214 research outputs found

    p-adic number theory and its applications in a cryptographic form

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    On a theory of the bb-function in positive characteristic

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    We present a theory of the bb-function (or Bernstein-Sato polynomial) in positive characteristic. Let ff be a non-constant polynomial with coefficients in a perfect field kk of characteristic p>0.p>0. Its bb-function bfb_f is defined to be an ideal of the algebra of continuous kk-valued functions on Zp.\mathbb{Z}_p. The zero-locus of the bb-function is thus naturally interpreted as a subset of Zp,\mathbb{Z}_p, which we call the set of roots of bf.b_f. We prove that bfb_f has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Musta\c{t}\u{a} and is in terms of DD-modules, where DD is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of bfb_f and relate it to the test ideals of f.f.Comment: Final versio

    On the complexity of algebraic number I. Expansions in integer bases

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    Let b≥2b \ge 2 be an integer. We prove that the bb-adic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion

    On K(1)-local SU-bordism

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    This paper works towards a K(1)-local multiplicative splitting of SU-bordism
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