I am teaching a course in commutative algebra, and it includes a project where the students research on a particular topic, solve a small problem and present it to the class.
I usually give my students some project proposals which they can choose from, in case they cannot come up with an interesting problem statement themselves. I try to be as varied as possible in the problems I propose, covering most areas of commutative algebra.
One such area I wanted to include is Gröbner bases and Elimination theory. I wanted to ask the Mathematics StackExchange community if they knew of any interesting polynomial systems of equations, perhaps arising from physics or chemistry, that are not completely straightforward but are also approachable for a graduate student of mathematics.