A surjection problem leading to the Ax-Grothendieck theorem

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 map $X\mapsto P(X)$ on $\mathcal{M}(2,\mathbb{C})$ is surjective. The ${\mathbb{C}}^4$-version of the famous Ax-Grothendieck theorem is obtained as an immediate Corollary.

KEYWORDS:

• Natural powers and exponentials of matrices
• functional calculus with matrices
• global problems concerning polynomials of matrices

AMS SUBJECT CLASSIFICATIONS:

• 30C15
• 33C50
• 15A60
• 65F15

Acknowledgements

A part of this paper was completed in the period August-October 2017, when C. Buşe has visited the Western Kentucky University, Bowling Green USA as Visiting Scholar. The support of the university is gratefully acknowledged.

Notes

No potential conflict of interest was reported by the authors.