[Author] A. L. Chistov [Title] A Bound for the Degree of a System of Equations Giving the Variety of Reducible Polynomials [AMS Subj-class] 14Q15 Higher-dimensional varieties 14M99 Algebraic geometry 12Y05 Computational aspects of field theory and polynomials 12E05 Polynomials (irreducibility, etc.) 13P05 Polynomials, factorization [Abstract] Consider the affine space ${\mathbb A}^N(\overline{K})$ of homogeneous polynomials of degree $d$ in $n+1$ variables with coefficients from an algebraic closure $\overline{K}$ of a field $K$ of arbitrary characteristic, so $N={n+d \choose n}$. We prove that the variety of all reducible polynomials from this affine space can be given by a system of polynomial equations of degree less than $56d^7$ in $N$ variables. Using this result we formulate an effective version of the first Bertini theorem for the case of a hypersurface. [Keywords] absolute irreducibility, lattices, the Bertini theorem [Comments] LaTeX, English, 18 pp. To appear in: "Алгебра и анализ" No.3, 2012, in Russian. In the second version of this preprint the upper bound $84d^7$ is improved. It is replaced by $56d^7$. [Contact e-mail] alch@pdmi.ras.ru