Combinatorics of Cremona monomial maps
Authors:
Aron Simis and Rafael H. Villarreal
Journal:
Math. Comp. 81 (2012), 18571867
MSC (2010):
Primary 14E05, 14E07, 15A51, 15A36
Published electronically:
October 24, 2011
MathSciNet review:
2904605
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Abstract: We study Cremona monomial maps using linear algebra, lattice theory and linear optimization methods. Among the results is a simple integer matrix theoretic proof that the inverse of a Cremona monomial map is also defined by monomials of fixed degree, and moreover, the set of monomials defining the inverse can be obtained explicitly in terms of the initial data. We present another method to compute the inverse of a Cremona monomial map based on integer programming techniques and the notion of a Hilbert basis. A neat consequence is drawn for the plane Cremona monomial group, in particular, the known result saying that a plane Cremona monomial map and its inverse have the same degree.
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 M. AlberichCarramiñana, Geometry of the Plane Cremona Maps, Lecture Notes in Mathematics, vol. 1769, 2002, SpringerVerlag, BerlinHeidelberg. MR 1874328 (2002m:14008)
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 G. Cornuéjols, Combinatorial optimization: Packing and covering, CBMSNSF Regional Conference Series in Applied Mathematics 74, SIAM (2001). MR 1828452 (2002e:90004)
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 G. GonzalezSprinberg and I. Pan, On the monomial birational maps of the projective space, An. Acad. Brasil. Ci nc. 75 (2003), no. 2, 129134. MR 1984551 (2004e:14026)
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 A. Simis and R. H. Villarreal, Linear syzygies and birational combinatorics, Results Math. 48 (2005), no. 34, 326343. MR 2215584 (2007a:14020)
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Additional Information
Aron Simis
Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco, 50740540 Recife, Pe, Brazil
Email:
aron@dmat.ufpe.br
Rafael H. Villarreal
Affiliation:
Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14–740, 07000 Mexico City, D.F.
Email:
vila@math.cinvestav.mx
DOI:
http://dx.doi.org/10.1090/S002557182011025561
Received by editor(s):
September 1, 2009
Received by editor(s) in revised form:
April 5, 2011
Published electronically:
October 24, 2011
Additional Notes:
The first author was partially supported by a grant of CNPq. He warmly thanks CINVESTAV for support during a visit. The second author was partially supported by CONACyT grant 49251F and SNI
Article copyright:
© Copyright 2011
American Mathematical Society
