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• W2029041540 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F colon double-struck upper C Superscript n Baseline right-arrow double-struck upper C Superscript n> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>F:mathbb {C}^n rightarrow mathbb {C}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a polynomial mapping in Yagzhev form, i.e. <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis equals left-parenthesis x 1 plus upper H 1 left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis comma ellipsis comma x Subscript n Baseline plus upper H Subscript n Baseline left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis right-parenthesis comma> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <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>n</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <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:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <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>n</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <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>n</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>F(x_1,ldots ,x_n)=(x_1+H_1(x_1,ldots ,x_n),ldots ,x_n+H_n(x_1,ldots ,x_n)),</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Subscript i> <mml:semantics> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>H_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are homogeneous polynomials of degree 3. We show that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper J normal a normal c left-parenthesis upper F right-parenthesis element-of double-struck upper C Superscript asterisk> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>J</mml:mi> <mml:mi mathvariant=normal>a</mml:mi> <mml:mi mathvariant=normal>c</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>mathrm {Jac}(F) in mathbb {C}^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Jacobian matrix of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding=application/x-tex>F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is symmetric, then the polynomials <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x Subscript i Baseline plus upper H Subscript i Baseline left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <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>n</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>x_i+H_i(x_1,ldots ,x_n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are irreducible as elements of the ring <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper C left-bracket x 1 comma ellipsis comma x Subscript n Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <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>n</mml:mi> </mml:msub> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {C}[x_1,ldots ,x_n]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W2029041540 created "2016-06-24" @default.
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• W2029041540 date "2010-03-10" @default.
• W2029041540 modified "2023-09-27" @default.
• W2029041540 title "The irreducibility of symmetric Yagzhev maps" @default.
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