Reciprocal Polynomials and Modular Invariant Theory

Abstract

Let p be a prime and let V be a finite-dimensional vector space over the field \(\mathbb{F}_p\). In this paper we introduce, and study some basic properties of, the algebra of reciprocal polynomials A(V). This can be regarded as a purely inseparable integral extension of the symmetric algebra S(V*) of the dual space V* and has a closely related modular invariant theory with a provable degree bound for invariants which is only conjectural in the symmetric algebra case. The graded \(\mathbb{F}_p\)-algebra A(V) turns out to be normal and Cohen--Macaulay, there is an analogue of Steenrod powers and also a "Karagueuzian and Symonds-type" finiteness theorem for its invariant theory, etc.