Polynomial Bound on the Local Betti Numbers of a Real Analytic Germ
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Bibliographic record
Abstract
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.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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