Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2023, , 128 - 141, 15.06.2023
https://doi.org/10.33205/cma.1252639

Öz

Kaynakça

  • 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.

Beyond Descartes’ rule of signs

Yıl 2023, , 128 - 141, 15.06.2023
https://doi.org/10.33205/cma.1252639

Öz

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.

Kaynakça

  • 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.
Toplam 17 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Vladimir Kostov 0000-0001-5836-2678

Erken Görünüm Tarihi 6 Haziran 2023
Yayımlanma Tarihi 15 Haziran 2023
Yayımlandığı Sayı Yıl 2023

Kaynak Göster

APA Kostov, V. (2023). Beyond Descartes’ rule of signs. Constructive Mathematical Analysis, 6(2), 128-141. https://doi.org/10.33205/cma.1252639
AMA Kostov V. Beyond Descartes’ rule of signs. CMA. Haziran 2023;6(2):128-141. doi:10.33205/cma.1252639
Chicago Kostov, Vladimir. “Beyond Descartes’ Rule of Signs”. Constructive Mathematical Analysis 6, sy. 2 (Haziran 2023): 128-41. https://doi.org/10.33205/cma.1252639.
EndNote Kostov V (01 Haziran 2023) Beyond Descartes’ rule of signs. Constructive Mathematical Analysis 6 2 128–141.
IEEE V. Kostov, “Beyond Descartes’ rule of signs”, CMA, c. 6, sy. 2, ss. 128–141, 2023, doi: 10.33205/cma.1252639.
ISNAD Kostov, Vladimir. “Beyond Descartes’ Rule of Signs”. Constructive Mathematical Analysis 6/2 (Haziran 2023), 128-141. https://doi.org/10.33205/cma.1252639.
JAMA Kostov V. Beyond Descartes’ rule of signs. CMA. 2023;6:128–141.
MLA Kostov, Vladimir. “Beyond Descartes’ Rule of Signs”. Constructive Mathematical Analysis, c. 6, sy. 2, 2023, ss. 128-41, doi:10.33205/cma.1252639.
Vancouver Kostov V. Beyond Descartes’ rule of signs. CMA. 2023;6(2):128-41.