Pigeonhole proof of integer Siegel lemma ($0<\Vert z{\Vert }_{\infty }\le \lfloor (2NA{)}^{M/(N-M)}\rfloor$)

For $M<N$ homogeneous equations with integer coefficients of absolute value at most $A\ge 1$, map the box $\{0,\dots ,Q{\}}^N$ to its $M$ equation values. There are $(Q+1{)}^N$ inputs and at most $(2NAQ+1{)}^M$ outputs. Taking $Q=\lfloor (2NA{)}^{M/(N-M)}\rfloor$ makes the first count larger, since $2NAQ+1\le 2NA(Q+1)$. The difference of two inputs with the same output is the required nonzero bounded integer solution. Clearing denominators also treats rational equations; expanding a number-field basis replaces each field equation by finitely many rational equations.