Eakin-Sathaye type theorems for joint reductions and good filtrations of ideals

Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for ${\mathbb{N}}^s$-graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for $\mathbb N$-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in $2$-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds.