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On solving large systems of polynomial equations appearing In discrete differential geometry

Published: 06 February 2009 Publication History

Abstract

The poster describes methods that combined find all solutions of a very large overdetermined polynomial algebraic system for only up to 22 unknowns but with billions of equations all homogeneous of degree 8 with on average millions of terms each. This problem resulted from the classification of all integrable 3-dimensional scalar discrete quasilinear equations Q3 = 0 on an elementary cubic cell of the lattice Z3.

References

[1]
V.E. Adler, A.I. Bobenko, and Yu.B. Suris. Classification of integrable equations on quad-graphs. The consistency approach. Comm. Math. Phys., 233:513--543, 2003.
[2]
B.G. Konopelchenko and W.K. Schief. Reciprocal figures, graphical statics and inversive geometry of the Schwarzian BKP hierarchy. Stud. Appl. Math., 109:89--124, 2002. also preprint at arXiv: nlin.SI/0107001.
[3]
J.J.C. Nimmo and W.K. Schief. An integrable discretization of a 2 +1-dimensional sine-gordon equation. Stud. Appl. Math., 100:295--309, 1998. also preprint at arXiv: nlin.SI/0107001.
[4]
S.P. Tsarev and T. Wolf. Classification of 3-dimensional integrable scalar discrete equations. Lett in Math Phys, page 19, 2008. also arXiv: 0706.2464.
[5]
T. Wolf. An online tutorial for the package CRACK. http://lie.math.brocku.ca/crack/demo, 2004.
[6]
T. Wolf. On Solving Large Systems of Polynomial Equations Appearing in Discrete Differential Geometry. Programming and Computer Software, 34(2):75--83, 2008. http://lie.math.brocku.ca/twolf/papers/PCS75.pdf.pdf.

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Published In

cover image ACM Communications in Computer Algebra
ACM Communications in Computer Algebra  Volume 42, Issue 3
September 2008
80 pages
ISSN:1932-2232
EISSN:1932-2240
DOI:10.1145/1504347
Issue’s Table of Contents

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 06 February 2009
Published in SIGSAM-CCA Volume 42, Issue 3

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