# Closest root of polynomial

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

• depends on $f$; e.g., if $f(x)=x^d$ then the shortest distance is zero. Feb 19, 2011 at 18:55
• Since the OP asks for distance between real roots we require that the polynomial has real roots (so forbidden say $f(x)=x^{2^d}+1$ etc. ). Assume then that the polynomial $f(x)$ has at least $2$ real roots : The bounds in the literature may be better or worst under the reality condition ? Feb 23, 2011 at 0:54

As you've seen from the examples that others have given, you need to know something about the degree and size of coefficients. Suppose that $f$ has distinct roots, degree $n$ and integer coefficients of absolute value at most $H$. So the roots have absolute value at most $nH$, so $2nH$ is an upper bound for the distance between two roots. To get a lower bound, use that the discriminant is at least one and the previous bounds, to get the lower bound $(2nH)^{-n(n-1)}$. I am sure this can be improved.
Consider the polynomial $n\cdot m\cdot (x-1/n)(x-1/m).$