Upper bounds for (geometric) Hilbert functions

Abstract

Let ℓ be a homogeneous prime (and not maximal) ideal in the ring C[x _o, ⋯, x_N1, 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 P_N. 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).

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).