Vectors of type II Hermite-Pad\'e approximations and a new linear independence criterion

TL;DR

This paper introduces a new linear independence criterion based on type II Hermite-Padé approximations, providing a novel approach to prove the linear independence of certain real numbers, especially when type I approximations are unavailable.

Contribution

It proposes a dual linear independence criterion utilizing type II Hermite-Padé approximations, expanding tools for proving independence of numbers in number theory.

Findings

Developed a new criterion for linear independence using type II approximations

Showed the relation between type I and II approximations at infinite places

Highlighted advantages of type II approximations at finite places

Abstract

We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of analytic functions, how to prove that the $\mathbb{Q}$-vector space spanned by $1$ and those three numbers has dimension at least 3, whenever we are unable to achieve full linear independence, by using simultaneous approximations, i.e. those usually arising from Hermite-Pad\'e approximations of type II and their suitable generalizations. It should be recalled that approximations of type I and II are related, at least in principle: when the numerical application consists in specializing actual functional constructions of the two types, they can be obtained, one from the other, as explained in a well-known paper by K.Mahler [34]. That relation is reflected in a relation between the asymptotic behavior of the approximations at the infinite place of $\mathbb{Q}$. Rather interestingly, the two view-points split away regarding the asymptotic behaviors at finite places (i.e. primes) of $\mathbb{Q}$, and this makes the use of type II more convenient for particular purposes. In addition, sometimes we know type II approximations to a given set of functions, for which type I approximations are not known explicitly. Our approach can be regarded as a dual version of the standard linear independence criterion, which goes back to Siegel.