On the Gauss algebra of toric algebras

  • Jürgen HerzogEmail author
  • Raheleh Jafari
  • Abbas Nasrollah Nejad


Let A be a K-subalgebra of the polynomial ring \(S=K[x_1,\ldots ,x_d]\) of dimension d, generated by finitely many monomials of degree r. Then, the Gauss algebra \({\mathbb {G}}(A)\) of A is generated by monomials of degree \((r-1)d\) in S. We describe the generators and the structure of \({\mathbb {G}}(A)\), when A is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree 2, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph G with one loop, the embedding dimension of \({\mathbb {G}}(A)\) is bounded by the complexity of the graph G.


Gauss map Gauss algebra Birational morphism Borel fixed algebra Squarefree Veronese algebra Edge ring 

Mathematics Subject Classification

13C15 14M25 05E40 05C50 14E05 



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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Jürgen Herzog
    • 1
    Email author
  • Raheleh Jafari
    • 2
  • Abbas Nasrollah Nejad
    • 3
  1. 1.Fachbereich MathematikUniversität Duisburg-EssenEssenGermany
  2. 2.Mosaheb Institute of MathematicsKharazmi University, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM)TehranIran
  3. 3.Department of MathematicsInstitute for Advanced Studies in Basic Sciences (IASBS)ZanjanIran

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