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• W4387342849 abstract "Abstract 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 the Jelonek set , is a subset of $$mathbb {K}^2$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , where a dominant polynomial map $$f: mathbb {K}^2 rightarrow mathbb {K}^2$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> is not proper; $$mathbb {K}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>K</mml:mi> </mml:math> could be either $$mathbb {C}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>C</mml:mi> </mml:math> or $$mathbb {R}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>R</mml:mi> </mml:math> . Unlike all the previously known approaches we make no assumptions on f whenever $$mathbb {K} = mathbb {R}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> ; 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 ." @default.
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• W4387342849 date "2023-10-04" @default.
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• W4387342849 title "Computing the Non-properness Set of Real Polynomial Maps in the Plane" @default.
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• W4387342849 doi "https://doi.org/10.1007/s10013-023-00652-0" @default.
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