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.
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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)
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)
Harris, J.: Algebraic Geometry, Graduate Texts in Mathematics, vol. 133. Springer, New York (1992)
Hauenstein, J.D., and Rodriguez, J.I.: Multiprojective witness sets and a trace test, (2015). arXiv:1507.07069
Hauenstein, J.D., Sommese, A.J.: Witness sets of projections. Appl. Math. Comput. 217(7), 3349–3354 (2010)
Jouanolou, J.: Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42. Birkhäuser Boston Inc., Boston (1983)
Lazarsfeld, R.: Positivity in Algebraic Geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 48. Springer, Berlin (2004)
Sommese, A.J., Verschelde, J.: Numerical homotopies to compute generic points on positive dimensional algebraic sets. J. Complex. 16(3), 572–602 (2000)
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)
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)
Sommese, A.J., Wampler, C.W.: The Numerical Solution of Systems of Polynomials. World Scientific Publishing Co. Pte. Ltd., Hackensack (2005)
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Research of Leykin supported in part by NSF grants DMS-1151297 and DMS-1719968. Research of Rodriguez supported in part by NSF Grant DMS-1402545. Research of Sottile supported in part by NSF Grant DMS-1501370.
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Leykin, A., Rodriguez, J.I. & Sottile, F. Trace Test. Arnold Math J. 4, 113–125 (2018). https://doi.org/10.1007/s40598-018-0084-3
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DOI: https://doi.org/10.1007/s40598-018-0084-3