Newton polygons and local integrability of negative powers of smooth functions in the plane

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x comma y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(x,y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be any smooth real-valued function with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis 0 comma 0 right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">f(0,0)=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For a sufficiently small neighborhood <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the origin, we study the number <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup left-brace right-brace colon epsilon colon greater-than greater-than integral integral upper U vertical-bar vertical-bar of ff left-parenthesis right-parenthesis comma x comma y minus minus epsilon infinity period"> <mml:semantics> <mml:mrow> <mml:mo movablelimits="true" form="prefix">sup</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi> ϵ </mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mo> ∫ </mml:mo> <mml:mi>U</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi> ϵ </mml:mi> </mml:mrow> </mml:msup> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sup \left \{\epsilon :\int _U |f(x,y)|^{-\epsilon }&gt;\infty \right \}.</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> It is known that sometimes this number can be expressed in a natural way using the Newton polygon of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We provide necessary and sufficient conditions for this Newton polygon characterization to hold. The behavior of the integral at the supremal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon"> <mml:semantics> <mml:mi> ϵ </mml:mi> <mml:annotation encoding="application/x-tex">\epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is also analyzed.

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