On the asymmetric additive energy of polynomials (Q6582249)

For a polynomial \(f(x)\) with integer coefficients let \N\[\NE_f(B;k)=\#\{(x_1,x_2,x_3,x_4)\in {\mathbb Z}_{>0}^4\cap [1,B]^4 : f(x_1)-f(x_2)=f(x_3)-f(x_4)+k\}.\N\]\NWhen \(k=0\), it is a result of Hooley that \(E_f(B,0)=2B^2+O(B^{2-\delta})\) for some \(\delta>0\) for \(f(x)=x^d\) with \(d\ge 3\) with the situation for other polynomials having been investigated by other authors. The main term comes from the diagonal solutions \((a,b,a,b)\) and \((a,b,b,a)\). In the paper under review the author proves that if \(d\ge 3\), and \(k\ne 0\), then \N\[\NE_f(B;k)\ll_f B^{2-1/(50d)}. \N\]\NThe implied constant above depends on \(f\) but not on \(k\). The above theorem follows from a more general result (Theorem 1.2), which might be of independent interest, and for which the reader is referred to the paper. As applications, the author proves two corollaries, one which completes work of \textit{C. Chen} et al. [Math. Ann. 385, No. 1--2, 309--355 (2023; Zbl 1520.11075)] concerning square-root cancellation in Weil sums almost always, and the other addressing the large sieve inequality for polynomial sequences thus completing a recent work by \textit{R. C. Baker} et al. [Mathematika 68, No. 2, 362--399 (2022; Zbl 1548.11111)].\N\NThe proofs use sieve methods for small degrees (\(d=3,~4\)), the determinant method of \textit{T. D. Browning} and \textit{D. R. Heath-Brown} [Bull. Lond. Math. Soc. 38, No. 3, 401--410 (2006; Zbl 1174.11051)] when \(d\ge 5\) as well as results of \textit{E. Bombieri} and \textit{J. Pila} [Duke Math. J. 59, No. 2, 337--357 (1989; Zbl 0718.11048)] on the number of integer solutions \((x,y)\) of height at most \(B\) to a polynomial equation \(F(x,y)=0\), where \(F(x,y)\in {\mathbb Z}[x,y]\) is irreducible of degree \(d\).