nontrivial

The term "nontrivial" is used primarily in mathematics, computer science, and related fields to describe an object, problem, or solution that is significant, not obvious, or requires considerable effort to understand or solve. It often contrasts with "trivial," which refers to cases that are simple, straightforward, or can be easily resolved with minimal effort or complexity.

In mathematics, a nontrivial element or solution is one that is not degenerate or trivial in nature. For example, a nontrivial solution to an equation is one that provides meaningful insight beyond obvious or trivial solutions, such as those that are zero or one. Similarly, a nontrivial group or algebraic structure possesses properties that are not immediately evident and carry substantial mathematical importance.

In computer science, the term frequently appears in the context of algorithms and problems, indicating that a task is not easily reducible to a simple case. For instance, a nontrivial problem requires more than trivial computation or reasoning, often implying complexity or difficulty. An example is the NP-complete class of problems, where solutions are considered nontrivial because straightforward approaches are computationally infeasible for large instances.

The use of "nontrivial" emphasizes the importance or complexity of a particular aspect, highlighting that further effort, analysis, or innovation is necessary to address it adequately. While its precise interpretation can vary depending on context, the word generally conveys that the subject in question is substantial and warrants attention beyond the trivial or superficial level.