Annihilating polynomials for quadratic forms and Stirling numbers of the second kind

Abstract

We present a set of generators of the full annihilator ideal for the Witt ring of an arbitrary field of characteristic unequal to two satisfying a non-vanishing condition on the powers of the fundamental ideal in the torsion part of the Witt ring. This settles a conjecture of Ongenae and Van Geel. This result could only be proved by first obtaining a new lower bound on the 2-adic valuation of Stirling numbers of the second kind.

The 2-adic valuation of a sequence arising from a rational integral

“…and they count the number of ways to partition a set with n elements into exactly k nonempty subsets. De Wannemacker [8] also established the inequality…”

On a conjecture of Wilf

“…Recent applications of S(n, k) include computing annihilating polynomials for quadratic forms [11]. Further information on these applications can be found in [10,12].…”

“…For more details and basic results on Stirling numbers of the second kind we refer the reader to [8], [23], or [46]. Recent applications of S(n, k) include computing annihilating polynomials for quadratic forms [11]. Further information on these applications can be found in [12].…”

A remarkable sequence of integers

“…This study was motivated by the results described in this section. The papers [15,16,22,21,33,34] contain information about 2-adic valuations of related sequences.…”