Combinatorics of Cremona monomial maps

Authors:
Aron Simis and Rafael H. Villarreal

Journal:
Math. Comp. **81** (2012), 1857-1867

MSC (2010):
Primary 14E05, 14E07, 15A51, 15A36

DOI:
https://doi.org/10.1090/S0025-5718-2011-02556-1

Published electronically:
October 24, 2011

MathSciNet review:
2904605

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Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Aron Simis**

Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco, 50740-540 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:
https://doi.org/10.1090/S0025-5718-2011-02556-1

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 49251-F and SNI

Article copyright:
© Copyright 2011
American Mathematical Society