1. The problem is to determine whether a system is convenient or not, and whether it is trivial or non-trivial. 2. In mathematics, especially in algebra and system theory, a system is called **convenient** if it satisfies certain conditions that make it well-behaved or easy to analyze. For example, in polynomial systems, convenience often means the system includes all variables in a way that prevents degenerate solutions. 3. A system is **trivial** if it has only the zero solution or a solution that does not provide meaningful information. Conversely, a **non-trivial** system has solutions other than the zero solution. 4. To determine if a system is trivial or non-trivial, check the solutions: - If the only solution is the zero vector (e.g., $x=0$, $y=0$, etc.), the system is trivial. - If there exist solutions where variables are not all zero, the system is non-trivial. 5. To check convenience, ensure the system includes all variables in a way that the system's equations are independent and cover the variable space adequately. For example, in polynomial systems, a system is convenient if the Newton polyhedron intersects all coordinate axes. 6. In summary: - **Trivial vs Non-trivial:** Analyze the solution set. - **Convenient vs Non-convenient:** Analyze the structure and coverage of variables in the system. This approach helps in understanding the nature and solvability of the system.