On the proportion of squarefree numbers among sums of cubic polynomials
\textit{C. Hooley} [Mathematika 14, 21--26 (1967; Zbl 0166.05203)] proved that if \(f(x)\in Z[x]\) is an irreducible polynomial degree \(d\ge 3\) then \(f(n)\) takes \(c_fx+o(x)\) \((d-1)\)-free values of \(n\le x\). \textit{G. Greaves} [Q. J. Math., Oxf. II. Ser. 43, No. 169, 45--65 (1992; Zbl 0768.11034)] obtained an analogue result for binary forms. Assuming the \(abc\)-conjecture, \textit{B. Poonen} [Duke Math. J. 118, No. 2, 353--373 (2003; Zbl 1047.11021)] proved a similar result for squarefree values of multivariable polynomials. In the paper under review, the author gives an asymptotic formula for the proportion of squarefree numbers among \(f_1(x_1)+ f_2(x_2)+\cdots+ f_s(x_s)\) where \(s\ge 3\), each \(f_i\) is a nonzero cubic polynomial with integer coefficients and \(x_i\in Z\), \(a_iB\le x_i\le b_iB\) \((i=1,\dots,s)\), \(B\to\infty\).
scientific article; zbMATH DE number 7344959