A surjection problem leading to the Ax-Grothendieck theorem. Issue 4 (12th March 2019)

• Record Type: Journal Article
• Authors: Buşe, Constantin; O'Regan, Donal; Saierli, Olivia
• Is Part Of: Applicable analysis. Volume 98:Issue 4(2019)
• Page Start: 843
• Page End: 850
• Publication Date: 2019-03-12
• Subjects: Natural powers and exponentials of matrices -- functional calculus with matrices -- global problems concerning polynomials of matrices; 30C15 -- 33C50 -- 15A60 -- 65F15; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2017.1403590
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: We give a complete functional calculus for matrices of order two. The key tools are two simple Lemmas (Lemma 2.1 and Lemma 2.2, below), which are elementary but are somewhat new. As an application, we find the set of all polynomials P for which the matrix mapX ↦ P ( X ) onM ( 2, C ) is surjective. TheC 4 -version of the famous Ax-Grothendieck theorem is obtained as an immediate Corollary.