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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.

MSC:
 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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