, Volume 35, Issue 2–4, pp 193–215

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

Article

## Abstract

We consider the problem of minimizing or maximizing the quotient
$$f_{m,n}(p,q):=\frac{\|{pq}\|}{\|{p}\|\|{q}\|} \ ,$$
where $$p=p_0+p_1x+\dots+p_mx^m$$, $$q=q_0+q_1x+\dots+q_nx^n\in{\mathbb K}[x]$$, $${\mathbb K}\in\{{\mathbb R},{\mathbb C}\}$$, are non-zero real or complex polynomials of maximum degree $$m,n\in{\mathbb N}$$ respectively and $$\|{p}\|:=(|p_0|^2+\dots+|p_m|^2)^{\frac{1}{2}}$$ 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 $${\mathbb K}={\mathbb C}$$ and maximizers p max, q max for arbitrary $${\mathbb K}$$ 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 $${\mathbb K}={\mathbb R}$$ 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.

## Keywords

Inequalities of polynomial products Eigenvalues and eigenvectors of autocorrelation Toeplitz matrices

15A42 26D05

## References

1. 1.
Beauzamy, B., Bombieri, E., Enflo, P., Montgomery, H.L.: Products of polynomials in many variables. J. Number Theory 36, 219–245 (1990)
2. 2.
Boyd, D.W.: Two sharp inequalities for the norm of a factor of a polynomial. Mathematika 39, 341–349 (1992)
3. 3.
Boyd, D.W.: Sharp inequalities for the product of polynomials. Bull. Lond. Math. Soc. 26, 449–454 (1994)
4. 4.
Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Springer, New York (1995)
5. 5.
Borwein, P.: Exact inequalities for the norms of factors of polynomials. Can. J. Math. 46(4), 687–698 (1994)
6. 6.
Carathéodory, C.: Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911)
7. 7.
Data, L., Morgera, S.D.: Comments and corrections on “On the eigenvectors of symmetric Toeplitz matrices”. IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-32, no. 2, 440,441 (1984)Google Scholar
8. 8.
Delsarte, P., Genin, Y.: Spectral properties of finite Toeplitz matrices. In: Fuhrmann, P.A. (ed.) Mathematical Theory of Networks and Systems: Proc. MTNS-83 Int. 17 Symp., Beer Sheva, Israel, June 1983. Lecture Notes in Control and Information Sciences, vol. 58, pp. 194–213. Springer, New York (1984)Google Scholar
9. 9.
Genin, Y.: A survey of the eigenstructure properties of finite Hermitian Toeplitz matrices. Integr. Equ. Oper. Theory 10, 621–639 (1987)
10. 10.
Grenander, U., Szegö, G.: Toeplitz Forms and Their Applications. University of California Press, Berkeley (1958)
11. 11.
Gueguen, C.: Linear prediction in the singular case and the stability of eigen models. In: Acoustics, Speech and Signal Processing, IEEE International Conference on ICASSP 6, pp. 881–885 (1981)Google Scholar
12. 12.
Kneser, H.: Das Maximum des Produktes zweier Polynome. In: Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl., pp. 426–431 (1934)Google Scholar
13. 13.
Makhul, J.: On the eigenvectors of symmetric Toeplitz matrices. IEEE Transactions on Acoustic, Speech and Signal Processing ASSP-29, no. 4, 868–872 (1981)Google Scholar
14. 14.
Pisarenko, V.F.: The retrieval of Harmonics from a Covariance Function. Geophys. J. R. Astron. Soc. 33, 347–366 (1973)
15. 15.
Reddi, S.S.: Eigenvector properties of Toeplitz matrices and their application to spectral analysis of time. Signal Process. 7, 45–56 (1984)
16. 16.
Robinson, E.: Statistical Communication and Detection. Griffin, London (1967)Google Scholar
17. 17.
Rump, S.M., Sekigawa, H.: The ratio between the Toeplitz and the unstructured condition number. Oper. Theory Adv. Appl. 199, 397–419 (2009)
18. 18.
Sheil-Small, T.: Complex Polynomials. Cambridge University Press, Cambridge (2002)