Lower bounds for norms of products of polynomials via Bombieri inequality

Pinasco, Damián

doi:10.1090/S0002-9947-2012-05403-1

Lower bounds for norms of products of polynomials via Bombieri inequality
HTML articles powered by AMS MathViewer

by Damián Pinasco PDF

Trans. Amer. Math. Soc. 364 (2012), 3993-4010 Request permission

Abstract:

In this paper we give a different interpretation of Bombieri’s norm. This new point of view allows us to work on a problem posed by Beauzamy about the behavior of the sequence $S_n(P)=\sup _{Q_n} [PQ_n]_2$, where $P$ is a fixed $m-$homogeneous polynomial and $Q_n$ runs over the unit ball of the Hilbert space of $n-$homogeneous polynomials. We also study the factor problem for homogeneous polynomials defined on $\mathbb {C}^N$ and we obtain sharp inequalities whenever the number of factors is no greater than $N$. In particular, we prove that for the product of homogeneous polynomials on infinite dimensional complex Hilbert spaces our inequality is sharp. Finally, we use these ideas to prove that any set $\{z_k\}_{k=1}^n$ of unit vectors in a complex Hilbert space for which $\sup _{\Vert z \Vert =1} \vert \langle z, z_1\rangle \cdots \langle z, z_n\rangle \vert$ is minimum must be an orthonormal system.

References

Vasilios A. Anagnostopoulos and Szilárd Gy. Révész, Polarization constants for products of linear functionals over ${\Bbb R}^2$ and ${\Bbb C}^2$ and Chebyshev constants of the unit sphere, Publ. Math. Debrecen 68 (2006), no. 1-2, 63–75. MR 2213542
J. Arias-de-Reyna, Gaussian variables, polynomials and permanents, Linear Algebra Appl. 285 (1998), no. 1-3, 107–114. MR 1653495, DOI 10.1016/S0024-3795(98)10125-8
G. Aumann, Satz über das Verhalten von Polynomen auf Kontinuen. Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. (1933), 926–931.
Keith M. Ball, The complex plank problem, Bull. London Math. Soc. 33 (2001), no. 4, 433–442. MR 1832555, DOI 10.1112/S002460930100813X
S. Banach, Über Homogene Polynome in $L^2$. Studia Math. 7 (1938), pp. 36-44.
Bernard Beauzamy, Extremal products in Bombieri’s norm, Rend. Istit. Mat. Univ. Trieste 28 (1996), no. suppl., 73–89 (1997). Dedicated to the memory of Pierre Grisvard. MR 1602318
Bernard Beauzamy, Products of many-variable polynomials: pairs that are maximal in Bombieri’s norm, J. Number Theory 55 (1995), no. 1, 129–143. MR 1361564, DOI 10.1006/jnth.1995.1132
Bernard Beauzamy, Products of polynomials and a priori estimates for coefficients in polynomial decompositions: a sharp result, J. Symbolic Comput. 13 (1992), no. 5, 463–472. MR 1170091, DOI 10.1016/S0747-7171(10)80006-9
Bernard Beauzamy, Enrico Bombieri, Per Enflo, and Hugh L. Montgomery, Products of polynomials in many variables, J. Number Theory 36 (1990), no. 2, 219–245. MR 1072467, DOI 10.1016/0022-314X(90)90075-3
Bernard Beauzamy and Jérôme Dégot, Differential identities, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2607–2619. MR 1277095, DOI 10.1090/S0002-9947-1995-1277095-1
B. Beauzamy and P. Enflo, Estimations de produits de polynômes, J. Number Theory 21 (1985), no. 3, 390–412 (French). MR 814012, DOI 10.1016/0022-314X(85)90062-9
Bernard Beauzamy, Jean-Louis Frot, and Christian Millour, Massively parallel computations on many-variable polynomials, Ann. Math. Artificial Intelligence 16 (1996), no. 1-4, 251–283. MR 1389850, DOI 10.1007/BF02127800
Carlos Benítez, Yannis Sarantopoulos, and Andrew Tonge, Lower bounds for norms of products of polynomials, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 395–408. MR 1636556, DOI 10.1017/S030500419800259X
E. Bombieri and J. Vaaler, On Siegel’s lemma, Invent. Math. 73 (1983), no. 1, 11–32. MR 707346, DOI 10.1007/BF01393823
Peter B. Borwein, Exact inequalities for the norms of factors of polynomials, Canad. J. Math. 46 (1994), no. 4, 687–698. MR 1289054, DOI 10.4153/CJM-1994-038-8
C. Boyd and R. A. Ryan, The norm of the product of polynomials in infinite dimensions, Proc. Edinb. Math. Soc. (2) 49 (2006), no. 1, 17–28. MR 2202139, DOI 10.1017/S0013091504000756
David W. Boyd, Bounds for the height of a factor of a polynomial in terms of Bombieri’s norms. I. The largest factor, J. Symbolic Comput. 16 (1993), no. 2, 115–130. MR 1253908, DOI 10.1006/jsco.1993.1036
David W. Boyd, Bounds for the height of a factor of a polynomial in terms of Bombieri’s norms. II. The smallest factor, J. Symbolic Comput. 16 (1993), no. 2, 131–145. MR 1253909, DOI 10.1006/jsco.1993.1037
David W. Boyd, Sharp inequalities for the product of polynomials, Bull. London Math. Soc. 26 (1994), no. 5, 449–454. MR 1308361, DOI 10.1112/blms/26.5.449
Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
J. D. Donaldson and Q. I. Rahman, Inequalities for polynomials with a prescribed zero, Pacific J. Math. 41 (1972), 375–378. MR 308371
Péter E. Frenkel, Pfaffians, Hafnians and products of real linear functionals, Math. Res. Lett. 15 (2008), no. 2, 351–358. MR 2385646, DOI 10.4310/MRL.2008.v15.n2.a12
Juan Carlos García-Vázquez and Rafael Villa, Lower bounds for multilinear forms defined on Hilbert spaces, Mathematika 46 (1999), no. 2, 315–322. MR 1832622, DOI 10.1112/S0025579300007774
A. O. Gel′fond, Transcendental and algebraic numbers, Dover Publications, Inc., New York, 1960. Translated from the first Russian edition by Leo F. Boron. MR 0111736
Jean-Pierre Kahane, Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin-New York, 1970 (French). MR 0275043
H. Kneser, Das Maximum des Produkts zweier Polynome. Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. (1934), pp. 429–431.
A. E. Litvak, V. D. Milman, and G. Schechtman, Averages of norms and quasi-norms, Math. Ann. 312 (1998), no. 1, 95–124. MR 1645952, DOI 10.1007/s002080050213
K. Mahler, An application of Jensen’s formula to polynomials, Mathematika 7 (1960), 98–100. MR 124467, DOI 10.1112/S0025579300001637
K. Mahler, On some inequalities for polynomials in several variables, J. London Math. Soc. 37 (1962), 341–344. MR 138593, DOI 10.1112/jlms/s1-37.1.341
Máté Matolcsi, A geometric estimate on the norm of product of functionals, Linear Algebra Appl. 405 (2005), 304–310. MR 2148177, DOI 10.1016/j.laa.2005.03.028
M. Matolcsi, The linear polarization constant of $\textbf {R}^n$, Acta Math. Hungar. 108 (2005), no. 1-2, 129–136. MR 2155246, DOI 10.1007/s10474-005-0214-y
Gustavo A. Muñoz, Yannis Sarantopoulos, and Andrew Tonge, Complexifications of real Banach spaces, polynomials and multilinear maps, Studia Math. 134 (1999), no. 1, 1–33. MR 1688213, DOI 10.4064/sm-134-1-1-33
Alexandros Pappas and Szilárd Gy. Révész, Linear polarization constants of Hilbert spaces, J. Math. Anal. Appl. 300 (2004), no. 1, 129–146. MR 2100242, DOI 10.1016/j.jmaa.2004.06.031
Damián Pinasco and Ignacio Zalduendo, Integral representation of holomorphic functions on Banach spaces, J. Math. Anal. Appl. 308 (2005), no. 1, 159–174. MR 2142411, DOI 10.1016/j.jmaa.2004.11.031
I. E. Pritsker and S. Ruscheweyh, Inequalities for products of polynomials. I, Math. Scand. 104 (2009), no. 1, 147–160. MR 2498378, DOI 10.7146/math.scand.a-15091
Igor E. Pritsker and Stephan Ruscheweyh, Inequalities for products of polynomials. II, Aequationes Math. 77 (2009), no. 1-2, 119–132. MR 2495722, DOI 10.1007/s00010-008-2953-7
Szilárd Gy. Révész and Yannis Sarantopoulos, Plank problems, polarization and Chebyshev constants, J. Korean Math. Soc. 41 (2004), no. 1, 157–174. Satellite Conference on Infinite Dimensional Function Theory. MR 2048707, DOI 10.4134/JKMS.2004.41.1.157
Raymond A. Ryan and Barry Turett, Geometry of spaces of polynomials, J. Math. Anal. Appl. 221 (1998), no. 2, 698–711. MR 1621703, DOI 10.1006/jmaa.1998.5942
Bruce Reznick, An inequality for products of polynomials, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1063–1073. MR 1119265, DOI 10.1090/S0002-9939-1993-1119265-2
A. E. Taylor, Additions to the theory of polynomials in normed linear spaces. Tohoku Math. J. 44 (1938), pp. 302–318.