Set-theoretic complete intersections on binomials

Let $V$ be an affine toric variety of codimension $r$over a field of any characteristic. We completely characterize the affine toric varieties that are set-theoretic complete intersections on binomials. In particular we prove that in the characteristic zero case, $V$is a set-theoretic complete intersection on binomials if and only if $V$ is a complete intersection. Moreover, if $F_1,\dots ,F_r$ are binomials such that $I(V) = rad(F_1,\dots, F_r)$, then $I(V) = (F_1,\dots,F_r)$. While in the positive characteristic $p$ case, $V$ is a set-theoretic complete intersection on binomials if and only if $V$ is completely $p$-glued.

These results improve and complete all known results on these topics.