programs* (MPPs) by using *finite difference tree* (FDT).The MPP model is an expressive class of loops with multiple paths, polynomial loop guards and assignments, that enables practical code abstraction and analysis. It was proved in [@bradley:polynomial] that the termination problem of MPPs is generally undecidable. In [@bradley:polynomial], the authors only considered a small class of MPPs, i.e., MPPs with polynomial behaviour. Similar idea was used for termination analysis of polynomial programs in [@cook:polynomial]. In [@LXZZ14], the authors considered another class of MPPs, whose loop guards are polynomial equations. According to their algebraic structures, the authors established sufficient conditions for termination and nontermination simultaneously for these MPPs, thus termination analysis can be conducted by checking these conditions in parallel, which is analogous to [@Harris10]. In [@LXZZ14], the authors raised an open problem whether the termination of this family of MPPs is decidable.

In this paper, we give a confirmative answer to the open problem raised in [@LXZZ14] that the termination problem of MPPs with equality guards is decidable. To the best of our knowledge, this family of polynomial programs should be the largest one with a decidable termination problem so far, noting that the program termination with inequality conditions is hardly 