Result: Zero-free neighborhoods around the unit circle for Kac polynomials

In this paper, we study how the roots of the Kac polynomials$$W_n(z) = \sum _{k=0}^{n-1} \xi _k z^k$$Wn(z)=∑k=0n-1ξkzkconcentrate around the unit circle when the coefficients of$$W_n$$Wnare independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as$$n\rightarrow \infty $$n→∞if and only if$${\mathbb {E}}[\log ( 1+ |\xi _0|)]E[log(1+|ξ0|)]<∞. Under the condition$${\mathbb {E}}[\xi ^2_0]E[ξ02]<∞, we show that there exists an annulus of width$${\text {O}}(n^{-2}(\log n)^{-3})$$O(n-2(logn)-3)around the unit circle which isfreeof roots with probability$$1-{\text {O}}({(\log n)^{-{1}/{2}}})$$1-O((logn)-1/2). The proof relies on small ball probability inequalities and the least common denominator used in [17].