On Integer Chebyshev Polynomials

We are concerned with the problem of minimizing the supremum norm on $\lbrack 0,1\rbrack$ of a nonzero polynomial of degree at most $n$ with integer coefficients. We use the structure of such polynomials to derive an efficient algorithm for computing them. We give a table of these polynomials for degree up to $75$ and use a value from this table to answer an open problem due to P. Borwein and T. Erdélyi and improve a lower bound due to Flammang et al.