Beyond Descartes’ rule of signs

Year 2023, Volume: 6 Issue: 2, 128 - 141, 15.06.2023

Vladimir Kostov

https://doi.org/10.33205/cma.1252639

Abstract

We consider real univariate polynomials with all roots real. Such a polynomial with c sign changes and p sign preservations in the sequence of its coefficients has c positive and p negative roots counted with multiplicity. Suppose that all moduli of roots are distinct; we consider them as ordered on the positive half-axis. We ask the question: If the positions of the sign changes are known, what can the positions of the moduli of negative roots be? We prove several new results which show how far from trivial the answer to this question is.

Keywords

real polynomial in one variable, hyperbolic polynomial, sign pattern, Descartes’ rule of signs

References

• F. Cajori: A history of the arithmetical methods of approximation to the roots of numerical equations of one unknown quantity. Colo. Coll. Publ. Sci. Ser., 12 (7) (1910), 171–215 .
• D. R. Curtiss: Recent extensions of Descartes’ rule of signs, Ann. of Math., 19 (4) (1918), 251–278.
• J.-P. de Gua de Malves: Démonstrations de la Règle de Descartes, Pour connoître le nombre des Racines positives &négatives dans les Équations qui n’ont point de Racines imaginaires, Memoires de Mathématique et de Physique tirés des registres de l’Académie Royale des Sciences, (1741), 72–96.
• The Geometry of René Descartes with a facsimile of the first edition, translated by D. E. Smith and M.L. Latham, New York, Dover Publications, 1954.
• J. Forsgård, V. P. Kostov and B. Shapiro: Could René Descartes have known this?, Exp. Math., 24 (4) (2015), 438–448.
• J. Forsgård, D. Novikov and B. Shapiro: A tropical analog of Descartes’ rule of signs, Int. Math. Res. Not. IMRN, 2017 (12), 3726–3750.
• J. Fourier: Sur l’usage du théorème de Descartes dans la recherche des limites des racines. Bulletin des sciences par la Société philomatique de Paris, (1820) 156–165, 181–187; oeuvres 2, 291–309, Gauthier-Villars, 1890.
• Y. Gati, V. P. Kostov and M. C. Tarchi: Sign patterns and rigid moduli orders, Grad. J. Math., 6 (1) (2021), 60–72.
• C. F. Gauss: Beweis eines algebraischen Lehrsatzes. J. Reine Angew. Math., 3 (1828), 1-4; Werke 3, 67–70, Göttingen, 1866.
• J. L. W. Jensen: Recherches sur la théorie des équations, Acta Math., 36 (1913), 181–195 .
• V. P. Kostov: Descartes’ rule of signs and moduli of roots, Publ. Math. Debrecen, 96 (1-2) (2020) 161–184,
• V. P. Kostov: Hyperbolic polynomials and canonical sign patterns, Serdica Math. J., 46 (2020) 135–150.
• V. P. Kostov: Which Sign Patterns are Canonical, Results Math., 77 (6) (2022), 235.
• V. P. Kostov: Hyperbolic polynomials and rigid moduli orders, Publ. Math. Debrecen, 100 (1-2) (2022), 119–128,
• V. P. Kostov: The disconnectedness of certain sets defined after uni-variate polynomials, Constr. Math. Anal., 5 (3) (2022), 119–133.
• E. Laguerre: Sur la théorie des équations numériques, Journal de Mathématiques pures et appliquées, s. 3, t. 9, 99–146 (1883); oeuvres 1, Paris, 1898, Chelsea, New-York, 1972, pp. 3–47.
• B. E. Meserve: Fundamental Concepts of Algebra, New York, Dover Publications, 1982.