Advertisement

Arnold Mathematical Journal

, Volume 4, Issue 1, pp 113–125 | Cite as

Trace Test

  • Anton Leykin
  • Jose Israel Rodriguez
  • Frank Sottile
Research Contribution
First Online:
Received:
Revised:
Accepted:
  • 40 Downloads

Abstract

The trace test in numerical algebraic geometry verifies the completeness of a witness set of an irreducible variety in affine or projective space. We give a brief derivation of the trace test and then consider it for subvarieties of products of projective spaces using multihomogeneous witness sets. We show how a dimension reduction leads to a practical trace test in this case involving a curve in a low-dimensional affine space.

Keywords

Trace test Witness set Numerical algebraic geometry 

Mathematics Subject Classification

65H10 
This is a preview of subscription content, log in to check access.

References

  1. Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of Algebraic Curves, Vol. I, Grundlehren der Mathematischen Wissenschaften, vol. 267. Springer, New York (1985)Google Scholar
  2. Brake, D. A., Hauenstein, J. D., and Liddell, A. C.: Decomposing solution sets of polynomial systems using derivatives, In: Greuel G.M., Koch T., Paule P., Sommese A. (eds.) Mathematical Software ICMS 2016. ICMS 2016. Lecture Notes in Computer Science, vol. 9725, pp. 127–135. Springer International Publishing, Cham (2016)Google Scholar
  3. Harris, J.: Algebraic Geometry, Graduate Texts in Mathematics, vol. 133. Springer, New York (1992)CrossRefGoogle Scholar
  4. Hauenstein, J.D., and Rodriguez, J.I.: Multiprojective witness sets and a trace test, (2015). arXiv:1507.07069
  5. Hauenstein, J.D., Sommese, A.J.: Witness sets of projections. Appl. Math. Comput. 217(7), 3349–3354 (2010)MathSciNetzbMATHGoogle Scholar
  6. Jouanolou, J.: Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42. Birkhäuser Boston Inc., Boston (1983)zbMATHGoogle Scholar
  7. Lazarsfeld, R.: Positivity in Algebraic Geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 48. Springer, Berlin (2004)Google Scholar
  8. Sommese, A.J., Verschelde, J.: Numerical homotopies to compute generic points on positive dimensional algebraic sets. J. Complex. 16(3), 572–602 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Sommese, A.J., Verschelde, J., Wampler, C.W.: Numerical decomposition of the solution sets of polynomial systems into irreducible components. SIAM J. Numer. Anal. 38(6), 2022–2046 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Sommese, A.J., Verschelde, J., Wampler, C.W.: Symmetric functions applied to decomposing solution sets of polynomial systems. SIAM J. Numer. Anal. 40(6), 2026–2046 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Sommese, A.J., Wampler, C.W.: The Numerical Solution of Systems of Polynomials. World Scientific Publishing Co. Pte. Ltd., Hackensack (2005)CrossRefzbMATHGoogle Scholar

Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2018

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.George Herbert Jones LaboratoryThe University of ChicagoChicagoUSA
  3. 3.Department of MathematicsTexas A&M UniversityTXUSA

Personalised recommendations

Cite article

Buy options