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Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function., Author(s): Evelyne Hubert [sup.1], Michael F. Singer [sup.2] Author Affiliations: (1) INRIA Méditerranée, , 06902, Sophia Antipolis, France (2) grid.40803.3f, 0000 0001 2173 6074, Department of Mathematics, North Carolina State [...]

We consider systems of linear differential and difference equations ∂Y(x)=A(x)Y(x),σY(x)=B(x)Y(x) with ∂=d/dx, σ a shift operator σ(x)=x+a, q-dilation operator σ(x)=qx or Mahler operator σ(x)=xp and systems of two linear difference equations σ1Y(x)=A(x)Y(x),σ2Y(x)=B(x)Y(x) with (σ1,σ2) a sufficiently independent pair of shift operators, pair of q-dilation operators or pair of Mahler operators. Here A(x) and B(x) are n×n matrices with rational function entries. Assuming a consistency hypothesis, we show that such system can be reduced to a system of a very simple form. Using this we characterize functions satisfying two linear scalar differential or difference equations with respect to these operators. We also indicate how these results have consequences both in the theory of automatic sets, leading to a new proof of Cobham's Theorem, and in the Galois theories of linear difference and differential equations, leading to hypertranscendence results. [ABSTRACT FROM AUTHOR]

We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending existing results, which were obtained for SL2 in the case of just one derivation. As an application of the above bound, we develop an algorithm that tests whether the parameterized differential Galois group of a system of linear differential equations is reductive and, if it is, calculates it. [ABSTRACT FROM AUTHOR]

We study the notion of regular singularities for parameterized complex ordinary linear differential systems, prove an analogue of the Schlesinger theorem for systems with regular singularities and solve both a parameterized version of the weak Riemann–Hilbert Problem and a special case of the inverse problem in parameterized Picard–Vessiot theory. [ABSTRACT FROM AUTHOR]