Hurwitz: checking decay without solving every polynomial root

What changed

Determinant sign conditions characterize strict left-half-plane roots without separately solving each root.

Scientific scope

Determinant sign conditions characterize strict left-half-plane roots without separately solving each root. 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.

Evidence

Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.

Contributors

• Adolf Hurwitz · SCIENCE: Original source-listed scientific author of this scoped work; byline does not determine effort fractions.

• Hurwitz, negative-real-part root criterion (1895), original 273–284
• ETH Library institutional archive portrait of Adolf Hurwitz
• ETH Automatic Control Laboratory: Stodola/Hurwitz historical paragraph

Provisional WRS assessment, edition 2026-10-16.1. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.

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