Almost primes represented by binary forms.

*(English)*Zbl 1269.11099Schinzel’s and Sierpiński’s hypothesis H [A. Schinzel and W. Sierpínski, Acta Arith. 4, 185–208 (1958; Zbl 0082.25802)] concerning distinct irreducible polynomials \(f_1,...,f_g\in \mathbb{Z}[x]\) conjectures that if there are no local obstructions, then there are infinitely many integers \(x\) such that \(f_1(x),...,f_g(x)\) are simultaneously prime. Until today, only one instance of Schinzel’s hypothesis has been established, namely that of \(g=1\) and \(f_1\) being a linear polynomial. A way to approach hypothesis H is to consider the problem of counting integers \(x\) such that \(f_1\cdots f_g(x)\) has at most \(r\) prime factors, i.e. \(f_1\cdots f_g(x)=P_r\). Using a higher-dimensional sieve, H. Diamond and H. Halberstam [Lond. Math. Soc. Lect. Note Ser. 237, 101–107 (1997; Zbl 0941.11034)] attacked this problem successfully. It is natural to extend these considerations to irreducible forms in several variables, i.e. to count the number of solutions \({\mathbf x}\in \mathbb{Z}^k\) of \(F_1\cdots F_g({\mathbf x})=P_r\), where \(F_1,...,F_g\) are irreducible forms with integer coefficients in \(k\) variables. Indeed, the author proves the infinitude of such solutions for binary forms \(F_1,...,F_g\) satisfying certain natural conditions if \(r\leq [h+(g+1)\log g]\), \(h\) being the degree of \(F_1\cdots F_g\). To this end, he employs the above-mentioned sieve of Diamond and Halberstam. As applications of his work, he also considers saturation numbers for \(n\)-variable polynomials \(F({\mathbf x})\) over the integers with \({\mathbf x}\) in orbits of a subgroup of \(\mathrm{GL}_n(\mathbb{Z})\), and the detection of almost primes among the roots of isotropic ternary quadratic forms.

Reviewer: Stephan Baier (Norwich)