[Paper Review] Equations in the Hadamard ring of rational functions

Andrea Ferretti, Umberto Zannier|ArXiv.org|Jan 26, 2007

Polynomial and algebraic computation8 references4 citations

TL;DR

This paper generalizes Pisot's conjecture on d-th roots in the Hadamard ring of rational functions to arbitrary monic polynomial equations in Y with coefficients in the Hadamard ring over a number field. Using exponential polynomial representations and Hilbert irreducibility techniques, it proves that if such an equation has a solution in the base field for all n, then a solution exists in the Hadamard ring over a finite extension of the field, thus unifying results on d-th roots and the Hadamard quotient theorem.

ABSTRACT

Let k be a number field. It is well known that the set of sequences composed by Taylor coefficients of rational functions over k is closed under component-wise operations, and so it can be equipped with a ring structure. A conjecture due to Pisot asks if (after enlarging the field) one can take d-th roots in this ring, provided d-th roots of coefficients can be taken in k. This was proved true in a preceding paper of the second author; in this article we generalize this result to more general equations, monic in Y, where the former case can be recovered for g(X,Y)=X^d-Y=0. Combining this with the Hadamard quotient theorem by Pourchet and Van der Poorten, we are able to get rid of the monic restriction, and have a theorem that generalizes both results.

Motivation & Objective

• To resolve a conjecture of Van der Poorten on solving monic polynomial equations in the Hadamard ring of rational functions over a number field.
• To extend the previously known result on d-th roots in the Hadamard ring to more general algebraic equations.
• To unify the results on d-th roots (Pisot's conjecture) and the Hadamard quotient theorem via a common framework.
• To establish the existence of a solution in the Hadamard ring over a finite extension of the base field, given that solutions exist pointwise in the base field.

Proposed method

• Representing elements of the Hadamard ring as exponential polynomials with coefficients in a number field and roots in the algebraic closure.
• Applying Hilbert irreducibility theorems to control the existence of solutions across arithmetic progressions.
• Using reduction techniques and Galois theory to analyze the irreducibility of polynomial factors over finite fields.
• Employing Chebotarev's density theorem to ensure the existence of primes with desired splitting behavior for constructing solutions.
• Applying estimates from analytic number theory to bound the Euler function φ(m)/m and control the size of extensions.
• Proving irreducibility of the resulting polynomial over finite fields to lift solutions to the global field via the Hasse principle and approximation theorems.

Experimental results

Research questions

• RQ1 Under what conditions can a monic polynomial equation in Y with coefficients in the Hadamard ring over a number field be solved globally in the ring, given that it has solutions pointwise in the base field?
• RQ2 Can the result on d-th roots in the Hadamard ring (Pisot's conjecture) be generalized to arbitrary monic polynomial equations?
• RQ3 How can the Hadamard quotient theorem and Pisot's conjecture be unified under a single framework for solving algebraic equations in the Hadamard ring?
• RQ4 What role does Hilbert irreducibility play in lifting local solutions to global solutions in the Hadamard ring?
• RQ5 Can effective bounds on the degree of the finite field extension be derived using analytic number theory tools?

Key findings

• The paper proves that if a monic polynomial equation in Y has a solution in the base field for every n, then there exists a finite extension of the base field such that a solution exists in the Hadamard ring over that extension.
• The solution is shown to be an exponential polynomial with coefficients in a finite extension of the base field, generalizing the d-th root case.
• The proof relies on reducing the problem to irreducibility of a polynomial over a finite field, using Chebotarev's density theorem to ensure the existence of suitable primes.
• The authors establish a condition under which a polynomial remains irreducible modulo a prime, based on the non-existence of roots in the residue field.
• The method allows for the removal of the monic restriction in the Hadamard quotient theorem by combining it with the generalized solution of the equation.
• The result extends to fields finitely generated over Q via standard reduction techniques, though the paper focuses on number fields.