Semi-Algebraic Functions with Non-Compact Critical Set
Let X ⊂ R n be a closed semi-algebraic set, F : R n → R be a C 2 semi-algebraic function and f = F | X : X → R n be the restriction of F to X . We define the global index of a critical value c i of f and prove an index formula for χ ( X ) that generalizes a result previously proved by the authors fo...
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Veröffentlicht in: | Boletim da Sociedade Brasileira de Matemática 2023-06, Vol.54 (2), Article 22 |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
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Zusammenfassung: | Let
X
⊂
R
n
be a closed semi-algebraic set,
F
:
R
n
→
R
be a
C
2
semi-algebraic function and
f
=
F
|
X
:
X
→
R
n
be the restriction of
F
to
X
. We define the global index of a critical value
c
i
of
f
and prove an index formula for
χ
(
X
)
that generalizes a result previously proved by the authors for the case of isolated critical points. We define also new indices at infinity and prove an alternative index formula for
χ
(
X
)
. |
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ISSN: | 1678-7544 1678-7714 |
DOI: | 10.1007/s00574-023-00340-0 |