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–Padé 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 Mahler (1968) Compos Math 19: 95–166. 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 essentially goes back to Siegel.
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Acknowledgements
The core of the present note, a short less-than-seven-pages draft without applications, was written several years ago partly during, partly after a stay at the Centre International de Rencontres Mathématiques de Luminy, France. However, my attention to the topic was recently refreshed by the papers [15] and [62], and specially by the proof of [62, Lemma 5.3]. Along the time, I had the pleasure of chatting on this topic with F.Amoroso, M.Laurent and W.Zudilin; a special thank to them, and to whom else made all this possible, in a way or another. I owe a debt of gratitude to an anonymous referee for a very careful reading. Some computations in Sect. 4 were made with the help of the free software Pari/GP [42].
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Dedicated to Wadim Zudilin, with warm wishes, on the occasion of his 50th birthday.
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Marcovecchio, R. Vectors of type II Hermite–Padé approximations and a new linear independence criterion. Annali di Matematica 200, 2829–2861 (2021). https://doi.org/10.1007/s10231-021-01104-7
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DOI: https://doi.org/10.1007/s10231-021-01104-7
Keywords
- Linear independence
- Matrices and determinants
- Orthogonal polynomials
- Difference equations
- Simultaneous approximations