Given is a positive integer n. A polynomial has all coefficients being integers whose absolute value does not exceed n. What is the smallest possible positive root, if there is any?
If the root is rational, then by the rational root theorem, it cannot be smaller than 1/n. The polynomial nx−1=0 has x=1/n as the only solution. But if a root is irrational, can it be smaller than 1/n?
Answer
Let the polynomial be f(x)=akxk+ak+1xk+1+⋯+amxm, where ak≠0. Note that if $0
(Note that this bound does not really require the coefficients of f(x) to be integers. It just requires them to all have absolute value ≤n, and that the first nonzero coefficient has absolute value ≥1.)
Conversely, we can find such polynomials f(x) with positive roots arbitrarily close to 1n+1. Namely, consider fm(x)=1−m∑j=1nxj.
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