Related papers: Computing the $p$-adic Canonical Quadratic Form in…

We provide a short proof of an algebraic identity. For integers $n\ge 2$ and variables $x,y,z$, it represents $(x^n+y^n-z^n)$ as a value of the quadratic form $\mathcal A^2+\mathcal B^2-\mathcal C^2$ after multiplication by an explicit…

Summation of the $p$-adic functional series $\sum \varepsilon^n \, n! \, P_k^\varepsilon (n; x)\, x^n ,$ where $P_k^\varepsilon (n; x)$ is a polynomial in $x$ and $n$ with rational coefficients, and $\varepsilon = \pm 1$, is considered. The…

We give explicit formulas as well as a quadratic time algorithm to solve (so called) generalized Vandermonde's systems of p linear equations and n variables. It allows in particular to find all (so called Lagrange's) interpolation polynoms…

Let $p$ be a fixed odd prime and $Q(x,y,z)=ax^2+bxy+cy^2+dxz+eyz+fz^2$ be a fixed quadratic form in $\mathbb{Z}[x,y,z]$ which is non-degenerate in $\mathbb{F}_p[x,y,z]$ and $(a(4ac-b^2),p)=1.$ Let $(x_0,y_0,z_0)$ be a fixed point in…

We construct algorithms that efficiently generate random factorisations of values $P(n)$ as products of two integers, where $P\in\mathbb{Z}[x]$ is a given quadratic or cubic monic polynomial. In other words, the algorithms produce random…

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility…

In this paper we consider the Witt's fprmula related to Carlitz's type q-Euler numbers and polynomials.

In the present paper, our objective is to treat a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order Dedekind-type sums with weight in connection with modified q-Genocchi polynomials with weight…

A unitary cyclotomic polynomial of order three is a polynomial of the form \[ \Phi^*_{PQR}(x)=\frac{(x^{PQR}-1)(x^P-1)(x^Q-1)(x^R-1)}{(x^{PQ}-1)(x^{QR}-1)(x^{RP}-1)(x-1)}, \] where $P$, $Q$ and $R$ are powers of three distinct primes $p$,…

The best deterministic unconditionally proven integer factorization algorithms have exponential running time complexities of O(N^(1/4)) arithmetic operations, and conditional on the Riemann hypothesis, there is a deterministic algorithm of…

A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the…

This paper proves that non-convex quadratically constrained quadratic programs can be solved in polynomial time when their underlying graph is acyclic, provided the constraints satisfy a certain technical condition. When this condition is…

Let $p$ be an odd prime. For any $p$-adic integer $a$ we let $\overline{a}$ denote the unique integer $x$ with $-p/2<x<p/2$ and $x-a$ divisible by $p$. In this paper we study some permutations involving quadratic residues modulo $p$. For…

Let $p$ be an odd prime. In this paper we investigate quadratic residues modulo $p$ and related permutations, congruences and identities. If $a_1<\ldots<a_{(p-1)/2}$ are all the quadratic residues modulo $p$ among $1,\ldots,p-1$, then the…

The Hasse principle and weak approximation is established for equations of the shape P(t)=N(x_1,x_2,x_3,x_4), where P is an irreducible quadratic polynomial in one variable and N is a norm form associated to a quartic extension of the…

Let n be a square-free polynomial over F_q, where q is an odd prime power. In this paper, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case…

The arithmetic partial derivative (with respect to a prime $p$) is a function from the set of integers that sends $p$ to 1 and satisfies the Leibniz rule. In this paper, we prove that the $p$-adic valuation of the sequence of higher order…

Let $a,k\in\mathbb{N}$. For the $k-1$-th iterate of the exponential function $x\mapsto a^x$, also known as tetration, we write \[ ^k a:=a^{a^{.^{.^{.^{a}}}}}. \] In this paper, we show how an efficient algorithm for tetration modulo natural…

We will show that the roots of a polynomial equation in one variable of degree n are related to the solutions of a symmetric quadratic form in n-1 variables with constant positive integer coefficients. The classic polynomial notation will…

Let $m\neq0,\pm1$ and $n\geq 2$ be integers. The ring of algebraic integers of the pure fields of type $\mathbb{Q}(\sqrt[n]{m})$ is explicitly known for $n=2,3,4$. It is well known that for $n=2$, an integral basis of the pure quadratic…