Number Theory Seminar

Speaker: Gauree Wathodkar, University of Mississippi

"Partition regularity in commutative rings."

Let A ∈ Mm×n(Z) be a matrix with integer coefficients. The system of equations A⃗x = ⃗0 is said to be partition regular over Z if for every finite partition Z \ {0} = ∪ri =1Ci, there exists a solution ⃗x ∈ Zn, all of whose components belonging to the same Ci. For example, the equation x + y − z = 0 is partition regular. In 1933 Rado characterized completely all partition regular matrices. He also conjectured that for any partition Z \ {0} = ∪ri =1Ci, there exists a partition class Ci that contains solutions to all partition regular systems. This conjecture was settled in 1975 by Deuber. We study the analogue of Rado’s conjecture in commutative rings, and prove that the same conclusion holds true in any integral domain.

MC 5403