Toward Effective Detection of the Bifurcation Locus of Real Polynomial Maps

Abstract

We answer to a problem raised by recent work of Jelonek and Kurdyka: how can one detect by rational arcs the bifurcation locus of a polynomial map R n → R p in case p > 1. We describe an effective estimation of the "nontrivial" part of the bifurcation locus.

Asymptotic critical value set and Newton polyhedron

“…Futher in this article we shall use the terminology "parametric length of a curve" to denote this number. Thus in our Examples 5.1, 5.2 the parametric length of a real rational curve has been reduced to 4 in comparison with 3601 proposed in [4]. In Example 5.3, case ii) the parametric length has been reduced to a number less than 2deg f W in comparison with (1 + 5deg f W )(5deg f W ) 2 + 1 proposed in [4].…”

“…Under the condition that the projective closure of the generic fibre of f in P n has only isolated singularities, Parusiński [12] proved that B(f ) = K ∞ (f ) ∪ f (Singf ). A precedent work [4] established a method to detect the bifurcation set in an efficient way. It gave an answer to a question raised in [9] and [5] about the detection of the bifurcation locus by rational curve with parametric representation.…”

“…More concretely, for a real polynomial f : R n → R of degree ≤ d, authors of [4] consider a real rational curve X(t), lim t→0 X(t) → ∞, with parametric representation with length (d + 1)d n−1 + 1 to attain the critical value lim t→0 f (X(t)) ∈ K ∞ (f ).…”

Existence of Pareto Solutions for Vector Polynomial Optimization Problems with Constraints

“…, +∞), the notations K ∞,≤ȳ (f, S), K ∞,≤ȳ (f, S) and T ∞,≤ȳ (f, S) will be written as K ∞ (f, S), K ∞ (f, S) and T ∞ (f, S), respectively. We would note here that all of the sets mentioned above can be computed effectively as shown recently in [16][17][18]28].…”

“…this is because there is no {x } ⊂ S with x → ∞ such that v(x ) → 0 as → ∞ by (17). Thus, (18) along with (8) implies that K ∞ (f, S) = ∅.…”