What's New in Maple 14: Number Theory
<Text-field style="Heading 1" layout="Heading 1">Description of Feature</Text-field> Efficiency improvements for numtheory[cyclotomic] for large orders. IntegerRelations[PSLQ], an algorithm to find integer linear dependencies between vectors of floating point numbers, was extended to support complex numbers.
<Text-field style="Heading 1" layout="Heading 1">Examples</Text-field> Cyclotomic polynomials: 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The 15th roots of unity 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 NyoiIiIiIiMiIiUiIigiIikiIzYiIzgiIzk= The 15th cyclotomic polynomial has the primitiva 15th roots of unity (corresponding to the blue dots in the diagram above) as roots: 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 KjIsJkkieEc2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiYqJiMiIiMiIzpGJl4jRiZGJkYmSSNQaUdGKkYmISIiRiYsJkYkRiYtRig2IyomKiYjIiIlRjFGJkYyRiZGJkYzRiZGNEYmLCZGJEYmLUYoNiMqJiomIyIiKUYxRiZGMkYmRiZGM0YmRjRGJiwmRiRGJi1GKDYjKiYqJiMiIzlGMUYmRjJGJkYmRjNGJkY0RiYsJkYkRiYtRig2IywkKiYsJEZHRiZGJkYzRiZGNEY0RiYsJkYkRiYtRig2IywkKiYsJEZARiZGJkYzRiZGNEY0RiYsJkYkRiYtRig2IywkKiYsJEY5RiZGJkYzRiZGNEY0RiYsJkYkRiYtRig2IywkKiYsJEYuRiZGJkYzRiZGNEY0RiY= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYkUSdleHBhbmRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiUtRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUSYmUGhpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy9GNVEnbm9ybWFsRictRiM2JC1JI21uR0YkNiVRIzE1RidGQkZFRjQvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y0RkJGRUY0RitGQi8lK2V4ZWN1dGFibGVHRkRGRQ== 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 LDAqJClJInhHNiIiIikiIiJGKCokKUYlIiIoRighIiIqJClGJSIiJkYoRigqJClGJSIiJUYoRiwqJClGJSIiJEYoRihGJUYsRihGKA== The resulting polynomial actually has integer coefficients. This is what the command numtheory[cyclotomic] computes: 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 LDAqJClJInhHNiIiIikiIiJGKCokKUYlIiIoRighIiIqJClGJSIiJkYoRigqJClGJSIiJUYoRiwqJClGJSIiJEYoRihGJUYsRihGKA== 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 JCIkQyIhIiQ= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JFEnZGVncmVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtSSVtc3ViR0YkNiUtRiw2JVEmJlBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvRjBRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYlUScyMzQ1NjdGJ0Y9RkBGLy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRi9GPUZALUYsNiNRIUYnRj0vJStleGVjdXRhYmxlR0Y/RkA= IidbTzo= In Maple 13, this took more than 200 times longer. Integer relations: Suppose we want to compute the minimal polynomial over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM4NDc0O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi9GMkY0 of the following algebraic number: 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 LCYqJCkiIiMjIiIiRiVGJ0YnKiQpISIiI0YnIiIkRidGJw== 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JFEqcG90ZW5jaWFzRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RicvJSdpdGFsaWNHUSZmYWxzZUYnL0YwUSdub3JtYWxGJy8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkstRiw2JFEmZXZhbGZGJ0YvLUkobWZlbmNlZEdGJDYlLUYjNiZGK0Y2LyUrZXhlY3V0YWJsZUdGOEY5RjZGOUY2RlZGOQ== 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 Compute an integer linear combination of these five potencias that is approximately 0: 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 NyciIigiIiMhIiIhIiMiIiI= 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 LCYkIiIjISIqISIiKiYkIiIiISIpRileI0YpRilGJg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYkUShtaW5wb2x5RicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEjOj1GJy8lJ2l0YWxpY0dRJmZhbHNlRicvRjVRJ25vcm1hbEYnLyUmZmVuY2VHRj0vJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUC1GIzYmLUYsNiRRKHVuYXBwbHlGJ0Y0LUY4Ni5RMCZBcHBseUZ1bmN0aW9uO0YnRjtGPkZARkJGREZGRkhGSkZML0ZPUSYwLjBlbUYnL0ZSRmZuLUkobWZlbmNlZEdGJDYlLUYjNidGKy1GIzYnRistRiM2JS1GLDYkUTBQb2x5bm9taWFsVG9vbHNGJ0Y0LUZpbjYnLUYjNiZGKy1GIzYkLUYsNiRRNEZyb21Db2VmZmljaWVudExpc3RGJ0Y0RjRGK0Y0RjtGPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y0RlgtRmluNiUtRiM2Ji1GLDYkUSJ1RidGNC1GODYuUSIsRidGO0Y+RkAvRkNRJXRydWVGJ0ZERkZGSEZKRkxGZW4vRlJRLDAuMzMzMzMzM2VtRictRiw2JFEieEYnRjRGNEY7Rj5GNEZqcEZhcUY0RjtGPkY0RitGNEYrRjsvJStleGVjdXRhYmxlR0Y9Rj4= Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCwiIigiIiIqJiIiI0YrRiRGK0YrKiQpRiRGLUYrISIiKiZGLUYrKUYkIiIkRitGMCokKUYkIiIlRitGK0YlRiVGJQ== Verify that this is actually the minimal polynomial: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JFEqcmFkbm9ybWFsRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JFEobWlucG9seUYnRi8tRjM2JS1GIzYmLUYsNiVRJyYjOTQ1O0YnLyUnaXRhbGljR1EmZmFsc2VGJy9GMFEnbm9ybWFsRidGQS8lK2V4ZWN1dGFibGVHRkNGREZBRkRGQUZGRkRGQUZERkFGRkZE IiIh LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JFEnZmFjdG9yRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JFEobWlucG9seUYnRi8tRjM2JS1GIzYmLUYsNiRRInhGJ0YvLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRkMvRjBRJ25vcm1hbEYnRkFGRkZBRkRGRkZBRkZGQUZERkY= LCwiIigiIiIqJiIiI0YkSSJ4RzYiRiRGJCokKUYnRiZGJCEiIiomRiZGJClGJyIiJEYkRisqJClGJyIiJUYkRiQ= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
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 O que h\303\241 de novo no Maple 14: Teoria dos N\303\272meros
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Descri\303\247\303\243o do recurso</Font></Text-field> Melhorias de efici\303\252ncia para numtheory[cyclotomic] para grandes encomendas. IntegerRelations[PSLQ], um algoritmo para encontrar depend\303\252ncias lineares inteiras entre vetores de n\303\272meros de ponto flutuante foi estendido para suportar n\303\272meros complexos.
<Text-field style="Heading 1" layout="Heading 1">Exemplos</Text-field> Cyclotomic polynomials: 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As 15\302\252 ra\303\255zes da unidade 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 NyoiIiIiIiMiIiUiIigiIikiIzYiIzgiIzk= The 15th cyclotomic polynomial has the primitiva 15th roots of unity (corresponding to the blue dots in the diagram above) as roots: 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LDAqJClJInhHNiIiIikiIiJGKCokKUYlIiIoRighIiIqJClGJSIiJkYoRigqJClGJSIiJUYoRiwqJClGJSIiJEYoRihGJUYsRihGKA== O polin\303\264mio resultante, na verdade, tem coeficientes inteiros. Este \303\251 o comando numtheory[cyclotomic] computa: 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 LDAqJClJInhHNiIiIikiIiJGKCokKUYlIiIoRighIiIqJClGJSIiJkYoRigqJClGJSIiJUYoRiwqJClGJSIiJEYoRihGJUYsRihGKA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYkUSV0aW1lRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JFEnYXNzaWduRidGNC1GODYlLUYjNiktSSNtb0dGJDYuUSInRicvJSdpdGFsaWNHUSZmYWxzZUYnL0Y1USdub3JtYWxGJy8lJmZlbmNlR0ZJLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnLUklbXN1YkdGJDYlLUYsNiVRJiZQaGk7RidGR0ZKLUYjNiQtSSNtbkdGJDYlUScyMzQ1NjdGJ0ZHRkpGNC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkMtRkQ2LlEiPUYnRkdGSkZMRk5GUEZSRlRGVkZYL0ZlblEsMC4yNzc3Nzc4ZW1GJy9GaG5GXXAtRiw2JFErY3ljbG90b21pY0YnRjQtRjg2JS1GIzYmRmJvLUZENi5RIixGJ0ZHRkpGTC9GT1EldHJ1ZUYnRlBGUkZURlZGWC9GZW5GaW4vRmhuUSwwLjMzMzMzMzNlbUYnLUYsNiRRInhGJ0Y0RjRGR0ZKRjRGR0ZKRjRGR0ZKRjRGK0ZHLyUrZXhlY3V0YWJsZUdGSUZK JCIkQyIhIiQ= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JFEnZGVncmVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtSSVtc3ViR0YkNiUtRiw2JVEmJlBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvRjBRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYlUScyMzQ1NjdGJ0Y9RkBGLy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRi9GPUZALUYsNiNRIUYnRj0vJStleGVjdXRhYmxlR0Y/RkA= IidbTzo= No Maple 13, isso demorou mais de 200 vezes. Rela\303\247\303\265es inteiras: Suponha que queremos calcular o volume m\303\255nimo polinomialr LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM4NDc0O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi9GMkY0 do seguinte n\303\272mero alg\303\251brico: 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JFEqcG90ZW5jaWFzRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RicvJSdpdGFsaWNHUSZmYWxzZUYnL0YwUSdub3JtYWxGJy8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkstRiw2JFEmZXZhbGZGJ0YvLUkobWZlbmNlZEdGJDYlLUYjNiZGK0Y2LyUrZXhlY3V0YWJsZUdGOEY5RjZGOUY2RlZGOQ== QyQtSSdydGFibGVHJSpwcm90ZWN0ZWRHNiU7IiIiIiImNyckRigiIiFeJCQiK2lOQDk+ISIqJCIrUVNEZycpISM1XiQkIitoTkA5SEYwJCIrWV5eOkxGMF4kJCIreW41MkZGMCQiK0xxUXEpKUYwXiQkISswKysrREYwJCIrU1FVSz4hIikvSShzdWJ0eXBlRzYkRiVJKF9zeXNsaWJHNiImSSdWZWN0b3JHRkY2I0knY29sdW1uR0ZIUS1fbm90ZXJtaW5hdGVGSA== Calcule uma combina\303\247\303\243o linear inteira dessas cinco pot\303\252ncias que \303\251 aproximadamente 0: 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 LCYkIiIjISIqISIiKiYkIiIiISIpRileI0YpRilGJg== 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 Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCwiIigiIiIqJiIiI0YrRiRGK0YrKiQpRiRGLUYrISIiKiZGLUYrKUYkIiIkRitGMCokKUYkIiIlRitGK0YlRiVGJQ== Verifique se este \303\251 realmente o polin\303\264mio m\303\255nimo: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JFEqcmFkbm9ybWFsRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JFEobWlucG9seUYnRi8tRjM2JS1GIzYmLUYsNiVRJyYjOTQ1O0YnLyUnaXRhbGljR1EmZmFsc2VGJy9GMFEnbm9ybWFsRidGQS8lK2V4ZWN1dGFibGVHRkNGREZBRkRGQUZGRkRGQUZERkFGRkZE IiIh LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JFEnZmFjdG9yRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JFEobWlucG9seUYnRi8tRjM2JS1GIzYmLUYsNiRRInhGJ0YvLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRkMvRjBRJ25vcm1hbEYnRkFGRkZBRkRGRkZBRkZGQUZERkY= LCwiIigiIiIqJiIiI0YkSSJ4RzYiRiRGJCokKUYnRiZGJCEiIiomRiZGJClGJyIiJEYkRisqJClGJyIiJUYkRiQ= JSFH
JSFH