Let $I \subseteq \mathbb{C}[x_1,\ldots,x_n]$ be an ideal generated by polynomials $f_1,\ldots,f_r$ of degree at most $d$. Is it possible to generate the radical $\sqrt{I}$ of this ideal with polynomials of degree at most $d$? If not, is there any other upper bound known in terms of $d$ for the degrees of a set of polynomials generating $\sqrt{I}$?