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Quasialgebraicity of Picard-Vessiot fields. (English) Zbl 1058.34120

Consider the Fuchsian system of linear differential equations (i.e., with logarithmic poles) on Riemann’s sphere \[ \dot{X}=\left(\sum _{j=1}^mA_j/(t-a_j)\right)X,~\tag{\(*\)} \] where the constant matrices \(A_j\) and the matrix \(X(t)\) are \(n\times n\). The sum of the norms \(\| A_1\| +\dots +\| A_m\| +\| A_{\infty}\| \), \(A_{\infty}=-A_1-\dots -A_m\) is called the height. Suppose that one has the simple loop spectral condition (SLSC): for any simple loop on Riemann’s sphere bypassing the poles \(a_j\) of system \((*)\), the eigenvalues of its monodromy operator belong to the unit circle. An integer-valued function of one or several integer arguments is primitive recursive if it can be defined by several inductive rules each involving induction in only one variable.
The authors prove the theorem: There exists a primitive recursive function \({\mathcal R}(n,m,r)\) of three natural arguments such that if system \((*)\) satisfies the SLSC and its height is \(\leq r\), then any linear combination of entries of a fundamental solution in any triangular domain \(T\) on Riemann’s sphere without the poles \(a_j\) of system \((*)\) has no more than \({\mathcal R}(n,m,r)\) isolated roots in \(T\). The theorem is generalized in terms of quasialgebraicity of Picard-Vessiot fields for Fuchsian systems.

MSC:

34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32S65 Singularities of holomorphic vector fields and foliations
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
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