Lexsegment Ideals and Their h-Polynomials


Let S = K[x1, . . . , xn] denote the polynomial ring in n variables over a field K with each deg xi = 1 and IS a homogeneous ideal of S with dim S/I = d. The Hilbert series of S/I is of the form hS/I(λ)/(1 − λ)d, where hS/I(λ) = h0 + h1λ + h2λ2 + ⋯ + hsλs with hs≠ 0 is the h-polynomial of S/I. Given arbitrary integers r ≥ 1 and s ≥ 1, a lexsegment ideal I of S = K[x1,…,xn], where n ≤ max{r,s} + 2, satisfying reg(S/I) = r and deg hS/I(λ) = s will be constructed.

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  1. 1.

    Conca, A., Herzog, J.: Castelnuovo-mumford regularity of products of ideals. Collect. Math. 54(2), 137–152 (2003)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer, London (2011)

    Google Scholar 

  3. 3.

    Hibi, T., Matsuda, K.: Regularity and h-polynomials of monomial ideals. Math. Nachr., in press

  4. 4.

    Vasconcelos, W.V.: Computational Methods in Commutative Algebra and Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 2. Springer, Berlin (1998)

    Google Scholar 

Download references


During the participation of the first author in the workshop New Trends in Syzygies, organized by Jason McCullough (Iowa State University) and Giulio Caviglia (Purdue University), Banff International Research Station for Mathematical Innovation and Discovery, Banff, Canada, June 24 – 29, 2018, a motive for writing the present paper arose from an informal conversation with Marc Chardin. Special thanks are due to the BIRS for providing the participants with a wonderful atmosphere for mathematics.


The first author is partially supported by JSPS KAKENHI 26220701. The second author is partially supported by JSPS KAKENHI 17K14165.

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Correspondence to Kazunori Matsuda.

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Hibi, T., Matsuda, K. Lexsegment Ideals and Their h-Polynomials. Acta Math Vietnam 44, 83–86 (2019). https://doi.org/10.1007/s40306-018-0297-5

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  • Castelnuovo–Mumford regularity
  • Lexsegment ideal
  • h-polynomial

Mathematics Subject Classification (2010)

  • 05E40
  • 13H10