This paper proves the existence of a bound on the sum of local Betti numbers of a real analytic germ by a polynomial function of the multiplicity of the germ. This result can be interpreted as a localization of the classical Oleinik–Petrovsky bound (aka. Thom–Milnor bound) on the sum of Betti numbers of a semi-algebraic set. The key elements of the proof are the tangent cone of the germ, the Thom–Mather topological trivialization theorem, the Oleinik–Petrovsky bound, and a result by D. Mumford and J. Heintz bounding the degrees of the generators of an ideal by a polynomial function of the geometric degree of its associated variety. Our result is then applied to yield bounds on known geometric invariants: the Lipschitz–Killing invariants, and the Vitushkin variations.
Cet article borne de manière explicite la somme des nombres de Betti locaux d’un germe analytique réel par un polynôme en la multiplicité du germe. Ce résultat peut être interprété comme une localisation de la borne classique d’Oleinik–Petrovsky (ou borne de Thom–Milnor) de la somme des nombres de Betti d’un ensemble semi-algébrique. Les éléments clefs de la preuve sont le cône tangent du germe, le théorème de trivialité topologique de Thom–Mather, la borne d’Oleinik–Petrovsky, et un résultat de D. Mumford et J. Heintz bornant le degré des générateurs d’un idéal par un polynôme en la multiplicité du degré géométrique de la variété qui lui est associée. Le résultat est ensuite utilisé pour borner des invariants géométriques connus : les invariants de Lipschitz–Killing et les variations de Vitushkin.
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Keywords: multiplicity, analytic germ, Betti number, Thom–Mather, topological triviality, Thom–Milnor, Lipschitz–Killing, Vitushkin
Mot clés : multiplicité, germe analytique, nombre de Betti, Thom–Mather, trivialité topologique, Thom–Milnor, Lipschitz–Killing, Vitushkin
Alberti, Lionel F. 1
@article{AIF_2017__67_1_367_0, author = {Alberti, Lionel F.}, title = {Polynomial {Bound} on the {Local} {Betti} {Numbers} of a {Real} {Analytic} {Germ}}, journal = {Annales de l'Institut Fourier}, pages = {367--396}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {67}, number = {1}, year = {2017}, doi = {10.5802/aif.3085}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3085/} }
TY - JOUR AU - Alberti, Lionel F. TI - Polynomial Bound on the Local Betti Numbers of a Real Analytic Germ JO - Annales de l'Institut Fourier PY - 2017 SP - 367 EP - 396 VL - 67 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3085/ DO - 10.5802/aif.3085 LA - en ID - AIF_2017__67_1_367_0 ER -
%0 Journal Article %A Alberti, Lionel F. %T Polynomial Bound on the Local Betti Numbers of a Real Analytic Germ %J Annales de l'Institut Fourier %D 2017 %P 367-396 %V 67 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3085/ %R 10.5802/aif.3085 %G en %F AIF_2017__67_1_367_0
Alberti, Lionel F. Polynomial Bound on the Local Betti Numbers of a Real Analytic Germ. Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 367-396. doi : 10.5802/aif.3085. https://aif.centre-mersenne.org/articles/10.5802/aif.3085/
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