Proceedings Mathematical Sciences

, Volume 102, Issue 2, pp 93–123 | Cite as

Combinatorial meaning of the coefficients of a Hilbert polynomial

  • M. R. Modak
Article

Abstract

In [1] Abhyankar defines an idealI(p, a) generated by certain minors of a matrixX, the entries ofX being independent indeterminates, and proves that the Hubert function ofI(p, a) coincides with its Hilbert polynomialF(V) and obtains it in the form
$$F(V) = \sum\limits_{D \geqslant 0} {( - 1)} ^D F_D (m,p,a)\left( \begin{gathered} c - D + V \hfill \\ V \hfill \\ \end{gathered} \right)$$
. He also proves thatF(V) is the number of certain “indexed” monomials of degreeV in the entries ofX and that the coefficientsF D (m,p,a) are non-negative integers and asks for their combinatorial meaning. In this paper we characterize the indexed monomials in terms of certain sets of lattice paths, called frames, and prove that the coefficientsF D (m, p, a) count certain families of such frames.

Keywords

Determinantal ideals Hilbert polynomial lattice paths binomial determinants 

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References

  1. [1]
    Abhyankar S S,Enumerative Combinatorics of Young Tableaux (New York: Marcel Dekker) (1988)MATHGoogle Scholar
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    Abhyankar S S and Kulkarni D M, On Hilbertian ideals,Linear Algebra 116 (1989) 53–79MATHCrossRefMathSciNetGoogle Scholar
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    Gessel I and Viennot G, Binomial Determinants, Paths and Hook Length Formulae,Adv. Math. 58 (1985) 300–321MATHCrossRefMathSciNetGoogle Scholar
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    Ghorpade Sudhir R, Abhyankar’s work on Young Tableaux and Some Recent Developments, to appear inProc. Conf. on Algebraic Geometry and applications (Purdue Univ. June 1990), (New York: Springer-Verlag) (1992?)Google Scholar
  5. [5]
    Udpikar S G, On Hilbert polynomial of certain determinantal ideals,Int. Math. Math. Sci. 14 (1991) 155–162MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Science 1992

Authors and Affiliations

  • M. R. Modak
    • 1
  1. 1.Department of MathematicsS. P. CollegePuneIndia

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