A noncommutative real nullstellensatz corresponds to a noncommutative real ideal: Algorithms. Issue 5 (3rd December 2012) - Cimprič, Jakob; Helton, J. William; McCullough, Scott; Nelson, Christopher
Abstract : This article extends the classical Real Nullstellensatz of Dubois and Risler to left ideals in free *‐algebras ℝ 〈 x, x * 〉 with x =( x 1, …, x n ). First, we introduce notions of the (noncommutative) zero set of a left ideal and of a real left ideal. We prove that every element from ℝ 〈 x, x * 〉 whose zero set contains the intersection of zero sets of elements from a finite subset S of ℝ 〈 x, x * 〉 belongs to the smallest real left ideal containing S . Next, we give an implementable algorithm, which for every finite S ⊂ℝ 〈 x, x * 〉, computes the smallest real left ideal containing S, and prove that the algorithm succeeds in a finite number of steps. Our definitions and some of our results also work for other *‐algebras. As an example, we treat real left ideals in M n (ℝ[ x 1 ]).