The Zeros and the Final Value of a Polynomial Form

ABSTRACT

A polynomial form f, is a not necessarily linear map, from an infinite module over a ring 𝔷 to a finite abelian group of exponent n satisfying some additional conditions. Denote the zeros of f by Ω_f. We show it satisfies a weak closure condition. Among all 𝔷-submodules of finite index, there is a submodule B such that |f (B)| (the order of the subset f (B)) is as small as possible. f (B) is called the final value of f and D. S. Passman asks if f (B) is necessarily a subgroup of S. This paper shows that if the degree of f ≤ 2 then the final value is a subgroup and if the form f has arbitrary degree from an finitely generated infinite abelian group, then the final value is 0.

Abstract

Added in Proof: D. S. Passmam has recently found a counterexample to the final value problem.

Key Words:

• Final value
• Polynomial form

Mathematics Subject Classification:

• Primary 16S34
• Secondary 20F50

ACKNOWLEDGMENT

The author thanks D. S. Passman and the referee for useful suggestions and ideas.

Notes

^# Communicated by J. Kuzmanovich.