A Few Riddles Behind Rolle's Theorem

First year undergraduates usually learn about classical Rolle’s theorem saying that between two consecutive zeros of a smooth univariate function f one can always find at least one zero of its derivative f . In this paper we study a generalization of Rolle’s theorem dealing with zeros of higher derivatives of smooth univariate functions enjoying a natural additional property. Namely, we call a smooth function whose nth derivative does not vanish on some interval I ⊆ R a polynomial-like function of degree n on I. We conjecture that for polynomial-like functions of degree n with n real distinct roots there exists a non-trivial system of inequalities completely describing the set of possible locations of zeros of such functions together with their derivatives of order up to n − 1. We describe the corresponding system of inequalities in the simplest non-trivial case n = 3. Getting started In what follows we only consider smooth real-valued functions defined on a subinterval on the real axis. Consider a smooth function f with n distinct real zeros x (0) 1 < x (0) 2 < . . . < x (0) n on some interval I ⊆ R. Then, by Rolle’s theorem, f ′ has at least (n− 1) zeros, f ′′ has at least (n− 2) zeros, ... , f (n−1) has at least one zero on the open interval (x (0) 1 , x (0) n ). In what follows we will be interested in smooth functions f with n real simple zeros on I such that, additionally, for all i = 1, . . . , n the ith derivative f (i) has on I exactly n− i real simple zeros denoted by x (i) 1 < x (i) 2 < ... < x (i) n−i. Note, in particular, that f (n) is non-vanishing on I. Main Definition. A smooth function f defined on an interval I is called polynomial-like of degree n if f (n) does not vanish on I. A polynomial-like function of degree n on I with n simple real zeros is called real-rooted. Notice that by Rolle’s theorem n is the maximal possible number of real zeros of a polynomial-like function of degree n. An obvious example of a real-rooted polynomial-like function of degree n on R is a usual real polynomial of degree n with all real and distinct zeros. Observe also that if a polynomial-like function f of degree n is real-rooted on I then for all i < n its derivatives f (i) are also real-rooted of degree n− i on the same interval. In the above notation the following system of inequalities holds: (1) x (i) l < x (j) l < x (i) l+j−i; i < j ≤ n− l. We call (1) the system of standard Rolle’s restrictions. With any real-rooted polynomial-like function f of degree n one can associate its configuration Af of ( n+1 2 ) zeros {x l } of f , i = 0, . . . , n− 1; 1 ≤ l ≤ n− i, say, by taking first all x (0) l , then all x (1) l etc. The main problem we address in this short note is as follows. Question. What additional restrictions besides (1) exist on configurations Af = {x l } coming from realrooted polynomial-like functions of a given degree n? Or, even more ambitiously, given a configuration A = {x l | i = 0, . . . , n−1; l = 1, . . . n− i} of ( n+1 2 ) real numbers satisfying standard Rolle’s restrictions is it possible to determine if there exists a real-rooted polynomial-like f of degree n such that Af = A? Our motivation to consider the class of real-rooted polynomial-like functions is twofold. Firstly, this class is a natural generalization of the well-studied and important class of real-rooted polynomials. Secondly, one can easily see that as soon as one allows several real zeros of f ′ between two consecutive zeros of f then no 1 2 B. SHAPIRO AND M. SHAPIRO interesting additional restrictions are possible. We will soon show that the fact that a smooth function f is real-rooted polynomial-like implies additional inequalities on the components of its Af . Let RRn(I) denote the set of all real-rooted polynomial-like functions of degree n on an interval I. (Since particular choice of I is unimportant in the formulation below we will often use RRn instead of RRn(I).) Our result below answers the posed question in the first non-trivial case n = 3. In order to simplify our notation we denote the three zeros of a real-rooted function f of degree 3 by x1 < x2 < x3, the two zeros of f ′ by y1 < y2, and the only zero of f ′′ by z1. Main Theorem. The configuration Af = (x1, x2, x3, y1, y2, z1) of any real-rooted polynomial-like function f of degree 3 satisfies the following inequalities: