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• W2023556890 abstract "Let<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R mathematical left-angle x mathematical right-angle><mml:semantics><mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>R</mml:mi></mml:mrow><mml:mo fence=false stretchy=false>⟨<!-- ⟨ --></mml:mo><mml:mi>x</mml:mi><mml:mo fence=false stretchy=false>⟩<!-- ⟩ --></mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>mathbb Rlangle x rangle</mml:annotation></mml:semantics></mml:math></inline-formula>denote the ring of polynomials in<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g><mml:semantics><mml:mi>g</mml:mi><mml:annotation encoding=application/x-tex>g</mml:annotation></mml:semantics></mml:math></inline-formula>freely noncommuting variables<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x equals left-parenthesis x 1 comma ellipsis comma x Subscript g Baseline right-parenthesis><mml:semantics><mml:mrow><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=false>(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…<!-- … --></mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>x=(x_1,dots ,x_g)</mml:annotation></mml:semantics></mml:math></inline-formula>. There is a natural involution<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=asterisk><mml:semantics><mml:mo>∗<!-- ∗ --></mml:mo><mml:annotation encoding=application/x-tex>*</mml:annotation></mml:semantics></mml:math></inline-formula>on<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R mathematical left-angle x mathematical right-angle><mml:semantics><mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>R</mml:mi></mml:mrow><mml:mo fence=false stretchy=false>⟨<!-- ⟨ --></mml:mo><mml:mi>x</mml:mi><mml:mo fence=false stretchy=false>⟩<!-- ⟩ --></mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>mathbb Rlangle x rangle</mml:annotation></mml:semantics></mml:math></inline-formula>determined by<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x Subscript j Superscript asterisk Baseline equals x Subscript j><mml:semantics><mml:mrow><mml:msubsup><mml:mi>x</mml:mi><mml:mi>j</mml:mi><mml:mo>∗<!-- ∗ --></mml:mo></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:annotation encoding=application/x-tex>x_j^*=x_j</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=left-parenthesis p q right-parenthesis Superscript asterisk Baseline equals q Superscript asterisk Baseline p Superscript asterisk><mml:semantics><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mi>p</mml:mi><mml:mi>q</mml:mi><mml:msup><mml:mo stretchy=false>)</mml:mo><mml:mo>∗<!-- ∗ --></mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>q</mml:mi><mml:mo>∗<!-- ∗ --></mml:mo></mml:msup><mml:msup><mml:mi>p</mml:mi><mml:mo>∗<!-- ∗ --></mml:mo></mml:msup></mml:mrow><mml:annotation encoding=application/x-tex>(pq)^*=q^* p^*</mml:annotation></mml:semantics></mml:math></inline-formula>, and a free polynomial<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p element-of double-struck upper R mathematical left-angle x mathematical right-angle><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈<!-- ∈ --></mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi mathvariant=double-struck>R</mml:mi></mml:mrow><mml:mo fence=false stretchy=false>⟨<!-- ⟨ --></mml:mo><mml:mi>x</mml:mi><mml:mo fence=false stretchy=false>⟩<!-- ⟩ --></mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>pin mathbb Rlangle x rangle</mml:annotation></mml:semantics></mml:math></inline-formula>is symmetric if it is invariant under this involution. If<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X equals left-parenthesis upper X 1 comma ellipsis comma upper X Subscript g Baseline right-parenthesis><mml:semantics><mml:mrow><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=false>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…<!-- … --></mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>X=(X_1,dots ,X_g)</mml:annotation></mml:semantics></mml:math></inline-formula>is a<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g><mml:semantics><mml:mi>g</mml:mi><mml:annotation encoding=application/x-tex>g</mml:annotation></mml:semantics></mml:math></inline-formula>tuple of symmetric<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n times n><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>×<!-- × --></mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:annotation encoding=application/x-tex>ntimes n</mml:annotation></mml:semantics></mml:math></inline-formula>matrices, then the evaluation<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p left-parenthesis upper X right-parenthesis><mml:semantics><mml:mrow><mml:mi>p</mml:mi><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>p(X)</mml:annotation></mml:semantics></mml:math></inline-formula>is naturally defined and further<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p Superscript asterisk Baseline left-parenthesis upper X right-parenthesis equals p left-parenthesis upper X right-parenthesis Superscript asterisk><mml:semantics><mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mo>∗<!-- ∗ --></mml:mo></mml:msup><mml:mo stretchy=false>(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>=</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>X</mml:mi><mml:msup><mml:mo stretchy=false>)</mml:mo><mml:mo>∗<!-- ∗ --></mml:mo></mml:msup></mml:mrow><mml:annotation encoding=application/x-tex>p^*(X)=p(X)^*</mml:annotation></mml:semantics></mml:math></inline-formula>. In particular, if<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding=application/x-tex>p</mml:annotation></mml:semantics></mml:math></inline-formula>is symmetric, then<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p left-parenthesis upper X right-parenthesis Superscript asterisk Baseline equals p left-parenthesis upper X right-parenthesis><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>X</mml:mi><mml:msup><mml:mo stretchy=false>)</mml:mo><mml:mo>∗<!-- ∗ --></mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>p</mml:mi><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>p(X)^*=p(X)</mml:annotation></mml:semantics></mml:math></inline-formula>. The main result of this article says if<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding=application/x-tex>p</mml:annotation></mml:semantics></mml:math></inline-formula>is symmetric,<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p left-parenthesis 0 right-parenthesis equals 0><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy=false>(</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>p(0)=0</mml:annotation></mml:semantics></mml:math></inline-formula>and for each<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>and each symmetric positive definite<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n times n><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>×<!-- × --></mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:annotation encoding=application/x-tex>ntimes n</mml:annotation></mml:semantics></mml:math></inline-formula>matrix<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>the set<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartSet upper X colon upper A minus p left-parenthesis upper X right-parenthesis succeeds 0 EndSet><mml:semantics><mml:mrow><mml:mo fence=false stretchy=false>{</mml:mo><mml:mi>X</mml:mi><mml:mo>:</mml:mo><mml:mi>A</mml:mi><mml:mo>−<!-- − --></mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>≻<!-- ≻ --></mml:mo><mml:mn>0</mml:mn><mml:mo fence=false stretchy=false>}</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>{X:A-p(X)succ 0}</mml:annotation></mml:semantics></mml:math></inline-formula>is convex, then<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding=application/x-tex>p</mml:annotation></mml:semantics></mml:math></inline-formula>has degree at most two and is itself convex, or<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=negative p><mml:semantics><mml:mrow><mml:mo>−<!-- − --></mml:mo><mml:mi>p</mml:mi></mml:mrow><mml:annotation encoding=application/x-tex>-p</mml:annotation></mml:semantics></mml:math></inline-formula>is a hermitian sum of squares." @default.
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• W2023556890 date "2014-05-13" @default.
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• W2023556890 title "Quasi-convex free polynomials" @default.
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