Classifying All Degrees Below $N^3$

TL;DR

This paper classifies the degrees of all cubic polynomials below $n^3$ within the framework of transducer degrees, answering an open question about the structure of polynomial transducer degrees.

Contribution

It provides a complete classification of the degrees of cubic polynomials below $n^3$, advancing understanding of the structure of polynomial transducer degrees.

Findings

Classified degrees of all cubic polynomials below $n^3$

Developed methods applicable to higher-order polynomials

Answered an open question in transducer degree theory

Abstract

We answer an open question in the theory of transducer degrees initially posed in [3], on the structure of polynomial transducer degrees, in particular the question of what degrees, if any, lie below the degree of $n^3$. Transducer degrees are the equivalence classes formed by word transformations which can be realized by a finite-state transducer. While there are no general techniques to tell if a word $w_1$ can be transformed into $w_2$ via an FST, the work of Endrullis et al. in [2] provides a test for the class of spiralling functions, which includes all polynomials. We classify fully the degrees of all cubic polynomials which are below $n^3$, and many of the methods can also be used to classify the degrees of polynomials of higher orders.