On gradient at infinity of semialgebraic functions

Didier D'Acunto, and Vincent Grandjean. "On gradient at infinity of semialgebraic functions." Annales Polonici Mathematici 87.1 (2005): 39-49. <http://eudml.org/doc/280497>.

@article{DidierDAcunto2005,
abstract = {Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number $ϱ_c ≤ 1$ such that |x|·|∇f| and $|f(x) - c|^\{ϱ_c\}$ are separated at infinity. If c is a regular value and $ϱ_c < 1$, then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.},
author = {Didier D'Acunto, Vincent Grandjean},
journal = {Annales Polonici Mathematici},
keywords = {semialgebraic functions; gradient trajectories; Łojasiewicz inequalities; Malgrange condition},
language = {eng},
number = {1},
pages = {39-49},
title = {On gradient at infinity of semialgebraic functions},
url = {http://eudml.org/doc/280497},
volume = {87},
year = {2005},
}

TY - JOUR
AU - Didier D'Acunto
AU - Vincent Grandjean
TI - On gradient at infinity of semialgebraic functions
JO - Annales Polonici Mathematici
PY - 2005
VL - 87
IS - 1
SP - 39
EP - 49
AB - Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number $ϱ_c ≤ 1$ such that |x|·|∇f| and $|f(x) - c|^{ϱ_c}$ are separated at infinity. If c is a regular value and $ϱ_c < 1$, then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.
LA - eng
KW - semialgebraic functions; gradient trajectories; Łojasiewicz inequalities; Malgrange condition
UR - http://eudml.org/doc/280497
ER -