Skip to main content
SpringerLink
Log in

Software for the Gale transform of fewnomial systems and a Descartes rule for fewnomials

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

We give a Descartes’-like bound on the number of positive solutions to a system of fewnomials that holds when its exponent vectors are not in convex position and a sign condition is satisfied. This was discovered while developing algorithms and software for computing the Gale transform of a fewnomial system, which is our main goal. This software is a component of a package we are developing for Khovanskii-Rolle continuation, which is a numerical algorithm to compute the real solutions to a system of fewnomials.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bates, D.J., Bihan, F., Sottile, F.: Bounds on real solutions to polynomial equations. Int. Math. Res. Notes, 2007:rnm114–7 (2007)

  2. Bates, D.J., Hauenstein, J.D., Niemerg, M.E., Sottile, F.: Singular tracking in Khovanskii-Rolle continuation (in preparation)

  3. Bates, D.J., Hauenstein, J.D., Niemerg, M.E., Sottile, F.: galeDuality: software for solving fewnomial systems. Available at www.matthewniemerg.com/software/ (2015)

  4. Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Numerically solving polynomial systems with Bertini. SIAM (2103)

  5. Bates, D.J., Sottile, F.: Khovanskii-Rolle continuation for real solutions. Found. Comput. Math. 11(5), 563–587 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bihan, F., Dickenstein, A.: Descartes’ rule of signs for polynomial systems supported on circuits. In preparation (2015)

  7. Bihan, F., Spaenlehauer, P.-J.: Sparse polynomial systems with many positive solutions from bipartite simplicial complexes. arXiv:1510.05622 (2015)

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

    MathSciNet  MATH  Google Scholar 

  9. Bihan, F., Sottile, F.: Gale duality for complete intersections. Ann. Inst. Fourier (Grenoble) 58(3), 877–891 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Craciun, G., García-Puente, L., Sottile, F.: Some geometrical aspects of control points for toric patches. In: Mathematical methods for curves and surfaces, volume 5862 of lecture notes in computer science, pp. 111–135. Springer, Berlin (2010)

    Google Scholar 

  11. Georg, K.: Improving the efficiency of exclusion algorithms. Adv. Geom. 1(2), 193–210 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Georg, K.: A new exclusion test. J. Comput. Appl. Math. 152(1–2), 147–160 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hauenstein, J.D.: Numerically computing real points on algebraic sets. Acta applicandae mathematicae 125(1), 105–119 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hauenstein, J.D., Sottile, F.: Algorithm 921: alphaCertified: Certifying Solutions to Polynomial Systems. ACM Trans. Math. Softw. 38(4), 28 (2012)

    Article  MathSciNet  Google Scholar 

  15. Hay, L., Joseph, R. D.: The number of orthants in n-space intersected by an s-dimensional subspace. Technical Memorandum 8, Project PARA, Cornell Aeronautical Laboratory (1960)

  16. Khovanskii, A.G.: Fewnomials. Trans. Math. Monographs, AMS 88 (1991)

  17. Lasserre, J.B., Laurent, M., Rostalski, P.: Semidefinite characterization and computation of zero-dimensional real radical ideals. Found. Comput. Math. 8(5), 607–647 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lenstra, A.K., Lenstra, H.W. Jr., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Meintjes, K., Morgan, A.P.: A methodology for solving chemical equilibrium systems. Appl. Math. Comput. 22(4), 333–361 (1987)

    MathSciNet  MATH  Google Scholar 

  20. Müller, S., Feliu, E., Regensburger, G., Conradi, C., Shiu, A., Dickenstein, A.: Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. arXiv:1311.5493 (2013)

  21. Rouillier, F., Roy, M.-F., Safey El Din, M.: Finding at least one point in each connected component of a real algebraic set defined by a single equation. J. Complexity 16(4), 716–750 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rouillier, F.: Solving Zero-dimensional systems through the rational univariate representation. Appl. Algebra Engrg. Comm. Comput. 9(5), 433–461 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sommese, A.J., Wampler, C.W., II: The numerical solution of systems of polynomials arising in engineering and science. World Scientific Publishing Co. Pte. Ltd., Hackensack (2005)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523-1874, USA

    Daniel J. Bates

  2. Department of Applied & Computational Mathematics & Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Jonathan D. Hauenstein

  3. Fields Institute for Research in Mathematical Sciences, University of Toronto, 222 College Street, Toronto, ON, M5T 3J1, Canada

    Matthew E. Niemerg

  4. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Frank Sottile

Authors
  1. Daniel J. Bates
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jonathan D. Hauenstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthew E. Niemerg
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Frank Sottile
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frank Sottile.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bates, D.J., Hauenstein, J.D., Niemerg, M.E. et al. Software for the Gale transform of fewnomial systems and a Descartes rule for fewnomials. Numer Algor 73, 281–304 (2016). https://doi.org/10.1007/s11075-015-0095-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-015-0095-2

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation