Comput..

[15]  Min Sha,et al.  Effective results on the Skolem Problem for linear recurrence sequences , 2015, Journal of Number Theory.

[16]  A. Dubickas.  On the Distance Between Two Algebraic Numbers , 2020, Bulletin of the Malaysian Mathematical Sciences Society.