Minimizing and maximizing the Euclidean norm of the product of two polynomials

Abstract

We consider the problem of minimizing or maximizing the quotient f m,n ( p, q) := pq p q ,, à ∈ {Ê, }, are non-zero real or complex polynomials of maximum degree m, n ∈ AE respectively and p := (| p 0 | 2 + · · · + |p m | 2 ) 1 2 is simply the Euclidean norm of the polynomial coefficients. Clearly f m,n is bounded and assumes its maximum and minimum values min f m,n = f m,n ( p min , q min ) and max f m,n = f ( p max , q max ). We prove that minimizers p min , q min for à = and maximizers p max , q max for arbitrary à fulfill deg( p min ) = m = deg( p max ), deg(q min ) = n = deg(q max ) and all roots of p min , q min , p max , q max have modulus one and are simple. For à = Ê we can only prove the existence of minimizers p min , q min of full degree m and n respectively having roots of modulus one. These results are obtained by transferring the optimization problem to that of determining extremal eigenvalues and corresponding eigenvectors of autocorrelation Toeplitz matrices. By the way we give lower bounds for min f m,n for real polynomials which are slightly better than the known ones and inclusions for max f m,n .

A model problem for global optimization

“…More precisely it is shown in [15] that for a fixed polynomial P of degree n − 1 the minimum of min{||P Q|| : deg(Q) ≤ n − 1 and ||Q|| = 1} (9) is achieved by a polynomial Q having all roots on the unit circle. It was noted in [5] that the so called Caratheodory-representation of the autocorrelation Toeplitz matrix Ψ p Ψ T p (deduced from a theorem of Caratheodory [6], Theorem 4.1) even implies that Q can be found such that all roots of Q are simple.…”

“…Quite a number of very interesting results on this optimization problem can be found in [5]. Bünger gives arguments supporting that there is a unique minimizer for the real problem, that the complex and real optimization problem have the same minimum and, up to scaling, the same minimizer.…”