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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Homogenes Polynom</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Ein <a href="Polynom#Polynome_in_mehreren_Unbestimmten" title="Polynom">(multivariables) Polynom</a> heißt <b>homogen</b>, falls alle <a href="Monom" title="Monom">Monome</a>, aus denen das Polynom besteht, den gleichen <a href="Grad_(Polynom)" title="Grad (Polynom)">Grad</a> haben. Homogene Polynome werden auch als <b><a href="Form_(Algebra)" title="Form (Algebra)">Formen</a></b> bezeichnet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> ein <a href="Kommutativer_Ring_mit_Eins" class="mw-redirect" title="Kommutativer Ring mit Eins">kommutativer Ring mit Eins</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[X_{1},\dotsc ,X_{n}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R[X_{1},\dotsc ,X_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a163f23c4ea9575b3da5875cedfae1da278689b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.357ex; height:2.843ex;" alt="{\displaystyle R[X_{1},\dotsc ,X_{n}]}" loading="lazy"></span> der <a href="Polynomring" title="Polynomring">Polynomring</a> über <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> Unbestimmten. Ein Monom ist dann ein Polynom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in R[X_{1},\dotsc ,X_{n}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>∈<!-- ∈ --></mo>
<mi>R</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p\in R[X_{1},\dotsc ,X_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35799dc90ab6fdb914c24d736d2555566d520289.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:18.457ex; height:2.843ex;" alt="{\displaystyle p\in R[X_{1},\dotsc ,X_{n}]}" loading="lazy"></span>, für das ein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo>∈<!-- ∈ --></mo>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha \in R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b09907bd4a5d9f895bddca8cc8d829c3e214b5e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.092ex; height:2.176ex;" alt="{\displaystyle \alpha \in R}" loading="lazy"></span> mit
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\alpha X_{1}^{i_{1}}\cdot \dots \cdot X_{n}^{i_{n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mi>α<!-- α --></mi>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msubsup>
<mo>⋅<!-- ⋅ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>⋅<!-- ⋅ --></mo>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p=\alpha X_{1}^{i_{1}}\cdot \dots \cdot X_{n}^{i_{n}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69e62b82a4d28f8cfb983b0a781830c269581406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:19.316ex; height:3.509ex;" alt="{\displaystyle p=\alpha X_{1}^{i_{1}}\cdot \dots \cdot X_{n}^{i_{n}}}" loading="lazy"></span></dd></dl>
<p>existiert. Der Grad dieses Monoms ist
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {deg} (p)=i_{1}+\dotsb +i_{n}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">g</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>+</mo>
<msub>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {deg} (p)=i_{1}+\dotsb +i_{n}.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b0a57c2815fe86f0f3742a711647dca051bb46e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.493ex; height:2.843ex;" alt="{\displaystyle \mathrm {deg} (p)=i_{1}+\dotsb +i_{n}.}" loading="lazy"></span></dd></dl>
<p>Ein Polynom in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[X_{1},\dotsc ,X_{n}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R[X_{1},\dotsc ,X_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a163f23c4ea9575b3da5875cedfae1da278689b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.357ex; height:2.843ex;" alt="{\displaystyle R[X_{1},\dotsc ,X_{n}]}" loading="lazy"></span> wird homogen genannt, wenn es eine Summe von Monomen gleichen Grades ist.
</p>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<ul><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in R[X_{1},\dotsc ,X_{n}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>∈<!-- ∈ --></mo>
<mi>R</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\in R[X_{1},\dotsc ,X_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d8aff92116d80288377baa4491a48dc79291745.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.476ex; height:2.843ex;" alt="{\displaystyle f\in R[X_{1},\dotsc ,X_{n}]}" loading="lazy"></span> ist genau dann homogen vom Grad <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>, wenn in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[X_{1},\dotsc ,X_{n}][T]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
<mo stretchy="false">[</mo>
<mi>T</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R[X_{1},\dotsc ,X_{n}][T]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a78586955e9c081dc160dd42c9248f96a44c868e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.287ex; height:2.843ex;" alt="{\displaystyle R[X_{1},\dotsc ,X_{n}][T]}" loading="lazy"></span> gilt:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(TX_{1},\dotsc ,TX_{n})=T^{k}\cdot f(X_{1},\dotsc ,X_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>T</mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>T</mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(TX_{1},\dotsc ,TX_{n})=T^{k}\cdot f(X_{1},\dotsc ,X_{n})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8a1a4cc24806cb73d94e580077ede5f209edc92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.633ex; height:3.176ex;" alt="{\displaystyle f(TX_{1},\dotsc ,TX_{n})=T^{k}\cdot f(X_{1},\dotsc ,X_{n})}" loading="lazy"></span></dd></dl></li>
<li>Seien <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> ein Integritätsring und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g,h\in R[X_{1},\dotsc ,X_{n}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>,</mo>
<mi>g</mi>
<mo>,</mo>
<mi>h</mi>
<mo>∈<!-- ∈ --></mo>
<mi>R</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f,g,h\in R[X_{1},\dotsc ,X_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e4fec1e3415804d45544c9c7bfb275e30975b2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.999ex; height:2.843ex;" alt="{\displaystyle f,g,h\in R[X_{1},\dotsc ,X_{n}]}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=gh}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>=</mo>
<mi>g</mi>
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f=gh}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c45f7ab876ebaa12a47a36917f6f9523291d827b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.832ex; height:2.509ex;" alt="{\displaystyle f=gh}" loading="lazy"></span>. Dann gilt die Implikation
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> sind homogen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \implies }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thickmathspace"></mspace>
<mo stretchy="false">⟹<!-- ⟹ --></mo>
<mspace width="thickmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \implies }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/913c2e89ea94dfa446f69b056d4bf505e01fcc5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.096ex; height:1.843ex;" alt="{\displaystyle \implies }" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> ist homogen.</dd></dl></li>
<li>Fordert man zusätzlich <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,h\neq 0\in R[X_{1},\dotsc ,X_{n}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>,</mo>
<mi>h</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
<mo>∈<!-- ∈ --></mo>
<mi>R</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g,h\neq 0\in R[X_{1},\dotsc ,X_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5df103679c74761f576c3293b7272b9919816e29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.948ex; height:2.843ex;" alt="{\displaystyle g,h\neq 0\in R[X_{1},\dotsc ,X_{n}]}" loading="lazy"></span>, ist sogar die Äquivalenz erfüllt:
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> sind homogen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Longleftrightarrow }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">⟺<!-- ⟺ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Longleftrightarrow }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74a3a1a17366e695966bae38466f8466653a43f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.317ex; height:1.843ex;" alt="{\displaystyle \Longleftrightarrow }" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> ist homogen.</dd></dl></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<ul><li>Jedes Monom ist homogen.</li>
<li>Die Menge aller homogenen Polynome in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[X]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R[X]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afeccd52deabb878398a8485755c3ceea80caf9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle R[X]}" loading="lazy"></span>, dem Polynomring in einer Variablen über <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>, ist gegeben durch
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{aX^{n}\;\mid \;a\in R,\;n\in \mathbb {N} \cup \{0\}\}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mi>a</mi>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mspace width="thickmathspace"></mspace>
<mo>∣<!-- ∣ --></mo>
<mspace width="thickmathspace"></mspace>
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<mi>R</mi>
<mo>,</mo>
<mspace width="thickmathspace"></mspace>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
<mo>∪<!-- ∪ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
<mo fence="false" stretchy="false">}</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{aX^{n}\;\mid \;a\in R,\;n\in \mathbb {N} \cup \{0\}\}.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4f4860a091396e4ab5078376c521bedd7c4a7dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.141ex; height:2.843ex;" alt="{\displaystyle \{aX^{n}\;\mid \;a\in R,\;n\in \mathbb {N} \cup \{0\}\}.}" loading="lazy"></span></dd></dl></li>
<li>Einfache Beispiele für homogene Polynome in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [X,Y]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X,Y]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc3c6bf9e1d05d96bc2ca6ee873498ac705b2516.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.631ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X,Y]}" loading="lazy"></span> (siehe <a href="Ganze_Zahl" title="Ganze Zahl">ganze Zahlen</a>):
<ul><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{4}-Y^{4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{4}-Y^{4}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab1c84ba6e6ab634e6a09ca71739bc0d3ecfe78a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.846ex; height:2.843ex;" alt="{\displaystyle X^{4}-Y^{4}}" loading="lazy"></span> ist homogen wegen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(X^{4})=\deg(Y^{4})=4.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(X^{4})=\deg(Y^{4})=4.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/590ee82f7d2d82b14715ad7fa4cc018e9809e7aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.605ex; height:3.176ex;" alt="{\displaystyle \deg(X^{4})=\deg(Y^{4})=4.}" loading="lazy"></span></li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{7}+5X^{3}Y^{4}+XY^{6}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>5</mn>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>+</mo>
<mi>X</mi>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{7}+5X^{3}Y^{4}+XY^{6}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f062fd09b92a233ab6f1b7ffaadba8ae01da476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.834ex; height:2.843ex;" alt="{\displaystyle X^{7}+5X^{3}Y^{4}+XY^{6}}" loading="lazy"></span> ist homogen wegen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(X^{7})=\deg(X^{3}Y^{4})=\deg(XY^{6})=7.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>7.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(X^{7})=\deg(X^{3}Y^{4})=\deg(XY^{6})=7.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8fa7479b72435d4d4b7e24b449f0abfc7c693a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.986ex; height:3.176ex;" alt="{\displaystyle \deg(X^{7})=\deg(X^{3}Y^{4})=\deg(XY^{6})=7.}" loading="lazy"></span></li></ul></li>
<li>Beispiele für nicht-homogene Polynome in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} [X,Y,Z]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Q</mi>
</mrow>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo>,</mo>
<mi>Y</mi>
<mo>,</mo>
<mi>Z</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} [X,Y,Z]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4b14b09a3dbd8b84b25a883655911563840292e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.603ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} [X,Y,Z]}" loading="lazy"></span> (siehe <a href="Rationale_Zahl" title="Rationale Zahl">rationale Zahlen</a>):
<ul><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{4}Z-{\frac {3}{4}}YZ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mi>Z</mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
</mrow>
<mi>Y</mi>
<msup>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{4}Z-{\frac {3}{4}}YZ^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8528aeffd1d76260b3556c51257f91432decd7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.106ex; height:5.176ex;" alt="{\displaystyle X^{4}Z-{\frac {3}{4}}YZ^{2}}" loading="lazy"></span> ist nicht homogen wegen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(X^{4}Z)=5\neq 3=\deg(YZ^{2}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mi>Z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>5</mn>
<mo>≠<!-- ≠ --></mo>
<mn>3</mn>
<mo>=</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<msup>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(X^{4}Z)=5\neq 3=\deg(YZ^{2}).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0c51b0472f51988d43e9d82b4bb1e4de510f998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.128ex; height:3.176ex;" alt="{\displaystyle \deg(X^{4}Z)=5\neq 3=\deg(YZ^{2}).}" loading="lazy"></span></li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{3}Y^{3}Z^{2}-3X^{2}Y^{6}-{\frac {7}{3}}Y^{5}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<msup>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>3</mn>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>7</mn>
<mn>3</mn>
</mfrac>
</mrow>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{3}Y^{3}Z^{2}-3X^{2}Y^{6}-{\frac {7}{3}}Y^{5}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fcc4188d69d520cd48a747b37cf22b7153184c13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.57ex; height:5.176ex;" alt="{\displaystyle X^{3}Y^{3}Z^{2}-3X^{2}Y^{6}-{\frac {7}{3}}Y^{5}}" loading="lazy"></span> ist nicht homogen wegen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(X^{3}Y^{3}Z^{2})=\deg(X^{2}Y^{6})=8}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<msup>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>8</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(X^{3}Y^{3}Z^{2})=\deg(X^{2}Y^{6})=8}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad818c78c239b72f2aeff1f68b87ce2f98289229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.726ex; height:3.176ex;" alt="{\displaystyle \deg(X^{3}Y^{3}Z^{2})=\deg(X^{2}Y^{6})=8}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(Y^{5})=5.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>5.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(Y^{5})=5.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c5c3efa88169f4970fe0afad329979c562f87f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.159ex; height:3.176ex;" alt="{\displaystyle \deg(Y^{5})=5.}" loading="lazy"></span></li></ul></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Graduierung">Graduierung</h2></div>
<p>Jedes Polynom lässt sich auf eindeutige Weise als Summe von homogenen Polynomen verschiedenen Grades schreiben, indem man alle Monome gleichen Grades zusammenfasst. Der Polynomring lässt sich also als eine <a href="Direkte_Summe" title="Direkte Summe">direkte Summe</a> schreiben:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[X_{1},\dotsc ,X_{n}]=\bigoplus _{d\geq 0}A_{d},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">]</mo>
<mo>=</mo>
<munder>
<mo>⨁<!-- ⨁ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mrow>
</munder>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R[X_{1},\dotsc ,X_{n}]=\bigoplus _{d\geq 0}A_{d},}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b0fe2bdf1b4ee123b05b4288b14bbd4229ed25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.835ex; height:5.843ex;" alt="{\displaystyle R[X_{1},\dotsc ,X_{n}]=\bigoplus _{d\geq 0}A_{d},}" loading="lazy"></span></dd></dl>
<p>wobei
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{d}=\bigoplus _{e_{1}+\dotsb +e_{n}=d,\ e_{i}\geq 0}R\cdot X_{1}^{e_{1}}\cdot \dots \cdot X_{n}^{e_{n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo>⨁<!-- ⨁ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>+</mo>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>d</mi>
<mo>,</mo>
<mtext> </mtext>
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mrow>
</munder>
<mi>R</mi>
<mo>⋅<!-- ⋅ --></mo>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msubsup>
<mo>⋅<!-- ⋅ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>⋅<!-- ⋅ --></mo>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{d}=\bigoplus _{e_{1}+\dotsb +e_{n}=d,\ e_{i}\geq 0}R\cdot X_{1}^{e_{1}}\cdot \dots \cdot X_{n}^{e_{n}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34299b2bc194e51ba24db4b386a9a2d24a41b3a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.112ex; height:5.843ex;" alt="{\displaystyle A_{d}=\bigoplus _{e_{1}+\dotsb +e_{n}=d,\ e_{i}\geq 0}R\cdot X_{1}^{e_{1}}\cdot \dots \cdot X_{n}^{e_{n}}}" loading="lazy"></span></dd></dl>
<p>die Menge der homogenen Polynome vom Grad <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span> zusammen mit dem Nullpolynom ist. Es gilt
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{d}\cdot A_{d'}\subseteq A_{d+d'},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>d</mi>
<mo>′</mo>
</msup>
</mrow>
</msub>
<mo>⊆<!-- ⊆ --></mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
<mo>+</mo>
<msup>
<mi>d</mi>
<mo>′</mo>
</msup>
</mrow>
</msub>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{d}\cdot A_{d'}\subseteq A_{d+d'},}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be0062964490a99d7ff85d4a56ecfd3ce2ae0a7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.134ex; height:2.843ex;" alt="{\displaystyle A_{d}\cdot A_{d'}\subseteq A_{d+d'},}" loading="lazy"></span></dd></dl>
<p>der Polynomring ist also ein <a href="Graduierter_Ring" title="Graduierter Ring">graduierter Ring</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerung">Verallgemeinerung</h2></div>
<p>Allgemein heißen in einem graduierten Ring
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigoplus _{d\geq 0}A_{d}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo>⨁<!-- ⨁ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mrow>
</munder>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \bigoplus _{d\geq 0}A_{d}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c59b81f0284fbc5d028ce0671babd62c4b959ea3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:6.733ex; height:5.843ex;" alt="{\displaystyle \bigoplus _{d\geq 0}A_{d}}" loading="lazy"></span></dd></dl>
<p>die Elemente aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{d}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{d}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e4c67aa7c46d8276c9805b1036cd4c2e9887d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.835ex; height:2.509ex;" alt="{\displaystyle A_{d}}" loading="lazy"></span> homogen vom Grad <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Homogene_Funktion" title="Homogene Funktion">Homogene Funktion</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Fischer: <i>Lehrbuch der Algebra.</i> 2013, S. 169, Lemma.</span>
</li>
</ol>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Gerd_Fischer_(Mathematiker)" title="Gerd Fischer (Mathematiker)">Gerd Fischer</a>: <i>Lehrbuch der Algebra.</i> 3. Auflage. Springer, Wiesbaden 2013, ISBN 978-3-658-02220-4, S. 169.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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