We develop an effective version of Kronecker's Theorem on the splitting of polynomials, based on asymptotic arguments proposed by the Chudnovsky brothers, coming from Hermite-Pade approximation. In conjunction with Honda's proof of the p-curvature conjecture for first-order equations with polynomial coefficients we use this to deduce an effective version of the Grothendieck p-curvature conjecture for first-order equations. More precisely, we bound the number of primes for which the p-curvature of a given differential equation has to vanish in terms of the height and the degree of the coefficients, in order to conclude it has a non-zero algebraic solution. Using this approach, we describe an algorithm that decides algebraicity of solutions of first-order differential equations using p-curvatures, and report on an implementation in SageMath.
We study the surface equations relating the sides, face diagonals and long diagonal of a rectangular box. It is an open problem whether a 'rational cuboid' exists, i.e., a rectangular box all of whose sides, face diagonals and long diagonal have (positive) rational length. The question is equivalent to the existence of non-trivial rational points on S & strns;. The surface S parametrizing cuboids: it is defined by the S has 48 isolated singular points. Let S be the minimal desingularization of S. The main result of this paper is the explicit determination of the Picard group of S, including its structure as a Galois module over Q. The main ingredient for showing that the known subgroup is actually the full Picard group is the use of the combined action of the Galois group and the geometric automorphism group of S (which we also determine) on the Picard group. This reduces the proof to checking that the hyperplane section is not divisible by 2 in the Picard group. We use our explicit knowledge of the Picard group, together with that of a K3 surface obtained as a quotient of S, to study curves of low degree on S & strns;. In this way, we completely classify all integral curves of degree at most 6 on S & strns;.
. We establish improved error bounds for the time-splitting Hermite pseudospectral method applied to the highly-oscillatory Gross-Pitaevskii equation on unbounded domains, which corresponds to the strongly confined Gross-Pitaevskii equation. By combining refined Hermite spectral approximation estimates with an extension of the regularity compensation oscillation technique to unbounded domains, we establish an improved error bound of O(N (d/6 - m/2) + epsilon(2)tau(2)) in L-2-norm up to the long time O(1/epsilon(2)). Here, 0< epsilon <<1 is a dimensionless parameter, N denotes the spatial parameter, tau is the time step size, d is the dimension and m is an integer depending on the spatial regularity of the exact solution. Compared to prior results O(N1+d/4- m/2 +tau(2)) in the work of Shen and Wang [Found. Comput. Math. 13 (2013), pp. 99-137], our work demonstrates improved temporal estimates and sharper spatial estimates. Extensions to higher order time-splitting schemes are also discussed. Finally, numerical results are provided to confirm the improved error bounds and demonstrate that they are sharp.
The learning or approximation of manifold-valued functions has attracted a lot of attention in recent years. The non-linear structure of the functions’ range requires techniques and methods which significantly differ from standard approximation schemes. In this paper, we concentrate on the approximation of functions with values in the class of positive definite, symmetric matrices. Though not a compact manifold, its specific structure allows us to globally employ the log-Euclidean approach to define a Hilbert space structure on these matrices. Combining this approach with a kernel-based approximation method for symmetric matrices leads to a complete theory, including a thorough error analysis for deterministic approximations.
