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

Abstract:

Let ${\mathbb A}^N(\bar {K})$ denote the affine space of homogeneous polynomials of degree $d$ in $n+1$ variables with coefficients from the algebraic closure $\bar {K}$ of a field $K$ of arbitrary characteristic; so $N=\binom {n+d}{n}$. 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.