On the polynomial-like behaviour of certain algebraic functions
Annales de l'Institut Fourier, Tome 44 (1994) no. 4, p. 1091-1179
Étant donné des entiers D>0,n>1,0<r<n et une constante C>0, on considère l’espace des r-uples P =(P 1 ...P r ) de polynômes réels à n variables, de degré D, à coefficients C en valeur absolue, et satisfaisant à det P i x i (0) 1i,jr =1. On étudie la famille {f|V} des fonctions algébriques, où f est un polynôme et V={|x|δ,P (x)=0},δ>0 ne dépendant que de n,D,C. Le résultat principal est un théorème quantitatif d’extension de ces fonctions qui est uniforme par rapport à P . Ce résultat est utilisé pour obtenir des inégalités, uniformes par rapport à P , du type de celle de Bernstein.La démonstration s’appuie sur des résultats quantitatifs concernant les idéaux de polynômes et sur la théorie des ensembles semi-algébriques.
Given integers D>0,n>1,0<r<n and a constant C>0, consider the space of r-tuples P =(P 1 ...P r ) of real polynomials in n variables of degree D, whose coefficients are C in absolute value, and satisfying det P i x i (0) 1i,jr =1. We study the family {f|V} of algebraic functions, where f is a polynomial, and V={|x|δ,P (x)=0},δ>0 being a constant depending only on n,D,C. The main result is a quantitative extension theorem for these functions which is uniform in P . This is used to prove Bernstein-type inequalities which are again uniform with respect to P .The proof is based on some quantitative results on ideals of polynomials and on the theory of semi-algebraic sets.
@article{AIF_1994__44_4_1091_0,
     author = {Feffermann, Charles and Narasimhan, Raghavan},
     title = {On the polynomial-like behaviour of certain algebraic functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {44},
     number = {4},
     year = {1994},
     pages = {1091-1179},
     doi = {10.5802/aif.1428},
     zbl = {0811.14046},
     mrnumber = {95k:32011},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1994__44_4_1091_0}
}
Feffermann, Charles; Narasimhan, Raghavan. On the polynomial-like behaviour of certain algebraic functions. Annales de l'Institut Fourier, Tome 44 (1994) no. 4, pp. 1091-1179. doi : 10.5802/aif.1428. https://www.numdam.org/item/AIF_1994__44_4_1091_0/

[A] F. Amoroso, The membership problem for smooth ideals, (to appear in Publ. de l'I.H.P., Sém. de théorie des nombres).

[BCR] J. Bochnak, M. Coste and M.-F. Roy, Géométrie algébrique réelle, Springer, 1987. | MR 90b:14030 | Zbl 0633.14016

[BT] C.A. Berenstein and B.A. Taylor, On the geometry of interpolating varieties, Sém. Lelong-Skoda (1980-1981), Springer Lecture Notes in Math., vol. 919, 1-25. | MR 83k:32004 | Zbl 0484.32004

[BY] C.A. Berenstein and A. Yger, Ideals generated by exponential polynomials, Advances in Math., vol. 60 (1986), 1-80. | MR 87i:32005 | Zbl 0586.32019

[C] P.J. Cohen, Decision Procedures for Real and p-adic Fields, Comm. Pure Appl. Math., vol. 22 (1969), 131-151. | MR 39 #5342 | Zbl 0167.01502

[FN] C. Fefferman and R. Narasimhan, Bernstein's Inequality on Algebraic Curves, Annales de l'Inst. Fourier, vol. 43-5 (1993), 1319-1348. | Numdam | MR 95e:32007 | Zbl 0842.26013

[FS] G.B. Folland and E.M. Stein, Estimates for the ∂b-complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., vol. 27 (1974), 429-522. | MR 51 #3719 | Zbl 0293.35012

[He] G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Annalen, vol. 93 (1926), 736-788. | JFM 52.0127.01

[H] L. Hörmander, An Introduction to Complex Analysis in Several Variables, 2nd edition, Amsterdam, North-Holland, 1973. | Zbl 0271.32001

[KT] J.J. Kelleher and B.A. Taylor, Finitely generated ideals in rings of analytic functions, Math. Annalen, vol. 193 (1971), 225-237. | MR 46 #2077 | Zbl 0207.12906

[M] D. Mumford, Algebraic Geometry I. Complex Projective Varieties, Springer, 1976. | Zbl 0356.14002

[NSW] A. Nagel, E.M. Stein and S. Wainger, Balls and metrics defined by vector fields I. Basic properties, Acta Math., vol. 155 (1985), 103-147. | MR 86k:46049 | Zbl 0578.32044

[P] A. Parmeggiani, Subunit Balls for Symbols of Pseudodifferential Operators, Princeton Doctoral Dissertation, 1992 (to appear in Advances in Math.). | Zbl 0940.35214

[RS] L. Rothschild and E.M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., vol. 137 (1976), 247-320. | MR 55 #9171 | Zbl 0346.35030

[W] H. Whitney, Elementary structure of real algebraic varieties, Annals of Math., vol. 66 (1957), 545-556. | MR 20 #2342 | Zbl 0078.13403