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\iteman{io-port 05598410}
\itemau{To, Wing-Keung; Yeung, Sai-Kee}
\itemti{Effective {\L}ojasiewicz inequality for arithmetically defined varieties and a geometric application to bihomogeneous polynomials.}
\itemso{Int. J. Math. 20, No. 7, 915-944 (2009).}
\itemab
Consider an Hermitian form $f$ of degree $m$ on the complex space $\Bbb C^n$ which is positive along the affine cone $M$ of a projective variety defined over the field of algebraic numbers. The authors obtain (Thm. 2.2 of the text) an upper bound for the smallest power $\ell$ such that $\Vert z\Vert^{2\ell}f(z)$ is a sum of squared norms of polynomials on $M$. This can be interpreted as an effective isometric embedding theorem for the indefinite Hermitian holomorphic line bundle occuring. This result rests on an argument of {\it W.-K. To} and {\it S.-K. Yeung} and a \L ojasiewicz type inequality (Thm. 2.1 of the text which in fact follows directly from the closest point property as stated in chapter 6, \S5, pages 88--89, of ``Introduction to algebraic independence theory.'' Lect. Notes Math. 1752 (2001; Zbl 0966.11032), together with classical B\'ezout estimates, ibid. \S4). Combining this {\L}ojasiewicz inequality with an effective Nullstellensatz (here the sharp estimate due to {\it T. Krick}, {\it L. M. Pardo} and {\it M. Sombra} [Duke Math. J. 109, No. 3, 521--598 (2001; Zbl 1010.11035)]), leads to a variant (Thm. 2.3 of the text) of the estimate of Thm. 2.1, in terms of the given defining polynomials of the variety.
\itemrv{Patrice Philippon (Paris)}
\itemcc{}
\itemut{\L ojasiewicz inequality; arithmetic variety; bihomogeneous polynomial}
\itemli{doi:10.1142/S0129167X09005534}
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