Polynomial invariants are mathematical expressions that remain unchanged under certain transformations or operations. They are often used in fields like algebraic geometry and topology to study the properties of shapes and spaces. By identifying these invariants, mathematicians can classify and understand complex structures more easily. In the context of algebraic topology, polynomial invariants can help distinguish between different types of geometric objects. For example, the Hodge polynomial is an important invariant that provides information about the topology of a space. These invariants play a crucial role in simplifying problems and revealing underlying patterns in mathematics.