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    "title": "Hurwitz: checking decay without solving every polynomial root",
    "summary": "For a real polynomial with positive leading coefficient, positivity of specified coefficient determinants characterizes whether every root has strictly negative real part.\n\nA linear model can grow or decay depending on where its characteristic roots lie. Finding every root can be inconvenient when coefficients contain design parameters. Hurwitz replaces that calculation with a finite list of determinant signs, obtaining a necessary and sufficient test for the stated polynomial problem. The engineering question came from Stodola; the proof develops inherited Cauchy-index and quadratic-form ideas. Earlier stability work, including Routh’s line, prevents treating this as the first conceivable test. Applying the criterion to a physical controller still requires a faithful model; passing a polynomial test alone is not proof that a nonlinear or uncertain machine will be safe.",
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    "year": 1895,
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        "description": "Real-coefficient degree-n polynomial with positive leading coefficient: all roots have strictly negative real part iff the specified Hurwitz determinants are positive. All positive coefficients are necessary, not sufficient by themselves. The general Cauchy-index count first excludes imaginary-axis roots; the final special half-count convention is not a strict-stability guarantee for boundary roots. Finite-degree coefficient mathematics does not certify a nonlinear, uncertain, delayed or discretized plant without its own model/conditions. Original application footnote is author-reported, not independently replayed engineering success. matrix entries, signs, coefficients and full proof not independently certified. Earlier Maxwell/Routh and later equivalent forms are not absorbed as first discoveries; manuscript 1894 differs from publication 1895.",
        "domain": "Control and polynomial stability"
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          "citation": "Hurwitz, negative-real-part root criterion (1895), original273–284",
          "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0046/LOG_0026.pdf"
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      "statement": "# Hurwitz: checking decay without solving every polynomial root\n\nFor a real polynomial with positive leading coefficient, positivity of specified coefficient determinants characterizes whether every root has strictly negative real part.\n\nA linear model can grow or decay depending on where its characteristic roots lie. Finding every root can be inconvenient when coefficients contain design parameters. Hurwitz replaces that calculation with a finite list of determinant signs, obtaining a necessary and sufficient test for the stated polynomial problem. The engineering question came from Stodola; the proof develops inherited Cauchy-index and quadratic-form ideas. Earlier stability work, including Routh’s line, prevents treating this as the first conceivable test. Applying the criterion to a physical controller still requires a faithful model; passing a polynomial test alone is not proof that a nonlinear or uncertain machine will be safe.\n\n## Scientific scope\n\nReal-coefficient degree-n polynomial with positive leading coefficient: all roots have strictly negative real part iff the specified Hurwitz determinants are positive. All positive coefficients are necessary, not sufficient by themselves. The general Cauchy-index count first excludes imaginary-axis roots; the final special half-count convention is not a strict-stability guarantee for boundary roots. Finite-degree coefficient mathematics does not certify a nonlinear, uncertain, delayed or discretized plant without its own model/conditions. Original application footnote is author-reported, not independently replayed engineering success. matrix entries, signs, coefficients and full proof not independently certified. Earlier Maxwell/Routh and later equivalent forms are not absorbed as first discoveries; manuscript 1894 differs from publication 1895.\n\n## People and systems\n\nAdolf Hurwitz\n\n\n\n## Original sources\n\n- [Hurwitz, negative-real-part root criterion (1895), original273–284](https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0046/LOG_0026.pdf)\n- [ETH Library institutional archive portrait of Adolf Hurwitz](https://library.ethz.ch/sammlungen-und-archive/kurzportraets/adolf-hurwitz-1859-1919.html)\n- [ETH Automatic Control Laboratory: Stodola/Hurwitz historical paragraph](https://control.ee.ethz.ch/news-and-events/archived-events/hurwitz-memorial-lecture-series-archive.html)\n",
      "summary": "For a real polynomial with positive leading coefficient, positivity of specified coefficient determinants characterizes whether every root has strictly negative real part.\n\nA linear model can grow or decay depending on where its characteristic roots lie. Finding every root can be inconvenient when coefficients contain design parameters. Hurwitz replaces that calculation with a finite list of determinant signs, obtaining a necessary and sufficient test for the stated polynomial problem. The engineering question came from Stodola; the proof develops inherited Cauchy-index and quadratic-form ideas. Earlier stability work, including Routh’s line, prevents treating this as the first conceivable test. Applying the criterion to a physical controller still requires a faithful model; passing a polynomial test alone is not proof that a nonlinear or uncertain machine will be safe.",
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