Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions

We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental

Dynamics of rational symplectic mappings and difference Galois theory

In this paper we study the relationship between the integrability of rational symplectic maps and difference Galois theory. We present a Galoisian condition, of Morales-Ramis type, ensuring the non-integrability of a rational symplectic map in the non-commutative sense (Mishchenko-Fomenko). As a particular case, we obtain a com- plete discrete analogue of Morales-Ramis Theorems for non-integrabi- lity in the sense of Liouville

On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps

International audienceUsing Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number p and not only for almost all primes. Along the way, using Christol's functions, we provide an explicit formula for the ''Eisenstein constant'' of any globally bounded hypergeometric series with rational parameters. As an application of these results, we obtain an arithmetic statement of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It essentially contains all the similar univariate integrality results in the litterature

Algebraic solutions of linear differential equations: an arithmetic approach

Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz $p$-curvature conjecture.Comment: 47 page

In vitro screening of probiotic lactic acid bacteria and prebiotic glucooligosaccharides to select effective synbiotics

Probiotics and prebiotics have been demonstrated to positively modulate the intestinal microflora and could promote host health. Although some studies have been performed on combinations of probiotics and prebiotics, constituting synbiotics, results on the synergistic effects tend to be discordant in the published works. The first aim of our study was to screen some lactic acid bacteria on the basis of probiotic characteristics (resistance to intestinal conditions, inhibition of pathogenic strains). Bifidobacterium was the most resistant genus whereas Lactobacillus farciminis was strongly inhibited. The inhibitory effect on pathogen growth was strain dependent but lactobacilli were the most effective, especially L. farciminis. The second aim of the work was to select glucooligosaccharides for their ability to support the growth of the probiotics tested. We demonstrated the selective fermentability of oligodextran and oligoalternan by probiotic bacteria, especially the bifidobacteria, for shorter degrees of polymerisation and absence of metabolism by pathogenic bacteria. Thus, the observed characteristics confer potential prebiotic properties on these glucooligosaccharides, to be further confirmed in vivo, and suggest some possible applications in synbiotic combinations with the selected probiotics. Furthermore, the distinctive patterns of the different genera suggest a combination of lactobacilli and bifidobacteria with complementary probiotic effects in addition to the prebiotic ones. These associations should be further evaluated for their synbiotic effects through in vitro and in vivo models