Eventual positivity of Hermitian polynomials and integral operators

Abstract

Quillen proved that if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D’Angelo and Varolin deduced this positivstellensatz of Quillen from the eventual positive-definiteness of an associated integral operator. Their arguments involve asymptotic expansions of the Bergman kernel. The goal of this article is to give an elementary proof of the positive-definiteness of this integral operator.

Keywords

Asymptotics / Polynomial / Positivity

References

[1] Artin E.. Über die Zerlegung Definiter Funktionen in Quadrate. Abh. Math. Sem. Univ, 1927, 5(1): 100-115

[2] Axler S., Bourdon P., Ramey W.. Harmonic Function Theory, 2001, New York: Springer-Verlag

[3] Catlin D. W., D’Angelo J. P.. Positivity conditions for bihomogeneous polynomials. Math. Res. Lett, 1997, 4: 555-567

[4] Catlin D. W., D’Angelo J. P.. An isometric embedding theorem for holomorphic bundles. Math. Res. Lett, 1999, 6: 43-60

[5] Drouot A., Zworski M.. A quantitative version of Catlin-D’Angelo-Quillen theorem. Anal. Math. Phys, 2013, 3: 1-19

[6] Knuth D. E.. Two notes on notation. The American Mathematical Monthly, 1992, 99(5): 403-422

[7] Putinar M., Scheiderer C.. Sums of Hermitian squares on pseudoconvex boundaries. Math. Res. Lett, 2010, 17(6): 1047-1053

[8] Quillen D. G.. On the representation of Hermitian forms as sums of squares. Inventiones Math, 1968, 5: 237-242

[9] Tian G.. On a set of polarized Kahler metrics on algebraic manifolds. J. Diff. Geom, 1990, 32: 99-130

[10] To W. K., Yeung S. K.. Effective isometric embeddings for certain holomorphic line bundles. J. London Math. Soc, 2006, 73(2): 607-624

[11] Varolin D.. Geometry of Hermitian algebraic functions. Quotients of squared norms. Amer. J. Math, 2008, 130: 291-315