On the difference \(f^3(x)-g^2(x)\) (Q594929)

The authors consider a problem of \textit{H. Davenport} [Norske Vid. Selsk. Forhdl. 38, 86--87 (1965; Zbl 0136.25204)] for finding polynomials \(f(x)\) of degree \(2k\) and \(g(x)\) of degree \(3k\) such that \(f^3(x)-g^2(x)\) has (minimal) degree \(k+1\). Previous examples (all in \(\mathbb Z[x]\)) which were given by Davenport for \(k=2\) and \textit{B. J. Birch, S. Chowla, M. Hall jun.} and \textit{A. Schinzel} [Norske Vid. Selsk. Forhdl. 38, 65--69 (1965; Zbl 0144.03901)] are rediscovered by more general methods (and are given parameters). For \(k=7\) and \(11\), examples with algebraic (irrational) coefficients are given. One method is to ``extract'' \(f\) and \(g\) from the high parts of the square and cube of an unknown polynomial \(h(x)\). Another method is to minimize the degree of \(3fg'-2gf'\). The symbolic manipulation program \texttt{REDUCE-2} was used.