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

摘要

Let A~N(K) denote the affine space of homogeneous polynomials of degree d in n + 1 variables with coefficients from the algebraic closure K 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 56d~7 in N variables. This result makes it possible to formulate an efficient version of the first Bertini theorem for the case of a hypersurface.