the general linear or polynomial programs, is impossible [@tiwari:terminate; @Muller-Olm; @bradley:polynomial; @braverman:terminate]. So, a practical way for termination analysis is conducted by providing sufficient conditions for termination and/or nontermination. Classical method for establishing termination of a program, either linear or polynomial, makes use of a well-founded domain together with a so-called *ranking function* that maps the state space of the program to the domain, which provides a sufficient condition for the termination of the program, e.g., [@Ben-AmramG14; @colon:synthesis; @podelski:ranking; @podelski:transition; @chen:ranking; @cousot:lagrangian]. In [@gupta:NT], the authors considered a sufficient condition for non-termination inputs, while in [@Harris10], the authors investigated sufficient conditions for termination and nontermination inputs respectively, and check the two conditions in parallel for termination analysis.

In contrast, Tiwari investigated this issue at a very fundamental level. He first noticed that the termination of a class of simple linear loops is related to the eigenvalues of assignment matrix and proved that the termination problem of these linear programs with input set $\mathbb{R}$ is decidable [@tiwari:terminate]. This theory was further developed in [@braverman:terminate; @XZ2010; @XYZZ2011].

Following this line, Bradley et al. [@bradley:polynomial] tried to investigate the termination problem of a family of polynomial programs, which are modeled as *multi-path polynomial 