Sums and differences of three \(k\)th powers (Q1019859)

The author studies representation of integers by integral ternary forms and concludes as follows. Let \(N\in\mathbb N\), \(k\in\mathbb N\), and \(k\geq 3\); let \(F(\vec x)\) be a nonsingular form of degree \(k\) with integral rational coefficients, \(\vec x:= (x_1,x_2,x_3)\). If \(N\ll_F B^{3/13}\), then all integer points \(\vec x\) for which \[ |F(\vec x)|\leq N,\;B/2< \max x_i\leq B,\;1\leq i\leq 3 \] lie on a union of \(O_F(B^{9/10}N^{1/10})\) plane projective conics \(C_i(\vec x)= 0\), with \(C_i(\vec x)\in\mathbb Z[\vec x]\). For \(F(\vec x):= x^k_1+ \varepsilon_2 x^k_2+ \varepsilon_3 x^k_3\) with \(\varepsilon_i\in\{\pm 1\}\), \(i= 2,3\), let \[ {\mathcal N}(B, N):= \#\left\{\vec x\mid\vec x\in\mathbb Z^3, F(\vec x)= N, B/2< \max x_i\leq B,\;N\not\in\left\{x^k_1, \varepsilon_2 x^k_2, \varepsilon_3 x^k_3\right\}\right\}. \] Then \({\mathcal N}(B,N)= O_\varepsilon(B^{9/10+\varepsilon} N^{1/10})\) as soon as \(N\ll_k B^{3/13}\), \(\varepsilon> 0\), and \({\mathcal N}(B, N)= O_k(B^{10/k})\) for \(N\ll_k B\). Consideration of the singular form \(x^{k-1}y- z\) allows the author to establish an asymptotic formula for \((k-1)\)-free values of \(p^k+ h\), when \(p\) runs over primes and \(h\in\mathbb Z\setminus\{0\}\), answering a question raised by C. Hooley. According to the author, these ``results should be seen as examples of the ``determinant method'' developed'' in [\textit{D. R. Heath-Brown}, Ann. Math. (2), 155, No. 2, 553--598 (2002; Zbl 1039.11044)].