Perturbing polynomials with all their roots on the unit circle

Given a monic real polynomial with all its roots on the unit circle, we ask to what extent one can perturb its middle coefficient and still have a polynomial with all its roots on the unit circle. We show that the set of possible perturbations forms a closed interval of length at most $4$, with $4$ achieved only for polynomials of the form $x^{2n}+cx^n+1$ with $c$ in $[-2,2]$. The problem can also be formulated in terms of perturbing the constant coefficient of a polynomial having all its roots in $[-1,1]$. If we restrict to integer coefficients, then the polynomials in question are products of cyclotomics. We show that in this case there are no perturbations of length $3$ that do not arise from a perturbation of length $4$. We also investigate the connection between slightly perturbed products of cyclotomic polynomials and polynomials with small Mahler measure. We describe an algorithm for searching for polynomials with small Mahler measure by perturbing the middle coefficients of products of cyclotomic polynomials. We show that the complexity of this algorithm is $O(C^{\sqrt{d}})$, where $d$ is the degree, and we report on the polynomials found by this algorithm through degree 64.