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• W3202001049 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the face ring of a simplicial complex of dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d minus 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> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper R left-parenthesis German n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>R</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>n</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal R}({mathfrak {n}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the Rees algebra of the maximal homogeneous ideal <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German n> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>n</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathfrak {n}}</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 R period> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>R.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We show that the generalized Hilbert-Kunz function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H upper K left-parenthesis s right-parenthesis equals script l left-parenthesis script upper R left-parenthesis German n right-parenthesis slash left-parenthesis German n comma German n t right-parenthesis Superscript left-bracket s right-bracket Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>R</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>n</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>n</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>n</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>[</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>HK(s)=ell ({mathcal {R}}({mathfrak {n}})/({mathfrak {n}}, {mathfrak {n}} t)^{[s]})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given by a polynomial for all large <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s period> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>s.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We calculate it in many examples and also provide a Macaulay2 code for computing <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H upper K left-parenthesis s right-parenthesis period> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mi>K</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>HK(s).</mml:annotation> </mml:semantics> </mml:math> </inline-formula>" @default.
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• W3202001049 date "2021-01-01" @default.
• W3202001049 modified "2023-09-25" @default.
• W3202001049 title "Generalized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex" @default.
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• W3202001049 doi "https://doi.org/10.1090/conm/773/15543" @default.
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