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Toric P-difference varieties. (English) Zbl 1453.12007
Summary: In this paper, we introduce the concept of P-difference varieties and study the properties of toric P-difference varieties. Toric P-difference varieties are analogues of toric varieties in difference algebraic geometry. The category of affine toric P-difference varieties with toric morphisms is shown to be antiequivalent to the category of affine \(P[x]\)-semimodules with \(P[x]\)-semimodule morphisms. Moreover, there is a one-to-one correspondence between the irreducible invariant P-difference subvarieties of an afne toric P-difference variety and the faces of the corresponding affine \(P[x]\)-semimodule. We also define abstract toric P-difference varieties by gluing affine toric P-difference varieties. The irreducible invariant P-difference subvariety-face correspondence is generalized to abstract toric P-difference varieties. By virtue of this correspondence, a divisor theory for abstract toric P-difference varieties is developed.
12H10 Difference algebra
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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[1] Cohn, R. M., Difference Algebra (1965), New York: Interscience Publishers, New York
[2] Cox, D.; Little, J.; Schenck, H., Toric Varieties (2010), New York: Springer-Verlag, New York
[3] Fulton, W., Introduction to Toric Varieties (1993), Princeton: Princeton University Press, Princeton
[4] Gao, X. S.; Huang, Z.; Wang, J., Toric difference variety, J Syst Sci Complex, 30, 173-195 (2017) · Zbl 1407.14049
[5] Gelfand, I. M.; Kapranov, M.; Zelevinsky, A., Discriminants, Resultants and Multidimensional Determinants (1994), Boston: Birkhäauser, Boston · Zbl 0827.14036
[6] Hartshorne, R., Algebraic Geometry (1977), New York: Springer-Verlag, New York
[7] Hrushovski, E., The elementary theory of the Frobenius automorphisms (2012)
[8] Jing, R. J.; Yuan, C. M., A modular algorithm to compute the generalized Hermite normal form for ℤ[x]-lattices, J Symbolic Comput, 81, 97-118 (2017) · Zbl 1357.13028
[9] Levin, A., Difference Algebra (2008), New Work: Springer-Verlag, New Work
[10] Li, W.; Li, Y. H., Difference chow form, J Algebra, 428, 67-90 (2015) · Zbl 1349.12004
[11] Li, W.; Yuan, C. M.; Gao, X. S., Sparse difference resultant, J Symbolic Comput, 68, 169-203 (2015) · Zbl 1328.65266
[12] Oda, T., Convex Bodies and Algebraic Geometry (1988), New York: Springer, New York
[13] Ritt, J. F.; Doob, J. L., Systems of algebraic difference equations, Amer J Math, 55, 505-514 (1933) · JFM 59.0456.01
[14] Rotman, J. J., An Introduction to Homological Algebra (2008), New Work: Springer-Verlag, New Work · Zbl 0441.18018
[15] Wibmer, M., Algebraic difference equations (2013)
[16] Wibmer, M., Affine difference algebraic groups (2014)
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