Revista de la
Unión Matemática Argentina
On the zeros of univariate E-polynomials
María Laura Barbagallo, Gabriela Jeronimo, and Juan Sabia
Volume 65, no. 1 (2023), pp. 33–46    

https://doi.org/10.33044/revuma.2305

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Abstract

We consider two problems concerning real zeros of univariate E-polynomials. First, we prove an explicit upper bound for the absolute values of the zeroes of an E-polynomial defined by polynomials with integer coefficients that improves the bounds known up to now. On the other hand, we extend the classical Budan–Fourier theorem for real polynomials to E-polynomials. This result gives, in particular, an upper bound for the number of real zeroes of an E-polynomial. We show this bound is sharp for particular families of these functions, which proves that a conjecture by D. Richardson is false.

References

  1. M. Barbagallo, G. Jeronimo, and J. Sabia, Zero counting for a class of univariate Pfaffian functions, J. Algebra 452 (2016), 549–573. MR 3461080.
  2. M. Barbagallo, G. Jeronimo, and J. Sabia, Decision problem for a class of univariate Pfaffian functions, Appl. Algebra Engrg. Comm. Comput. (2022). https://doi.org/10.1007/s00200-022-00545-8.
  3. S. Basu, R. M. Pollack, and M.-F. Roy, Algorithms in Real Algebraic Geometry, second edition, Algorithms and Computation in Mathematics, 10, Springer, Berlin, 2006. MR 2248869. Online version available at http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted3.html.
  4. F. D. Budan de Boislaurent, Nouvelle méthode pour la résolution des équations numériques d'un degré quelconque, [second edition], Paris, 1822. Digitized version available at https://gallica.bnf.fr/ark:/12148/bpt6k1108332.
  5. M. Coste, T. Lajous-Loaeza, H. Lombardi, and M.-F. Roy, Generalized Budan-Fourier theorem and virtual roots, J. Complexity 21 (2005), no. 4, 479–486. MR 2152717.
  6. J.-B. J. Fourier, Analyse des équations déterminées, Firmin Didot frères, Paris, 1831. Digitized version available at http://gallica.bnf.fr/ark:/12148/bpt6k1057816b.
  7. A. G. Khovanskii, A class of systems of transcendental equations, Dokl. Akad. Nauk SSSR 255 (1980), no. 4, 804–807. MR 0600749. English translation: Soviet Math. Dokl. 22 (1980), 762–765.
  8. A. G. Khovanskii, Fewnomials, Translations of Mathematical Monographs, 88, Amer. Math. Soc., Providence, RI, 1991. MR 1108621.
  9. A. Maignan, Solving one and two-dimensional exponential polynomial systems, in Proc. ISSAC'98, 215–221, Association for Computing Machinery, New York, 1998. https://doi.org/10.1145/281508.281616.
  10. S. McCallum and V. Weispfenning, Deciding polynomial-transcendental problems, J. Symbolic Comput. 47 (2012), no. 1, 16–31. MR 2854845.
  11. M. Mignotte and D. Ştefănescu, Polynomials. An Algorithmic Approach. Springer Series in Discrete Mathematics and Theoretical Computer Science, Springer, Singapore, 1999. MR 1690362.
  12. D. Richardson, Towards computing non algebraic cylindrical decompositions, in Proc. ISSAC '91, 247–255, Association for Computing Machinery, New York, 1991. https://doi.org/10.1145/120694.120732.
  13. A. Tarski, A Decision Method for Elementary Algebra and Geometry, The Rand Corporation, Santa Monica, CA, 1948. MR 0028796.
  14. L. van den Dries, Exponential rings, exponential polynomials and exponential functions, Pacific J. Math. 113 (1984), no. 1, 51–66. MR 0745594.
  15. H. Wolter, On the “problem of the last root” for exponential terms, Z. Math. Logik Grundlag. Math. 31 (1985), no. 2, 163–168. MR 0786292.