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Upper bounds for (geometric) Hilbert functions

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1752)

Abstract

Let ℓ be a homogeneous prime (and not maximal) ideal in the ring C[x o, ⋯, xN1, and let X be the corresponding (irreducible) algebraic subvariety in the complex projective space P N. Recall that, for any integer t ≥ 0, the Hilbert function of X (or if you prefer, of :ℓ) provides the maximal number H(X,t) := H(J,t) of homogeneous polynomials of degree t which are linearly independent over C modulo ℓ. For large values of t, H(X, t) coincide with a polynomial, whose highest term is given by deg(X) t d/d!, where d denotes the dimension of X, and deg(X) its degree in PN. Thus, the asymptotic behaviour of H(X, t) is well-known. But the bounds needed for application to algebraic independence (see, for instance, Chapter 10 ) must be valid for all Vs. It is this type of bound that we here describe.

In the first section of these Chapter, we give simple geometric proofs of the following upper bounds for H(X,t).

Keywords

  • Projective Space
  • Hilbert Function
  • Hyperplane Section
  • General Hyperplane
  • Algebraic Independence

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Chapter’s author : Daniel BERTRAND.

[Cha] M. Chardin. Une majoration de la fonction de Hilbert et ses conséquences pour l’interpolation algébrique. Bull Soc. Math. France 117, (1989), 305-318 (see also Chapter I of his TUse de doctorat, Université de Paris 6, 1990).

[SOM] M. Sombra. Bounds for the Hilbert functions of polynomial ideals and for the degrees in the Nullstellensatz, J. Math. Appl. Algebra 117-118, (1997), 565-599.

[Kol2] J. Koll6r. Letters to the author (1/8/88 and 18/1/89).

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). Upper bounds for (geometric) Hilbert functions. In: Nesterenko, Y.V., Philippon, P. (eds) Introduction to Algebraic Independence Theory. Lecture Notes in Mathematics, vol 1752. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44550-1_9

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  • DOI: https://doi.org/10.1007/3-540-44550-1_9

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41496-4

  • Online ISBN: 978-3-540-44550-0

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