Extensions of real regular mappings and the Łojasiewicz exponent at infinity

Abstract

Let V ⊂ R n be an algebraic set of positive dimension and let L ∞ (F ) be the Łojasiewicz exponent at infinity of a regular mapping F : V → R m . We prove that F has a polynomial extension G : R n → R m such that L ∞ (G) = L ∞ (F ). Moreover, we give an estimate of the degree of this extension. Additionally, we prove that if dim V < n − 2, then for any β ∈ Q, β < L ∞ (F ) the mapping F has a polynomial extension G with L ∞ (G) = β. We also estimate its degree.

Sum of squares and the Łojasiewicz exponent at infinity

“…As is shown in the proof of [19,Proposition 2.11], the affine subspace A and the polynomial f in the above assertion can be effectively determined. More precisely, after choosing an appropriate coordinate system (using for instance a Gröbner basis), one can choose a nonzero polynomial g ∈…”

“…In what follows, we will use the Euclidean norm. The exponent L ∞ (F ) is an important tool in the study of properness and injectivity of polynomial mappings, in the effective Nullstellensatz and in optimization (for references see for instance [19]).…”