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Register a new userFast evaluation of some p-adic transcendental functions
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Added by
Xavier Caruso
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Xavier Caruso
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We design algorithms for computing values of many p-adic elementary and special functions, including logarithms, exponentials, polylogarithms, and hypergeometric functions. All our algorithms feature a quasi-linear complexity with respect to the target precision and most of them are based on an adaptation to the-adic setting of the binary splitting and bit-burst strategies.
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We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating a MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate Haar MRA. We also suggest a method of constructing sets of wavelet functions and prove that any set of wavelet functions generates a $p$-adic wavelet frame.
We develop a theory of log adic spaces by combining the theories of adic spaces and log schemes, and study the Kummer etale and pro-Kummer etale topology for such spaces. We also establish the primitive comparison theorem in this context, and deduce from it some related cohomological finiteness or vanishing results.
We present a non-commutative algorithm for the multiplication of a 2 x 2 block-matrix by its adjoint, defined by a matrix ring anti-homomorphism. This algorithm uses 5 block products (3 recursive calls and 2 general products)over C or in positive characteristic. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its adjoint to general matrix product, improving by a constant factor previously known reductions. We prove also that there is no algorithm derived from bilinear forms using only four products and the adjoint of one of them. Second we give novel dedicated algorithms for the complex field and the quaternions to alternatively compute the multiplication taking advantage of the structure of the matrix-polynomial arithmetic involved. We then analyze the respective ranges of predominance of the two strategies. Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a prime field.
Let $Pin Bbb Q_p[x,y]$, $sin Bbb C$ with sufficiently large real part, and consider the integral operator $ (A_{P,s}f)(y):=frac{1}{1-p^{-1}}int_{Bbb Z_p}|P(x,y)|^sf(x) |dx| $ on $L^2(Bbb Z_p)$. We show that if $P$ is homogeneous then for each character $chi$ of $Bbb Z_p^times$ the characteristic function $det(1-uA_{P,s,chi})$ of the restriction $A_{P,s,chi}$ of $A_{P,s}$ to the eigenspace $L^2(Bbb Z_p)_chi$ is the $q$-Wronskian of a set of solutions of a (possibly confluent) $q$-hypergeometric equation. In particular, the nonzero eigenvalues of $A_{P,s,chi}$ are the reciprocals of the zeros of such $q$-Wronskian.
To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $p$-adic counterpart of the GKZ hypergeometric system. The $p$-adic GKZ hypergeometric complex is a twisted relative de Rham complex of over-convergent differential forms with logarithmic poles. It is an over-holonomic object in the derived category of arithmetic $mathcal D$-modules with Frobenius structures. Traces of Frobenius on fibers at Techmuller points of the GKZ hypergeometric complex define the hypergeometric function over the finite field introduced by Gelfand and Graev. Over the non-degenerate locus, the GKZ hypergeometric complex defines an over-convergent $F$-isocrystal. It is the crystalline companion of the $ell$-adic GKZ hypergeometric sheaf that we constructed before. Our method is a combination of Dworks theory and the theory of arithmetic $mathcal D$-modules of Berthelot.
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