Skip to main content
SpringerLink
Log in

Optimal Descartes’ rule of signs for systems supported on circuits

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We present an optimal version of Descartes’ rule of signs to bound the number of positive real roots of a sparse system of polynomial equations in n variables with \(n+2\) monomials. This sharp upper bound is given in terms of the sign variation of a sequence associated to the exponents and the coefficients of the system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Bihan, F., Dickenstein, A.: Descartes’ rule of signs for polynomial systems supported on circuits. Int. Math. Res. Not. 22, 6867–6893 (2017)

    MathSciNet  MATH  Google Scholar 

  2. Bihan, F., Dickenstein, A., Giaroli, M.: Lower bounds for positive roots and regions of multistationarity in chemical reaction networks. J. Algebra 542, 367–411 (2020)

    Article  MathSciNet  Google Scholar 

  3. Bihan, F., Sottile, F.: New fewnomial upper bounds from Gale dual polynomial systems. Mosc. Math. J. 7(3), 387–407 (2007)

    Article  MathSciNet  Google Scholar 

  4. El Hilany, B.: Characterization of circuits supporting polynomial systems with the maximal number of positive solutions. Discrete Comput. Geom. 58(2), 355–370 (2017)

    Article  MathSciNet  Google Scholar 

  5. Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants, Mathematics: Theory & Applications. Birkhauser Boston Inc, Boston (1994)

    Book  Google Scholar 

  6. Huber, B., Rambau, J., Santos, F.: The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings. J. Eur. Math. Soc. (JEMS) 2(2), 179–198 (2000)

    Article  MathSciNet  Google Scholar 

  7. Itenberg, I., Roy, M.F.: Multivariate Descartes’ rule. Beitr. Algebra Geom. 37(2), 337–346 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Phillips, G.: Interpolation and Approximation by Polynomials. CMS books in Mathematics. Springer, Berlin (2003)

    Book  Google Scholar 

  9. Pólya, G., Szegő, G.: Problems and Theorems in Analysis, vol. II. Springer, New York (1976)

    Book  Google Scholar 

  10. Sturmfels, B.: Viro’s theorem for complete intersections. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4(3), 377–386 (1994)

    MathSciNet  MATH  Google Scholar 

  11. Sturmfels, B.: On the Newton polytope of the resultant. J. Algebr. Combin. 3, 207–236 (1994)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are grateful to the referee for several interesting comments; in particular, for pointing out that inequality (20) could be interpreted as a discrete version of Rolle’s theorem on \(\mathbb {R}/k \mathbb {Z}\).

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Université de Savoie, 73376, Le Bourget-du-Lac Cedex, France

    Frédéric Bihan

  2. Dto. de Matemática, FCEN, Universidad de Buenos Aires, and IMAS (UBA-CONICET), Ciudad Universitaria, Pab. I, C1428EGA, Buenos Aires, Argentina

    Alicia Dickenstein

  3. Department of Mathematics, One UTSA Circle, University of Texas at San Antonio, San Antonio, TX, 78249, USA

    Jens Forsgård

Authors
  1. Frédéric Bihan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alicia Dickenstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jens Forsgård
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frédéric Bihan.

Additional information

Communicated by Jean-Yves Welschinger.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

AD was partially supported by UBACYT 20020100100242, CONICET PIP 20110100580, and ANPCyT PICT 2013-1110, Argentina. FB was partially supported by grant ANR-18-CE40-0009 “ENUMGEOM” of Agence Nationale de Recherche. JF was funded by SNSF grant #159240 “Topics in tropical and real geometry,” the NCCR SwissMAP project, and NWO-grant TOP1EW.15.313.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bihan, F., Dickenstein, A. & Forsgård, J. Optimal Descartes’ rule of signs for systems supported on circuits. Math. Ann. 381, 1283–1307 (2021). https://doi.org/10.1007/s00208-021-02216-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-021-02216-4

Search

Navigation