Related papers: Using the smoothness of p-1 for computing roots mo…

We give a deterministic algorithm that, given a composite number $N$ and a target order $D \ge N^{1/6}$, runs in time $D^{1/2+o(1)}$ and finds either an element $a \in \mathbb{Z}_N^*$ of multiplicative order at least $D$, or a nontrivial…

Let p be prime and Zpn the degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast algorithms for common operations in Zpn modulo p^N. Combining existing methods with recent work…

We prove the integrality of the open instanton numbers in two examples of counting holomorphic disks on local Calabi-Yau threefolds: the resolved conifold and the degenerate $ \P \times \P $. Given the B-model superpotential, we extract by…

We present an unconditional CRT algorithm to compute the modular polynomial $\Phi_\ell(X,Y)$ in quasi-linear time. The main ingredients of our algorithm are: the embedding of $\ell$-isogenies in smooth-degree isogenies in higher dimension,…

We study the uniform distribution of the polynomial sequence $\lambda(P)=(\lfloor P(k) \rfloor )_{k\geq 1}$ modulo integers, where $P(x)$ is a polynomial with real coefficients. In the nonlinear case, we show that $\lambda(P)$ is uniformly…

Given a non-zero polynomial $f$ in a polynomial ring $R$ with coefficients in a finite field of prime characteristic $p$, we present an algorithm to compute a differential operator $\delta$ which raises $1/f$ to its $p$th power. For some…

Recently, Ko\c{c} proposed a neat and efficient algorithm for computing \[ x = a^{-1} \pmod {p^k} \] for a prime $p$ based on the exact solution of linear equations using $p$-adic expansions. The algorithm requires only addition and right…

A time-dependent monic polynomial in the z variable with N distinct roots such that exactly one root has multiplicity m>=2 is considered. For k=1,2, the k-th derivatives of the N roots are expressed in terms of the derivatives of order j<=…

We obtain asymptotic formulas for sums over arithmetic progressions of coefficients of polynomials of the form $$\prod_{j=1}^n\prod_{k=1}^{p-1}(1-q^{pj-k})^s,$$ where $p$ is an odd prime and $n, s$ are positive integers. Let us denote by…

We give improvements of the deformation method for computing the zeta function of a generic projective hypersurface in characteristic~$p$ that either reduce the dependence on~$p$ of the time complexity to $\tilde{O}(p^{1/2})$ or that of the…

We address complexity issues for linear differential equations in characteristic $p>0$: resolution and computation of the $p$-curvature. For these tasks, our main focus is on algorithms whose complexity behaves well with respect to $p$. We…

In the present paper we provide a probabilistic polynomial time algorithm that reduces the complete factorization of any squarefree integer $n$ to counting points on elliptic curves modulo $n$, succeeding with probability $1-\varepsilon$,…

A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: for any prime $p$, reduce its coefficients mod $p$ and consider its action on the field $\mathbb{F}_p$. We say a subset of…

We study the question of when the coefficients of a hypergeometric series are p-adically unbounded for a given rational prime p. Our first main result is a necessary and sufficient criterion (applicable to all but finitely many primes) for…

For finitely generated subgroups $W_1, \ldots , W_t$ of $\mathbb{Q}^{\times}$, integers $k_1, \ldots , k_t$, a Galois extension $F$ of $\mathbb{Q}$ and a union of conjugacy classes $C \subset \text{Gal}(F/\mathbb{Q})$, we develop methods…

Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence $$p^{\deg f}\sum_{k=r(mod p^a)}\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\sum_{i=a}^{\infty}[n/p^i]})$$ (motivated by…

Let $G$ be a semisimple algebraic group over a field of characteristic $p > 0$. We prove that the dual Weyl modules for $G$ all have $p$-filtrations when $p$ is not too small. Moreover, we give applications of this theorem to…

The main objective of this article is to establish the $p$-adic Artin formalism for the algebraic $p$-adic $L$-functions attached to the adjoint representations of Coleman families of modular forms. In particular, we prove a factorization…

A polynomial is expansive if all of its roots lie outside the unit circle. We define some special determinants involving the coefficients of a real polynomial and formulate necessary and sufficient conditions for expansivity using these…

The $L$-function of exponential sums associated to the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is studied. A polygon called the Frobenius polygon of the generic polynomial of degree $d$ in…