A note on classical and p-adic Fr\'{e}chet functional equations with restrictions

Given X,Y two Q-vector spaces, and f:X -> Y, we study under which conditions on the sets $B_k\subseteq X$, k=1,…,s, if ${\mathrm{\Delta }}_{h_1{h}_2\mathrm{.}\mathrm{.}\mathrm{.}{h}_s}f(x)=0$ for all x in X and h_k in B_k, k=1,2,…,s, then ${\mathrm{\Delta }}_{h_1{h}_2\mathrm{.}\mathrm{.}\mathrm{.}{h}_s}f(x)=0$ for all (x,h_1,…,h_s) in X^{s+1}.