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Noncommutative Cauchy Bound and Noncommutative Montel Bound for Roots of Polynomials

Primarily human-written textHuman understanding: all partsmath.OA — Operator Algebrasmath.CV — Complex Variablesmath.FA — Functional Analysis

Contributed by K. MAHESH KRISHNA ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

In 1829, Cauchy derived an upper bound for every root of a complex polynomial using the maximum of the absolute values of the coefficients. In 1931, Montel derived an upper bound using the sum of the absolute values of the coefficients. We derive noncommutative versions of the Cauchy and Montel bounds.

References

  1. A. L. Cauchy. Sur la resolution des equations numeriques et sur la theorie de l'elimination. Oeuvres , 9(2):87--161, 1829.
  2. Holly P. Hirst and Wade T. Macey. Bounding the roots of polynomials. The College Mathematics Journal , 28(4):292--295, 1997.
  3. Morris Marden. Geometry of polynomials. 2nd ed. , volume 3 of Math. Surv. American Mathematical Society (AMS), Providence, RI, 1966.
  4. P. Montel. Sur la limite sup é rieure des modules des z é ros des polynomes. C. R. Acad. Sci., Paris , 193:974--976, 1931.
  5. Q. I. Rahman and G. Schmeisser. Analytic theory of polynomials , volume 26 of Lond. Math. Soc. Monogr., New Ser. Oxford: Oxford University Press, 2002.
  6. M. Takesaki. Theory of operator algebras I . , volume 124 of Encycl. Math. Sci. Berlin: Springer, 2002.

Version history

  1. v1Submitted by K. MAHESH KRISHNAInitial depositCurrentNov 01, 2026