The level sets of a cubic polynomial in four or more variables tends to have many integer solutions, while ones in two variables a limited number of solutions. Very little is known in case of three variables. For cubics which are character varieties (thus carrying a nonlinear group of morphisms) a Diophantine analysis has been developed and we will describe it. Passing from solutions in integers to integers in say a real quadratic field there is a fundamental change which is closely connected to challenging questions about one-commutators in SL2 over such rings.
This video is part of the Number Theory Web Seminar series.
