I found myself unable to answer the following question:
Let $I,J$ be two ideals in $K[x_1, \dots, x_n]$ such that $I$ has a set of generators of degree $\leq d_I$ and $J$ has a set of generators of degree $\leq d_J$. Is it true that $I \cap J$ has a set of generators of degree $\leq d_I + d_J$?
I think this feels correct and I would wish to show that $I\cap J = (f \in I \cap J| \deg f \leq d_I + d_J)$. One approach could be to choose generators for $I$ and $J$ and try to obtain a generating set for $I \cap J$. Hence suppose $I = (f_1, \dots, f_r)$ and $J = (g_1, \dots, g_s)$ (note that we are in a noetherian setting). At first I thought that perhaps $I \cap J = (\operatorname{lcm}(f_i, g_j))_{i,j},$ but this fails: A counterexample is $(x,y) \cap (x+y) \neq (x(x+y), y(x+y)$. Perhaps a better candidate for a true statement is something like $$I \cap J = (f_i \mid f_i \in J) + (g_j \mid g_j \in I) + (\operatorname{lcm}(f_i, g_j))_{i,j},$$ but working with this feels janky.