The road from Kac's matrix to Kac's random polynomials

This paper tells the story of a matrix and a problem both of which are associated with the name Mark Kac, though most likely he never made the connection. This matrix appears as the Clement matrix in Higham's toolbox, and in the Ehrenfest urn model of diffusion. Our immediate interest in this matrix arises in the study of roots of random polynomials. 1 A!---?B This report may be thought of as a path with two endpoints, both of which may be associated with the name Mark Kac. No on second thought, the focus is not so much on the path, but rather the two endpoints themselves as there are many paths between the two points, some more scenic than others. We have no reason to suspect that Kac ever took the path from A to B as his itinerary took him to many other destinations. Endpoint "A" is the so-called Kac matrix, a simple matrix with many hidden treasures. Endpoint "B" is a Kac problem on the average number of real roots of a random algebraic equation. The matrix ("A") appeared in 1946 i..