@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-SNTKWPJM-D
  skos:prefLabel "polynôme"@fr, "polynomial"@en ;
  a skos:Concept ;
  skos:narrower psr:-BMFQ647R-D .

psr:-BMFQ647R-D
  skos:prefLabel "alternating polynomial"@en, "polynôme alterné"@fr ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Alternating_polynomial> ;
  dc:created "2023-08-18"^^xsd:date ;
  skos:inScheme psr: ;
  skos:broader psr:-SNTKWPJM-D, psr:-LP057SP3-B ;
  skos:definition """In algebra, an <b>alternating polynomial</b> is a polynomial <span class="mwe-math-element"><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(x_{1},\\\\dots ,x_{n})}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <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(x_{1},\\\\dots ,x_{n})}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bf4c0e141cb899fcfb90f39d8eee053074e9331" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.198ex; height:2.843ex;" alt="{\\\\displaystyle f(x_{1},\\\\dots ,x_{n})}"></span> such that if one switches any two of the variables, the polynomial changes sign:  <dl><dd><span class="mwe-math-element"><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(x_{1},\\\\dots ,x_{j},\\\\dots ,x_{i},\\\\dots ,x_{n})=-f(x_{1},\\\\dots ,x_{i},\\\\dots ,x_{j},\\\\dots ,x_{n}).}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <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>j</mi>           </mrow>         </msub>         <mo>,</mo>         <mo>…<!-- … --></mo>         <mo>,</mo>         <msub>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>           </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>         <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>i</mi>           </mrow>         </msub>         <mo>,</mo>         <mo>…<!-- … --></mo>         <mo>,</mo>         <msub>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>j</mi>           </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>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f(x_{1},\\\\dots ,x_{j},\\\\dots ,x_{i},\\\\dots ,x_{n})=-f(x_{1},\\\\dots ,x_{i},\\\\dots ,x_{j},\\\\dots ,x_{n}).}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd00ba4e566c762987e9e06acabe32c9f30e7bb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:61.4ex; height:3.009ex;" alt="{\\\\displaystyle f(x_{1},\\\\dots ,x_{j},\\\\dots ,x_{i},\\\\dots ,x_{n})=-f(x_{1},\\\\dots ,x_{i},\\\\dots ,x_{j},\\\\dots ,x_{n}).}"></span></dd></dl> Equivalently, if one permutes the variables, the polynomial changes in value by the sign of the permutation:  <dl><dd><span class="mwe-math-element"><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\\\\left(x_{\\\\sigma (1)},\\\\dots ,x_{\\\\sigma (n)}\\
ight)=\\\\mathrm {sgn} (\\\\sigma )f(x_{1},\\\\dots ,x_{n}).}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>f</mi>         <mrow>           <mo>(</mo>           <mrow>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mi>σ<!-- σ --></mi>                 <mo stretchy="false">(</mo>                 <mn>1</mn>                 <mo stretchy="false">)</mo>               </mrow>             </msub>             <mo>,</mo>             <mo>…<!-- … --></mo>             <mo>,</mo>             <msub>               <mi>x</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mi>σ<!-- σ --></mi>                 <mo stretchy="false">(</mo>                 <mi>n</mi>                 <mo stretchy="false">)</mo>               </mrow>             </msub>           </mrow>           <mo>)</mo>         </mrow>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="normal">s</mi>           <mi mathvariant="normal">g</mi>           <mi mathvariant="normal">n</mi>         </mrow>         <mo stretchy="false">(</mo>         <mi>σ<!-- σ --></mi>         <mo stretchy="false">)</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>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f\\\\left(x_{\\\\sigma (1)},\\\\dots ,x_{\\\\sigma (n)}\\
ight)=\\\\mathrm {sgn} (\\\\sigma )f(x_{1},\\\\dots ,x_{n}).}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34cfc4411bb83eaff5e6ed1290026b7c8c545bd1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.799ex; height:3.343ex;" alt="{\\\\displaystyle f\\\\left(x_{\\\\sigma (1)},\\\\dots ,x_{\\\\sigma (n)}\\
ight)=\\\\mathrm {sgn} (\\\\sigma )f(x_{1},\\\\dots ,x_{n}).}"></span></dd></dl> More generally, a polynomial <span class="mwe-math-element"><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(x_{1},\\\\dots ,x_{n},y_{1},\\\\dots ,y_{t})}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <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>,</mo>         <msub>           <mi>y</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>1</mn>           </mrow>         </msub>         <mo>,</mo>         <mo>…<!-- … --></mo>         <mo>,</mo>         <msub>           <mi>y</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>t</mi>           </mrow>         </msub>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f(x_{1},\\\\dots ,x_{n},y_{1},\\\\dots ,y_{t})}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a328701fccfd6622c73b7528ab37e8da8fcc1a1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.569ex; height:2.843ex;" alt="{\\\\displaystyle f(x_{1},\\\\dots ,x_{n},y_{1},\\\\dots ,y_{t})}"></span> is said to be <i>alternating in</i> <span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5afdbc2d248d8fa9ba2c4f5188d946a0537e753" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\\\\displaystyle x_{1},\\\\dots ,x_{n}}"></span> if it changes sign if one switches any two of the <span class="mwe-math-element"><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_{i}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle x_{i}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\\\\displaystyle x_{i}}"></span>, leaving the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle y_{j}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>y</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>j</mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle y_{j}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8df4e372390588acb968986cfc388e50b930b3a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.049ex; height:2.343ex;" alt="{\\\\displaystyle y_{j}}"></span> fixed 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Alternating_polynomial">https://en.wikipedia.org/wiki/Alternating_polynomial</a>)"""@en ;
  a skos:Concept .

psr:-LP057SP3-B
  skos:prefLabel "fonction symétrique"@fr, "symmetric function"@en ;
  a skos:Concept ;
  skos:narrower psr:-BMFQ647R-D .