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• W1569354372 abstract "Let f:K^n-->K^p be a C^2 semi-algebraic mapping for K=R and a polynomial mapping for K=C. It is well-known that f is a locally trivial topological fibration over the complement of the bifurcation set B(f). In this work, we consider the t-regularity and rho-regularity to study B(f). We show that t-regularity is equivalent to regularity conditions of Rabier (1997), Gaffney (1999), Kurdyka, Orro, Simon (2000) and Jelonek (2003). We prove that t-regularity implies rho-regularity. From rho-regularity, we define the set of non rho-regular values S(f), and the set A_{rho}, which is the union of f(Sing f) and S(f). We prove a structure theorem for S(f) and A_{rho}. We also obtain that B(f) is contained in A_{rho}. We study also two classes of maps, the fair maps and the Newton non-degenerate maps. For fair maps, we give an interpretation of t-regularity in terms of integral closure of modules, which is a real counterpart of Gaffney's result (1999). For non-degenerate maps, we obtain an approximation for B(f) through a set which depends on the Newton polyhedron of f (results like this have been obtained by Nemethi and Zaharia (1990) and by Chen and Tibar (2012)). To finish, we discuss some consequences of our work: the t-regularity for maps f: X-->K^p, where X is a smooth affine variety; the problem of bijectivity of semi-algebraic maps; and a formula to compute the Euler characteristic of regular fibers of f:R^n-->R^{n-1}. The above results are also extensions of some results obtained, for polynomial functions f:K^n-->K, by Nemethi and Zaharia (1990), Siersma and Tibar (1995), Paunescu and Zaharia (1997), Parusinski (1995) and Tibar (1998)." @default.
• W1569354372 created "2016-06-24" @default.
• W1569354372 creator A5043993966 @default.
• W1569354372 date "2015-11-19" @default.
• W1569354372 modified "2023-09-26" @default.
• W1569354372 title "Regularity at infinity and global fibrations of real algebraic maps" @default.
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• W1569354372 doi "https://doi.org/10.11606/t.55.2013.tde-30042013-163011" @default.
• W1569354372 hasPublicationYear "2015" @default.
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