Noether's forms for the study of non-composite rational functions and their spectrum

TL;DR

This paper investigates the spectrum and decomposability of multivariate rational functions using Noether's irreducibility theorem, providing explicit bounds and results for reductions modulo prime integers.

Contribution

It introduces effective bounds for the spectrum and decomposability of rational functions modulo primes, extending Noether's theorem to multivariate cases with explicit criteria.

Findings

Reduction of spectrum modulo p matches spectrum of reduced function for large primes

Explicit bounds depend on degree, height, and number of variables

Results apply to polynomials with coefficients in polynomial rings

Abstract

In this paper, the spectrum and the decomposability of a multivariate rational function are studied by means of the effective Noether's irreducibility theorem given by Ruppert. With this approach, some new effective results are obtained. In particular, we show that the reduction modulo p of the spectrum of a given integer multivariate rational function r coincides with the spectrum of the reduction of r modulo p for p a prime integer greater or equal to an explicit bound. This bound is given in terms of the degree, the height and the number of variables of r. With the same strategy, we also study the decomposability of r modulo p. Some similar explicit results are also provided for the case of polynomials with coefficients in a polynomial ring.