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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Grad (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>Der <b>Grad</b> eines <a href="Polynom" title="Polynom">Polynoms</a> in einer Variablen ist in der <a href="Mathematik" title="Mathematik">Mathematik</a> der größte Exponent in dessen Standarddarstellung als Summe von <a href="Monom" title="Monom">Monomen</a>. Beispielsweise ist der Grad des 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 2X^{5}-X^{3}+7X^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>7</mn>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2X^{5}-X^{3}+7X^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c77df4e5ccd8b21d4030e1af61ca8ff03a5a0c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.159ex; height:2.843ex;" alt="{\displaystyle 2X^{5}-X^{3}+7X^{2}}" loading="lazy"></span> gleich 5, nämlich der Exponent des Monoms <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 2X^{5}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2X^{5}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/224aa6d8e58974e401fe1c62aeebbee5cbfd0799.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.214ex; height:2.676ex;" alt="{\displaystyle 2X^{5}}" loading="lazy"></span>. Bei Polynomen in mehreren Variablen ist der Grad eines Monoms definiert als die Summe der Exponenten der enthaltenen Variablenpotenzen und der Grad eines Polynoms (auch <b>Totalgrad</b> genannt) als das <a href="Gr%C3%B6%C3%9Ftes_Element" class="mw-redirect" title="Größtes Element">Maximum</a> der Grade der Monome, aus denen das Polynom besteht. So haben zum Beispiel das Monom <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^{2}Y^{3}Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mi>Z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{2}Y^{3}Z}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2209522e293e2befe7c30aedac010d4389cb9e2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.686ex; height:2.676ex;" alt="{\displaystyle X^{2}Y^{3}Z}" loading="lazy"></span> und damit auch das 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 -3X^{2}Y^{3}Z+7X^{4}Y+XYZ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<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>3</mn>
</mrow>
</msup>
<mi>Z</mi>
<mo>+</mo>
<mn>7</mn>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mi>Y</mi>
<mo>+</mo>
<mi>X</mi>
<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 -3X^{2}Y^{3}Z+7X^{4}Y+XYZ^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6149a1f69e7af33d321ebd01b54776624987b547.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.84ex; height:2.843ex;" alt="{\displaystyle -3X^{2}Y^{3}Z+7X^{4}Y+XYZ^{2}}" loading="lazy"></span> den Grad 6.<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>
</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" class="mw-redirect" title="Kommutativer Ring">kommutativer Ring</a>, <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>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n>0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27a6a5d982d54202a14f111cb8a49210501b2c96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}" loading="lazy"></span> eine <a href="Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürliche Zahl</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},\dots ,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},\dots ,X_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/007f4babedee03ea87429dad95262c1a49217dbb.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},\dots ,X_{n}]}" loading="lazy"></span> der <a href="Polynomring" title="Polynomring">Polynomring</a> in den Variablen <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_{1},\dots ,X_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1},\dots ,X_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38ed92ce88f900210607bbb8f4d66e14d52d7a17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\dots ,X_{n}}" loading="lazy"></span>. 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 0\neq m:=X_{1}^{e_{1}}X_{2}^{e_{2}}\cdots X_{n}^{e_{n}}\in R[X_{1},\dots ,X_{n}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>≠<!-- ≠ --></mo>
<mi>m</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>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msubsup>
<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>
<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 0\neq m:=X_{1}^{e_{1}}X_{2}^{e_{2}}\cdots X_{n}^{e_{n}}\in R[X_{1},\dots ,X_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d2fabdf5a50ca4e7708066ad657c5273e07c357.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.356ex; height:3.176ex;" alt="{\displaystyle 0\neq m:=X_{1}^{e_{1}}X_{2}^{e_{2}}\cdots X_{n}^{e_{n}}\in R[X_{1},\dots ,X_{n}]}" loading="lazy"></span></dd></dl>
<p>ein <a href="Monom" title="Monom">Monom</a> 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 e_{1},\dots ,e_{n}\in \mathbb {N} \cup \{0\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<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>
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e_{1},\dots ,e_{n}\in \mathbb {N} \cup \{0\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41317f844a23f2741fa1bfa3214b04ee28973268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.207ex; height:2.843ex;" alt="{\displaystyle e_{1},\dots ,e_{n}\in \mathbb {N} \cup \{0\}}" loading="lazy"></span>, so ist der Grad von <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 m}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> definiert als
</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 \deg(m):=e_{1}+\ldots +e_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(m):=e_{1}+\ldots +e_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84cb0d509ff677196e5e187c4da8cb9cd390e97a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.926ex; height:2.843ex;" alt="{\displaystyle \deg(m):=e_{1}+\ldots +e_{n}}" loading="lazy"></span>.</dd></dl>
<p>Sei nun
</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 0\neq f=a_{1}m_{1}+\ldots +a_{r}m_{r}\in R[X_{1},\dots ,X_{n}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>≠<!-- ≠ --></mo>
<mi>f</mi>
<mo>=</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mo>…<!-- … --></mo>
<mo>+</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
<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 0\neq f=a_{1}m_{1}+\ldots +a_{r}m_{r}\in R[X_{1},\dots ,X_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72e35a964784b2230e35c448d5e3c338e4e7897c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.836ex; height:2.843ex;" alt="{\displaystyle 0\neq f=a_{1}m_{1}+\ldots +a_{r}m_{r}\in R[X_{1},\dots ,X_{n}]}" loading="lazy"></span></dd></dl>
<p>ein Polynom 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 r\in \mathbb {N} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r\in \mathbb {N} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f7b1b20ccd888e84ebffce5446840cc882b3dc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.567ex; height:2.176ex;" alt="{\displaystyle r\in \mathbb {N} }" 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 a_{1},\dots ,a_{r}\in R\setminus \{0\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mi>R</mi>
<mo class="MJX-variant">∖<!-- ∖ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{1},\dots ,a_{r}\in R\setminus \{0\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d35d7d10d14e56059bee670f0f044c74fa2a726.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.953ex; height:2.843ex;" alt="{\displaystyle a_{1},\dots ,a_{r}\in R\setminus \{0\}}" loading="lazy"></span> und Monomen <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 m_{1},\dots ,m_{r}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m_{1},\dots ,m_{r}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d987c03bcf56483694d31dd9f64d00a579465380.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.287ex; height:2.009ex;" alt="{\displaystyle m_{1},\dots ,m_{r}}" loading="lazy"></span>. Dann ist der Grad oder Totalgrad von <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> definiert als
</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 \deg(f):=\max _{j=1,\dots ,r}\deg(m_{j})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<munder>
<mo movablelimits="true" form="prefix">max</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>r</mi>
</mrow>
</munder>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(f):=\max _{j=1,\dots ,r}\deg(m_{j})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef1f30ff1ba6cf4ca0a4cb40c23f7b5fa306ca98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.314ex; height:4.343ex;" alt="{\displaystyle \deg(f):=\max _{j=1,\dots ,r}\deg(m_{j})}" loading="lazy"></span>.</dd></dl>
<p>Es gibt verschiedene Konventionen zur Definition des Grades von <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 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span>. In der <a href="Algebra" title="Algebra">Algebra</a> ist es üblich, <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(0):=-\infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(0):=-\infty }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f83aefd87213c474e02cffd2fe2afe48f606948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.336ex; height:2.843ex;" alt="{\displaystyle \deg(0):=-\infty }" loading="lazy"></span> zu setzen. Dagegen wird in den Bereichen der Mathematik, die sich mit der Lösung von algebraischen Problemen mit Hilfe von Computern befassen, häufig die Definition <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(0):=-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(0):=-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62834930f495c929490e4e40749a448330dcdc78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.175ex; height:2.843ex;" alt="{\displaystyle \deg(0):=-1}" loading="lazy"></span> bevorzugt.
</p><p>Bemerkung: Da Monome nur aus endlich vielen Faktoren bestehen, lässt sich die Definition des Grads eines Monoms und somit auch die Definition des Grads eines Polynoms direkt auf Polynomringe in beliebig vielen Variablen erweitern.
</p>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<p>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 f,g\in R[X_{1},\dots ,X_{n}]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>,</mo>
<mi>g</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\in R[X_{1},\dots ,X_{n}]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f62f14bb624407ce463f8ffcb158d432399a7ba7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.626ex; height:2.843ex;" alt="{\displaystyle f,g\in R[X_{1},\dots ,X_{n}]}" loading="lazy"></span> Polynome 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>. Dann gilt
</p>
<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 \deg(fg)\leq \deg(f)+\deg(g)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mo>≤<!-- ≤ --></mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(fg)\leq \deg(f)+\deg(g)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/101bcf877876b47480ce38bbce907149411d9e86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.618ex; height:2.843ex;" alt="{\displaystyle \deg(fg)\leq \deg(f)+\deg(g)}" loading="lazy"></span> und</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 \deg(f+g)\leq \max(\deg(f),\deg(g))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo>+</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mo>≤<!-- ≤ --></mo>
<mo movablelimits="true" form="prefix">max</mo>
<mo stretchy="false">(</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(f+g)\leq \max(\deg(f),\deg(g))}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87ac388e46f856f839f8ebb20c347a09e8ae72db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.787ex; height:2.843ex;" alt="{\displaystyle \deg(f+g)\leq \max(\deg(f),\deg(g))}" loading="lazy"></span>.</li></ul>
<p>Für den Fall <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(f)\neq \deg(g)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>≠<!-- ≠ --></mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(f)\neq \deg(g)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8daf5a4b0ed9d279613a531dbcd4757912bf511f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.086ex; height:2.843ex;" alt="{\displaystyle \deg(f)\neq \deg(g)}" loading="lazy"></span> erhält man sogar <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(f+g)=\max(\deg(f),\deg(g))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo>+</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo movablelimits="true" form="prefix">max</mo>
<mo stretchy="false">(</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(f+g)=\max(\deg(f),\deg(g))}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc00ac31eab032eccfd53a7c89e4aa9bf3f16f67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.787ex; height:2.843ex;" alt="{\displaystyle \deg(f+g)=\max(\deg(f),\deg(g))}" loading="lazy"></span>.
</p><p>Ist <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="Integrit%C3%A4tsring" title="Integritätsring">Integritätsring</a>, so gilt sogar
</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 \deg(fg)=\deg(f)+\deg(g)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(fg)=\deg(f)+\deg(g)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13914b30debf3dc7e2139787cac109e625bca7b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.618ex; height:2.843ex;" alt="{\displaystyle \deg(fg)=\deg(f)+\deg(g)}" loading="lazy"></span></dd></dl>
<p>für alle <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\in R[X_{i}\;|\;i\in I]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>,</mo>
<mi>g</mi>
<mo>∈<!-- ∈ --></mo>
<mi>R</mi>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mspace width="thickmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mspace width="thickmathspace"></mspace>
<mi>i</mi>
<mo>∈<!-- ∈ --></mo>
<mi>I</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f,g\in R[X_{i}\;|\;i\in I]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5638e516fb083f0bb31dce639c93c0f0739d954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.803ex; height:2.843ex;" alt="{\displaystyle f,g\in R[X_{i}\;|\;i\in I]}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<p>Betrachte 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,Z]}">
<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>,</mo>
<mi>Z</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X,Y,Z]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8eeb4c4d84cebe105870c9190b3a3ad4c5b0c41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.346ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X,Y,Z]}" loading="lazy"></span> (siehe <a href="Ganze_Zahlen" class="mw-redirect" title="Ganze Zahlen">ganze Zahlen</a>). Es gilt
</p>
<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 \deg(X^{5})=5}">
<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>5</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>5</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(X^{5})=5}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8de71b0c6a113a838676a26e9411072906bef964.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.609ex; height:3.176ex;" alt="{\displaystyle \deg(X^{5})=5}" 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 \deg(X^{2}Y^{3}Z^{4})=2+3+4=9}">
<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>2</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>4</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mn>3</mn>
<mo>+</mo>
<mn>4</mn>
<mo>=</mo>
<mn>9</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(X^{2}Y^{3}Z^{4})=2+3+4=9}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32118a7ca750d6dfa96b9357e3d8775133479cc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.592ex; height:3.176ex;" alt="{\displaystyle \deg(X^{2}Y^{3}Z^{4})=2+3+4=9}" 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 \deg(X^{7}Z^{2}+3X^{3}Y^{3}-XY^{4}Z+5YZ)=\deg(X^{7}Z^{2})=9}">
<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>
<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>3</mn>
</mrow>
</msup>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mi>X</mi>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mi>Z</mi>
<mo>+</mo>
<mn>5</mn>
<mi>Y</mi>
<mi>Z</mi>
<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>7</mn>
</mrow>
</msup>
<msup>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>9</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(X^{7}Z^{2}+3X^{3}Y^{3}-XY^{4}Z+5YZ)=\deg(X^{7}Z^{2})=9}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a4750a1f7f26ce381d7ae0edb9988a3efad784c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.5ex; height:3.176ex;" alt="{\displaystyle \deg(X^{7}Z^{2}+3X^{3}Y^{3}-XY^{4}Z+5YZ)=\deg(X^{7}Z^{2})=9}" loading="lazy"></span> und</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 \deg(3X^{4}Y^{4}-X^{2}Y^{3}Z^{3}+3Y^{4}Z)=\deg(X^{4}Y^{4})=\deg(X^{2}Y^{3}Z^{3})=8}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>−<!-- − --></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>3</mn>
</mrow>
</msup>
<msup>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>3</mn>
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mi>Z</mi>
<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>4</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>
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</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>3</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>8</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(3X^{4}Y^{4}-X^{2}Y^{3}Z^{3}+3Y^{4}Z)=\deg(X^{4}Y^{4})=\deg(X^{2}Y^{3}Z^{3})=8}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5662c746b49f3914f97b314729466d9550722265.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.536ex; height:3.176ex;" alt="{\displaystyle \deg(3X^{4}Y^{4}-X^{2}Y^{3}Z^{3}+3Y^{4}Z)=\deg(X^{4}Y^{4})=\deg(X^{2}Y^{3}Z^{3})=8}" loading="lazy"></span>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Graduierung_(Algebra)" title="Graduierung (Algebra)">Graduierung (Algebra)</a></li>
<li><a href="Bewertungstheorie" class="mw-redirect" title="Bewertungstheorie">Bewertungstheorie</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"><a rel="nofollow" class="external text" href="https://www.spektrum.de/lexikon/mathematik/grad-eines-polynoms/3562">Spektrum.de</a></span>
</li>
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