Let F be a field and let C be the algebraic closure of F. Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$w \in M_n(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mo>∈</mo> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be a matrix whose minimal polynomial is of the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$g(x)=(x-a_1)(x-a_2) \in C[x]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi>C</mi> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$a_1 \ne a_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>≠</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d(x):=wx-xw$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mo>=</mo> <mi>w</mi> <mi>x</mi> <mo>-</mo> <mi>x</mi> <mi>w</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$x \in M_n(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Then <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$S_{2n-1}(d(x_1),...,d(x_{2n-1}))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is an identity for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$M_n(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

On a multilinear generalized polynomial identity in $M_n(F)$, where F is a Field

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2026年2月20日

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Marcos Ligiero

摘要

Let F be a field and let C be the algebraic closure of F. Let $w \in M_n(F)$ $w \in M_{n} (F)$ be a matrix whose minimal polynomial is of the form $g(x)=(x-a_1)(x-a_2) \in C[x]$ $g (x) = (x - a_{1}) (x - a_{2}) \in C [x]$ , where $a_1 \ne a_2$ $a_{1} \neq a_{2}$ . Let $d(x):=wx-xw$ $d (x) : = w x - x w$ for all $x \in M_n(F)$ $x \in M_{n} (F)$ . Then $S_{2n-1}(d(x_1),...,d(x_{2n-1}))$ $S_{2 n - 1} (d (x_{1}), . . ., d (x_{2 n - 1}))$ is an identity for $M_n(F)$ $M_{n} (F)$ .

On a multilinear generalized polynomial identity in \(M_n(F)\), where F is a Field

摘要