Sums and differences of two cubic polynomials
Let \(f(x)\) be a cubic polynomial with integer coefficients. The purpose of the present paper is to investigate the frequency of nontrivial solutions to the equation \[ f(x_1)+ f(x_2)= f(x_3)+ f(x_4), \tag \(*\) \] with integers \(x_i\) in the interval \(|x_i|\leq P\). It is shown that there are \(O(P^{5/3+ \varepsilon})\) such solutions, for any fixed \(\varepsilon> 0\). As a corollary it is shown that there are \(O(x^{5/9+ \varepsilon})\) integers \(n\) with \(|n|\leq x\), that have two or more essentially distinct representations as \(n= f(x_1)\pm f(x_2)\). These results are extensions of the author's work [Int. Math. Res. Not. 1995, 181-185 (1995; Zbl 0821.11049)] on the case \(f(x)= x^3\), in which an affine slicing method was used to reduce the problem to one involving binary quadratic equations. One can reduce considerations to polynomials of the form \(f(x)= ax^3+ bx\). Solutions of \((*)\) lying on the hyperplane \(x_1+ x_2- x_3- x_4= h\) then take a particularly simple form, on which the argument is able to capitalize. In contrast the method of the reviewer [Acta Arith. 79, 17-30 (1997; Zbl 0863.11021)], which is superior in the simple case \(f(x)= x^3\), appears unsuited to this inhomogeneous problem.
scientific article; zbMATH DE number 1423223