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Article Dans Une Revue Vietnam Journal of Mathematics Année : 2023

Computing the non-properness set of real polynomial maps in the plane

Résumé

We introduce novel mathematical and computational tools to develop a complete algorithm for computing the set of non-properness of polynomials maps in the plane. In particular, this set, which we call \emph{the Jelonek set}, is a subset of $\mathbb{K}^2$ where a dominant polynomial map $f:\mathbb{K}^2\to\mathbb{K}^2$ is not proper; $\mathbb{K}$ could be either $\mathbb{C}$ or $\mathbb{R}$. Unlike all the previously known approaches we make no assumptions on $f$ whenever $\mathbb{K} = \mathbb{R}$; this is the first algorithm with this property. The algorithm takes into account the Newton polytopes of the polynomials. As a byproduct we provide a finer representation of the set of non-properness as a union of semi-algebraic curves, that correspond to edges of the Newton polytopes, which is of independent interest. Finally, we present a precise Boolean complexity analysis of the algorithm and a prototype implementation in Maple.
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Dates et versions

hal-04223513 , version 1 (29-09-2023)

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Boulos El Hilany, Elias Tsigaridas. Computing the non-properness set of real polynomial maps in the plane. Vietnam Journal of Mathematics, inPress, ⟨10.1007/s10013-023-00652-0⟩. ⟨hal-04223513⟩
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