Suppose $f$ is uni-variate degree d polynomial have integer coefficient.


What will be shortest distance between any two real root of polynomial.


Can we compute this exact if not then upper and lower bound for the same.


Correction :Here I am assuming all roots are distinct



Closest root of polynomial
There is considerable literature on this question, much of which can be revealed by doing a google search on "root separation". In particular, Wolfram MathWorld's "root separation" article is brief but to the point, and also see:


Polynomial Minimum Root Separation
George E. Collins
Department of Computer and Information Sciences, University of Delaware, Newark, DE 19716, U.S.A.
Received 21 June 2001; accepted 7 July 2001. Available online 28 February 2002. 
Abstract
There is a well-known lower bound, due to Mignotte, for the minimum root separation of a squarefree integral polynomial, but no evidence for the sharpness of this bound. This paper provides massive computational evidence for a conjectured much larger bound, one that is approximately the square root of Mignotte’s bound.
References


(this is journal of symbolic computation).


