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• W4283705555 abstract "Given a closed semi-algebraic set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X subset-of double-struck upper R Superscript n> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>X subset mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a continuous semi-algebraic mapping <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G colon upper X right-arrow double-struck upper R Superscript m> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>:<!-- : --></mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>G colon X to mathbb {R}^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, it will be shown that there exists an open dense semi-algebraic subset <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper U> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=script>U</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathscr {U}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L left-parenthesis double-struck upper R Superscript n Baseline comma double-struck upper R Superscript m Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L(mathbb {R}^n, mathbb {R}^m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the space of all linear mappings from <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript n> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript m> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, such that for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F element-of script upper U> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=script>U</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>F in mathscr {U}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the image <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper F plus upper G right-parenthesis left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>+</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(F + G)(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a closed (semi-algebraic) set in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript m> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. To do this, we study the tangent cone at infinity <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Subscript normal infinity Baseline upper X> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>C_infty X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E Subscript normal infinity Baseline upper X subset-of upper C Subscript normal infinity Baseline upper X> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> <mml:mi>X</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>E_infty X subset C_infty X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of (unit) exceptional directions at infinity of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Specifically we show that the set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E Subscript normal infinity Baseline upper X> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>E_infty X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is nowhere dense in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Subscript normal infinity Baseline upper X intersection double-struck upper S Superscript n minus 1> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> <mml:mi>X</mml:mi> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>C_infty X cap mathbb {S}^{n - 1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W4283705555 created "2022-06-30" @default.
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• W4283705555 date "2022-05-20" @default.
• W4283705555 modified "2023-10-17" @default.
• W4283705555 title "Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings" @default.
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• W4283705555 doi "https://doi.org/10.1090/proc/15827" @default.
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