Arithmetic properties related to the shuffle-product

TL;DR

This paper explores a quadratic form linked to the shuffle product over fields of characteristic 2, showing it preserves certain series and conjecturally induces bijections between classes of formal power series.

Contribution

It introduces a new quadratic form related to the shuffle product and conjectures its bijective correspondence between rational and algebraic power series.

Findings

Quadratic form preserves rational and algebraic series

Restriction to series with constant term 1 is bijective

Conjecture: bijection restricts to rational and algebraic series

Abstract

Properties of the shuffle product suggest the definition of a quadratic form with domain and values in formal power series over a field of characteristic 2. This quadratic form preserves rational (respectively algebraic) power series and its restriction to the affine subspace of series with constant term 1 is bijective. Conjecturally, this bijection restricts to a bijection of rational (respectively algebraic) formal power series.