The Betti polynomials of powers of an ideal

For an ideal I in a regular local ring or a graded ideal I in the polynomial ring we study the limiting behavior of as k goes to infinity. By Kodiyalam’s result it is known that β_i(S/I^k) is a polynomial for large k. We call these polynomials the Kodiyalam polynomials and encode the limiting behavior in their generating polynomial. It is shown that the limiting behavior depends only on the coefficients on the Kodiyalam polynomials in the highest possible degree. For these we exhibit lower bounds in special cases and conjecture that the bounds are valid in general. We also show that the Kodiyalam polynomials have weakly descending degrees and identify a situation where the polynomials all have the highest possible degree.     

Regularity of the powers of an ideal

For a homogeneous ideal I of S = A[X_o,... ,X_n] we compare the Castelnuovo-Mumford regularities of the powers of I to the regularity reg I of I itself.     

Every Banach ideal of polynomials is compatible with an operator ideal

We show that for each Banach ideal of homogeneous polynomials, there exists a (necessarily unique) Banach operator ideal compatible with it. Analogously, we prove that any ideal of n-homogeneous polynomials belongs to a coherent sequence of ideals of k-homogeneous polynomials.     

Every Banach ideal of polynomials is compatible with an operator ideal

We show that for each Banach ideal of homogeneous polynomials, there exists a (necessarily unique) Banach operator ideal compatible with it. Analogously, we prove that any ideal of n-homogeneous polynomials belongs to a coherent sequence of ideals of k-homogeneous polynomials.     

Equations defining the multi-Rees algebras of powers of an ideal

In this paper we describe the defining equations of the multi-Rees algebra $R[{\{I^{n_i}{t}_i\}}_{1\le i\le r}]$ over a Noetherian ring R when I is an ideal of linear type. This generalizes a result of Ribbe and recent work of Lin–Polini and Sosa.     

Index of lattices and Hilbert polynomials

An upper bound for the index of a sublattice, which arises in relation to various versions of zero lemmas in the theory of linear forms in logarithms of algebraic numbers, in terms of the Hilbert polynomial is found. Simultaneously, a lower bound for the values of this polynomial is obtained.     

Sums of powers of Fibonacci polynomials

Using the explicit (Binet) formula for the Fibonacci polynomials, a summation formula for powers of Fibonacci polynomials is derived straightforwardly, which generalizes a recent result for squares that appeared in Proc. Ind. Acad. Sci. (Math. Sci.) 118 (2008) 27–41.     

Hilbert and Hilbert—Samuel polynomials and partial differential equations

Systems of linear partial differential equations with constant coefficients are considered. The spaces of formal and analytic solutions of such systems are described by algebraic methods. The Hilbert and Hilbert—Samuel polynomials for systems of partial differential equations are defined.     

The depth of an ideal with a given Hilbert function

Let denote the polynomial ring in variables over a field with each . Let be a homogeneous ideal of with and the Hilbert function of the quotient algebra . Given a numerical function satisfying for some homogeneous ideal of , we write for the set of those integers such that there exists a homogeneous ideal of with and with . It will be proved that one has either for some or .

     

Hilbert and Hilbert—Samuel polynomials and partial differential equations

Systems of linear partial differential equations with constant coefficients are considered. The spaces of formal and analytic solutions of such systems are described by algebraic methods. The Hilbert and Hilbert—Samuel polynomials for systems of partial differential equations are defined.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 141–151.Original Russian Text Copyright © 2005 by A. G. Khovanskii, S. P. Chulkov.This revised version was published online in April 2005 with a corrected issue number.     

Orthogonal polynomials and rigged Hilbert space

A duality correspondence is set up between polynomials satisfying a recurrence relation and the generalized eigenvectors of a Jacobi operator in a rigged Hilbert space. In the case of a symmetric recurrence a ready consequence of that correspondence is an alternative proof of Favard's Theorem and the Representation Theorem.     

Hilbert polynomials and Arveson's curvature invariant

We define and study Hilbert polynomials for certain holomorphic Hilbert spaces. We obtain several estimates for these polynomials and their coefficients. Our estimates inspire us to investigate the connection between the leading coefficients of Hilbert polynomials for invariant subspaces of the symmetric Fock space and Arveson's curvature invariant for coinvariant subspaces. We are able to obtain some formulas relating the curvature invariant with other invariants. In particular, we prove that Arveson's version of the Gauss-Bonnet-Chern formula is true when the invariant subspaces are generated by any polynomials.     

Quantum Hilbert matrices and orthogonal polynomials

Using the notion of quantum integers associated with a complex number q≠0, we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q-Jacobi polynomials when |q|<1, and for the special value they are closely related to Hankel matrices of reciprocal Fibonacci numbers called Filbert matrices. We find a formula for the entries of the inverse quantum Hilbert matrix.     

Hilbert polynomials of non-standard bigraded algebras

This paper investigates Hilbert polynomials of bigraded algebras which are generated by elements of bidegrees $(1,0), (d_1,1),\ldots,(d_r,1)$, where $d_1,\ldots,d_r$ are non-negative integers. The obtained results can be applied to study Rees algebras of homogeneous ideals and their diagonal subalgebras. Mathematics Subject Classification (1991):13D40, 13H15.The authors are partially supported by the National Basic Research Program of\break Vietnam.