Chevalley Theorem

Chevalley's Theorem states that if you have a polynomial equation describing a set of solutions (an algebraic variety) within a larger space, and then you look at this set within a smaller part (like a subset or subspace), the collection of its solutions in this smaller part will still be describable using polynomial equations. In other words, the structure of solutions is preserved when projecting or restricting to smaller spaces, ensuring that algebraic sets behave nicely under such mappings, which is fundamental for understanding how complex geometric shapes behave under transformations.