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Let AN (K̄) denote the affine space of homogeneous polynomials of degree d in n + 1 variables with coefficients from the algebraic closure K̄ of a field K of arbitrary characteristic; so N = (n+d n ) . It is proved that the variety of all reducible polynomials in this affine space can be given by a system of polynomial equations of degree less than 56d7 in N variables. This result makes it possible to formulate an efficient version of the first Bertini theorem for the case of a hypersurface.

A bound for the degree of a system of equations determining the variety of reducible polynomials