A Finite Moment-Cone Criterion for Derivative-Root Configurations

We give a necessary and sufficient finite criterion for prescribed numerical positions of all derivative roots of a real-rooted polynomial-like function. Polynomial-like degree n means that the nth derivative is nowhere zero; the function need not be an ordinary polynomial of degree n. Eliminating integration constants produces m=n(n-1)/2 explicit piecewise polynomial Peano kernels. Realizability is equivalent to zero being interior to their essential-trace convex hull, and to representing an explicit baseline vector by at most m nonnegative trace terms. This gives a finite semialgebraic criterion and a terminating decision procedure for algebraic coordinates. Permitted cross-order coincidences are included. Every feasible array also has an ordinary-polynomial realization of unspecified larger degree.