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Almost primes represented by binary forms. (English) Zbl 1269.11099
Schinzel’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.

11N36 Applications of sieve methods
11E76 Forms of degree higher than two
11N05 Distribution of primes
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11R42 Zeta functions and \(L\)-functions of number fields
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