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• W2990582066 abstract "In an earlier work, the authors prove Stillman’s conjecture in all characteristics and all degrees by showing that, independent of the algebraically closed field <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or the number of variables, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> forms of degree at most <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> in a polynomial ring <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> over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are contained in a polynomial subalgebra of <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> generated by a regular sequence consisting of at most <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Superscript eta Baseline upper B left-parenthesis n comma d right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> </mml:mrow> <mml:mi>η<!-- η --></mml:mi> </mml:msup> <mml:mspace width=negativethinmathspace /> <mml:mi>B</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{}^eta !B(n,d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> forms of degree at most <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>; we refer to these informally as “small” subalgebras. Moreover, these forms can be chosen so that the ideal generated by any subset defines a ring satisfying the Serre condition R<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Subscript eta> <mml:semantics> <mml:msub> <mml:mi /> <mml:mi>η<!-- η --></mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>_eta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A critical element in the proof is to show that there are functions <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Superscript eta Baseline upper A left-parenthesis n comma d right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> </mml:mrow> <mml:mi>η<!-- η --></mml:mi> </mml:msup> <mml:mspace width=negativethinmathspace /> <mml:mi>A</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{}^eta !A(n,d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the following property: in a graded <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vector subspace <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</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> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spanned by forms of degree at most <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>, if no nonzero form in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is in an ideal generated by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Superscript eta Baseline upper A left-parenthesis n comma d right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> </mml:mrow> <mml:mi>η<!-- η --></mml:mi> </mml:msup> <mml:mspace width=negativethinmathspace /> <mml:mi>A</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{}^eta !A(n,d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> forms of strictly lower degree (we call this a <italic>strength</italic> condition), then any homogeneous basis for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding=application/x-tex>V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an R<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Subscript eta> <mml:semantics> <mml:msub> <mml:mi /> <mml:mi>η<!-- η --></mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>_eta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sequence. The methods of our earlier work are not constructive. In this paper, we use related but different ideas that emphasize the notion of a <italic>key function</italic> to obtain the functions <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Superscript eta Baseline upper A left-parenthesis n comma d right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> </mml:mrow> <mml:mi>η<!-- η --></mml:mi> </mml:msup> <mml:mspace width=negativethinmathspace /> <mml:mi>A</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{}^eta !A(n,d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in degrees 2, 3, and 4 (in degree 4 we must restrict to characteristic not 2, 3). We give bounds in closed form for the key functions and the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Superscript eta Baseline upper A underbar> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> </mml:mrow> <mml:mi>η<!-- η --></mml:mi> </mml:msup> <mml:mspace width=negativethinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:munder> <mml:mi>A</mml:mi> <mml:mo>_<!-- _ --></mml:mo> </mml:munder> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{}^eta !{underline {A}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> functions, and explicit recursions that determine the functions <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Superscript eta Baseline upper B> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> </mml:mrow> <mml:mi>η<!-- η --></mml:mi> </mml:msup> <mml:mspace width=negativethinmathspace /> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{}^eta !B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Superscript eta Baseline upper A underbar> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> </mml:mrow> <mml:mi>η<!-- η --></mml:mi> </mml:msup> <mml:mspace width=negativethinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:munder> <mml:mi>A</mml:mi> <mml:mo>_<!-- _ --></mml:mo> </mml:munder> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{}^eta !{underline {A}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> functions. In degree 2, we obtain an explicit value for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Superscript eta Baseline upper B left-parenthesis n comma 2 right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> </mml:mrow> <mml:mi>η<!-- η --></mml:mi> </mml:msup> <mml:mspace width=negativethinmathspace /> <mml:mi>B</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{}^eta !B(n,2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that gives the best known bound in Stillman’s conjecture for quadrics when there is no restriction on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, for an ideal <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> generated by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> quadrics, the projective dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R slash upper I> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>R/I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is at most <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 Superscript n plus 1 Baseline left-parenthesis n minus 2 right-parenthesis plus 4> <mml:semantics> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <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:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>2^{n+1}(n - 2) + 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2990582066 title "Strength conditions, small subalgebras, and Stillman bounds in degree ≤4" @default.
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