02145nas a2200157   4500008004100000245008600041210006900127260003000196300001500226490000800241520163000249100002001879700002001899700002101919856004701940       2019                        eng d00aIsomonodromy deformations at an irregular singularity with coalescing eigenvalues0 aIsomonodromy deformations at an irregular singularity with coale  bDuke University Pressc04  a967–11080 v1683 a<p>We consider an n×n linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at z=∞, holomorphically depending on parameter t within a polydisc in Cn centred at t=0. The eigenvalues of the leading matrix at z=∞ coalesce along a locus Δ contained in the polydisc, passing through t=0. Namely, z=∞ is a resonant irregular singularity for t∈Δ. We analyse the case when the leading matrix remains diagonalisable at Δ. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as t varies in the polydisc, and their limits for t tending to points of Δ. When the deformation is isomonodromic away from Δ, it is well known that a fundamental matrix solution has singularities at Δ. When the system also has a Fuchsian singularity at z=0, we show under minimal vanishing conditions on the residue matrix at z=0 that isomonodromic deformations can be extended to the whole polydisc, including Δ, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed t=0. Conversely, if the t-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting Δ, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that Δ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.</p>1 aCotti, Giordano1 aDubrovin, Boris1 aGuzzetti, Davide  uhttps://doi.org/10.1215/00127094-2018-0059