Question about toric ideal

Question

In the proposition 1.2 contained in the following http://www.math.harvard.edu/~jbland/ma232b2_notes.pdf, I can't understand why a monomial satisfying (1.7) exists. Can you help me? Thanks.

Answer 1

Since $x^\alpha$ was chosen to be minimal w.r.t. to the ordering, and $f(t^{m_1},\cdots,t^{m_s})=0$, a cancellation must occur. In other words, there must exist another monomial $x^\beta$ such that cancels $\prod (t^{m_i})^{a_i}$ in the expression for $f(t^{m_1},\cdots,t^{m_s})$.