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• W2134038245 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper A comma script upper M right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>( A,mathcal {M})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a local ring of positive dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-primary ideal. We denote the reduction number of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r left-parenthesis upper I right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>r(I)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is the smallest integer <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding=application/x-tex>r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I Superscript r plus 1 Baseline equals upper J upper I Superscript r> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>I</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>J</mml:mi> <mml:msup> <mml:mi>I</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>I^{r+1}=JI^r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some reduction <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper J> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding=application/x-tex>J</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 I period> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>I.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> In this paper we give an upper bound on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r left-parenthesis upper I right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>r(I)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of numerical invariants which are related with the Hilbert coefficients of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Cohen-Macaulay. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d equals 1> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>d=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, it is known that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r left-parenthesis upper I right-parenthesis less-than-or-equal-to e left-parenthesis upper I right-parenthesis minus 1> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>e</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>r(I) le e(I) -1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e left-parenthesis upper I right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>e(I)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the multiplicity of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I period> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>I.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d less-than-or-equal-to 2 comma> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>d le 2,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in Corollary 1.5 we prove <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r left-parenthesis upper I right-parenthesis less-than-or-equal-to e 1 left-parenthesis upper I right-parenthesis minus e left-parenthesis upper I right-parenthesis plus lamda left-parenthesis upper A slash upper I right-parenthesis plus 1> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:msub> <mml:mi>e</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>e</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>r(I) le e_1(I) - e(I) + lambda (A/I) + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e 1 left-parenthesis upper I right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>e</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>e_1(I)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the first Hilbert coefficient of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I period> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>I.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> From this bound several results follow. Theorem 1.3 gives an upper bound on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r left-parenthesis upper I right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>r(I)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a more general setting." @default.
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• W2134038245 date "1999-10-05" @default.
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• W2134038245 title "A bound on the reduction number of a primary ideal" @default.
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