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**Toric ideals for high Veronese subrings of toric algebras.**
*(English)*
Zbl 1273.13050

Veronese subrings and Koszul algebras are very interesting topics. In a nice paper [Adv. Math. 109, No. 2, 168–187 (1994; Zbl 0839.13013)], D. Eisenbud, A. Reeves and B. Totaro proved that if \(S\) is a graded polynomial ring over a field \(K\), and \(I\) a homogeneous ideal, then up to generic coordinates the initial ideal of \(n\)-Veronese transform of the ideal \(I\), \(\text{in}(V_n(I)) \) is generated by quadratic monomials for \(n \) large enough. An effective bound was given in terms of the Castelnuovo-Mumford regularity of \(I\). In the paper under review, the author considers a toric standard graded ideal \(I\) in a polynomial ring, and proves that for some monomial order \(\prec \), he can define a monomial order such that \(\text{in}(V_n(I)) \) is generated by quadratic monomials for \(n \) large enough, a bound is given in terms of \(\text{in}_{\prec }(I), \) this result seems to be effective and follows from a more general result.

Reviewer: Marcel Morales (Saint-Martin-d’Hères)

### MSC:

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |

### Citations:

Zbl 0839.13013### References:

[1] | J. Backelin, On the rates of growth of the homologies of Veronese subrings , Algebra, algebraic topology, and their interactions (J.-E. Roos, ed.), Springer Lect. Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 79-100. · Zbl 0588.13011 |

[2] | S. Barcanescu and N. Manolache, Betti numbers of Segré-Veronese singularities , Rev. Roumaine Math. Pures Appl. 26 (1982), 549-565. · Zbl 0465.13006 |

[3] | W. Bruns, J. Gubeladze and N. V. Trung, Normal polytopes, triangulations, and Koszul algebras , J. Reine Angew. Math. 485 (1997), 123-160. · Zbl 0866.20050 |

[4] | D. Cox, J. Little and D. O’Shea, Ideals, varieties and algorithms , Springer-Verlag, New York, 1992. |

[5] | D. Cox, J. Little and D. O’Shea, Using algebraic geometry , Springer-Verlag, New York, 1998. · Zbl 0920.13026 |

[6] | E. De Negri, Toric rings generated by special stable sets of monomials , Math. Nachr. 203 (1999), 31-45. · Zbl 0954.13012 |

[7] | D. Eisenbud, A. Reeves and B. Totaro, Initial ideals, Veronese subrings, and rates of algebras , Adv. Math. 109 (1994), 168-187. · Zbl 0839.13013 |

[8] | R. Fröberg, Determination of a class of Poincaré series , Math. Scand. 37 (1975), 29-39. |

[9] | B. Sturmfels, Gröbner bases and convex polytopes , Univ. Lecture Ser., vol. 8, Amer. Math. Soc., Providence, 1996. · Zbl 0856.13020 |

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