On polynomial-like functions

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A polynomial-like function (PLF) of degree n is a smooth function F whose nth derivative never vanishes. A PLF has ⩽ n real zeros; in case of equality it is called hyperbolic; F ( i ) has ⩽ n − i real zeros. We consider the arrangements of the n ( n + 1 ) / 2 distinct real numbers x k ( i ) , i = 0 , … , n − 1 , k = 1 , … , n − i , which satisfy the conditions x k ( i ) < x k ( j ) < x k + j − i ( i ) . We ask the question whether all such arrangements are realizable by the roots of a hyperbolic PLF and its derivatives. We show that for n ⩾ 5 the answer is negative.