Sparse squares of polynomials

Abstract:

We answer a question left open in an article of Coppersmith and Davenport which proved the existence of polynomials whose powers are sparse, and in particular polynomials whose squares are sparse (i.e., the square has fewer terms than the original polynomial). They exhibit some polynomials of degree $12$ having sparse squares, and ask whether there are any lower degree complete polynomials with this property. We answer their question negatively by reporting that no polynomial of degree less than $12$ has a sparse square, and explain how the substantial computation was effected using the system CoCoA.