# On regularity conditions at infinity

@inproceedings{Dias2014OnRC,
title={On regularity conditions at infinity},
author={Luis Renato G. Dias},
year={2014}
}
Let f : X → Kp be a restriction of a polynomial mapping on X, where X ⊂ Kn is a smooth affine variety. We prove the equivalence of regularity conditions at infinity, which are useful to control the bifurcation set of f .
2 Citations
Global Euler obstruction, global Brasselet numbers and critical points
• Mathematics
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 2020
Abstract Let X ⊂ ℂn be an equidimensional complex algebraic set and let f: X → ℂ be a polynomial function. For each c ∈ ℂ, we define the global Brasselet number of f at c, a global counterpart of the
Toward Effective Detection of the Bifurcation Locus of Real Polynomial Maps
• Mathematics, Computer Science
Found. Comput. Math.
• 2017
An effective estimation of the “non-trivial” part of the bifurcation locus of a polynomial map is described.

## References

SHOWING 1-10 OF 30 REFERENCES
Asymptotic behaviour of families of real curves
• Mathematics
• 1999
Abstract:We consider the family of fibres of a polynomial function f on a smooth noncompact algebraic real surface and we characterise the regular fibres of f which are atypical due to their
Topology at infinity of polynominal mappings and Thom regularity condition
We consider polynomial mappings which have atypical fibres due to the asymptotic behavior at infinity. Fixing some proper extension of the polynomial mapping, we study the localizability at infinity
ON GRADIENT AT INFINITY OF SEMIALGEBRAIC FUNCTIONS
• Mathematics
• 2005
Let f : U R n ! R be a C 2 semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number c6 1 such thatjxj jrfj andjf(x) cj c are
On the generalized critical values of a polynomial mapping
Abstract Let be a polynomial dominant mapping and let deg fi≤d. We prove that the set K(f) of generalized critical values of f is contained in the algebraic hypersurface of degree at most
Fibers of Polynomial Mappings at Infinity and a Generalized Malgrange Condition
Let f be a complex polynomial mapping. We relate the behaviour of f ‘at infinity’ to the characteristic cycle associated to the projective closures of fibres of f. We obtain a condition on the
Regularity at infinity of real mappings and a Morse–Sard theorem
• Mathematics
• 2012
We prove a new Morse-Sard type theorem for the asymptotic critical values of semi-algebraic mappings and a new fibration theorem at infinity for $C^2$ mappings. We show the equivalence of three
SEMIALGEBRAIC SARD THEOREM FOR GENERALIZED CRITICAL VALUES
• Mathematics
• 2000
We prove that a semialgebraic differentiable mapping has a generalized critical values set of measure zero. Moreover, if the mapping is C 2 we obtain, bya generalisation of Ehresmann’s fibration
Bifurcation values and monodromy of mixed polynomials
• Mathematics
• 2010
We study the bifurcation values of real polynomial maps $f: \bR^{2n} \to \bR^2$ which reflect the lack of asymptotic regularity at infinity. We formulate real counterparts of some structure results
Geometry of real polynomial mappings
Abstract. In this paper we study the set of points at which a real polynomial mapping is not proper.
Singularities at infinity and their vanishing cycles
• Mathematics
• 1995
Let f C n C be any polynomial function By using global polar methods we introduce models for the bers of f and we study the monodromy at atypical values of f including the value in nity We construct