Very hyperbolic polynomials (Q883694)

A real polynomial in one variable is said to be hyperbolic (resp., strictly hyperbolic) if all its roots are real (resp., real and distinct). A polynomial \(f\) is called a primitive of order \(k\) of a polynomial \(g\) if \(f^{(k)}=g\). A hyperbolic polynomial is said to be very hyperbolic if it has hyperbolic primitives of all orders. Every hyperbolic polynomial of degree at most 3 is very hyperbolic. In this paper, the author gives a geometric description of the set of very hyperbolic polynomials of degree 4. He also shows that for every integer \(n\geq 1\), there is a strictly hyperbolic and very hyperbolic polynomial \(g\) of degree \(n\) and integration constants \(c_i\) such that the primitives of all orders of \(g\) with integration constants \(c_i\) are strictly hyperbolic and very hyperbolic.