Classes of functions in the set of circulant matrices and their characterization

What is it about?

I would like to explain in simple terms what this paper is about. Professor B. Q. Li published two papers in the American Mathematical Monthly, in which he established the necessary and sufficient conditions for entire function to be polynomial function and for meromorphic function to be rational function. The conditions are very simple, and their further extension might seem natural. However, such extension for the case of the functions of many complex variables is impossible. Such techniques as integration of functions of many complex variables are unknown or may be extremely hard. My idea was to obtain the required extension for a suitable algebraic structure that contains many complex variables. Such structure is the set of d times d circulant matrices with complex entries. Circulant matrices commute and form commutative ring. The operations over circulant matrices are the same as with complex vectors. In my recent paper published in the American Mathematical Monthly, I have just studied the matrix polynomial with circulant matrices and proved a version of the fundamental theorem of algebra. So, using my previous ideas I could solve the required problem and extend the results of B. Q. Li to the many dimensional case.

Featured Image

Why is it important?

This is the first paper that provides a characterization of important classes of functions such as entire function, polynomial function, meromorphic function and rational function in many dimensional case.

Perspectives