Integrators of matrices

The notion of matrix integration was introduced in 2007 by Bhat and Mukherjee [1], as a natural counterpart to the classical notion of matrix differentiation [2]. It has several motivations in various areas, including the geometry of polynomials -- an extensive topic with many beautiful open problems, for example, Sendov's conjecture.

Not every matrix has an integral, and the problem of existence of integrable and non-integrable matrices with a given Jordan structure has remained open since [1]. We present a complete solution to this problem. Moreover, we have established an easy-to-check criterion for matrix integrability. Our solution is based on special types of polynomials, namely Shabat polynomials and conservative polynomials. Shabat polynomials are complex polynomials with at most 2 finite critical values, and conservative polynomials are polynomials leaving their critical points fixed [3]. To construct the necessary polynomials of both types we rely on their relation with bicolored plane trees, see [4, 5].

Additionally, we explore the properties of matrix integrals and their applications in the geometry of polynomials.

The talk is based on several joint works with Alexander Guterman, Elena Kreines, Patrick Ng, and Fedor Pakovich.

References:
[1] B. V. R. Bhat and M. Mukherjee, Integrators of matrices, Linear Algebra Appl. 426 (2007), 71--82.
[2] C. Davis, Eigenvalues of compressions, Bull. Math. Soc. Sci. Math. Phys. RPR 51 (1959), 3--5.
[3] A. Kostrikin, Conservative polynomials, in ``Stud. Algebra Tbilisi'' (1984). 115--129.
[4] F. Pakovich, Conservative polynomials and yet another action of Gal(Q/Q) on plane trees, J. Théor. Nr. Bordx. 20 (2008), 205--218.
[5] G. B. Shabat and V. A. Voevodsky, Drawing curves over number fields, The Grothendieck Festschrift 3 (1990), 199--227.