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 $\sqrt{I}$?
In general, one can not bound the degree of the radical $\sqrt{I}$ using only $d$, even in the polynomial ring $R=\mathbb C[x,y,z,t]$. Consider the (complete intersection) ideal $I= (x^mty^mz, z^{n+2}xt^{n+1})$, generated in degrees $m+1$ and $n+2$. Then the radical of $I$ has a generator of degree $mn+2$, as shown in Lemma 2.4 of this paper.

$\begingroup$ So the the CastelnuovoMumford regularity of an ideal is equal to the tight degree bound of the generators? Meaning any polynomial set of degree less than $reg(I)$ can not generate $I$? $\endgroup$ – Pew Mar 30 '19 at 2:53

1$\begingroup$ The regularity is in general an upper bound on the max degree of generators. However in that example they actually compute the degrees of the generators, not just regularity. $\endgroup$ – Hailong Dao Mar 30 '19 at 3:39

$\begingroup$ I see, they compute the generators in example 2.1. By the way, are you aware of any similar results in the real radical case? $\endgroup$ – Pew Mar 31 '19 at 4:22