Related papers: $p$-adic Zeros of Quintic Forms

We offer some elementary characterisations of group and round quadratic forms. These characterisations are applied to establish new (and recover existing) characterisations of Pfister forms. We establish "going-up" results for group and…

Let R be a regular semi-local domain containing a field such that all the residue fields are infinite. Let K be the fraction field of R. Let q be a quadratic space over R on a free rank n R-module P such that the projective quadric q=0 is…

To our knowledge at the time of writing, the maximum Waring rank for the set of all ternary forms of degree $d$ (with coefficients in an algebraically closed field of characteristic zero) is known only for $d\le 4$. The best upper bound…

The existence of non trivial zeros off the critical line for a function obtained by analytic continuation of a particular Dirichlet series is studied. Contrary to what has been presumed for a long time, we prove that such zeros cannot…

Using an adelic approach we simultaneously consider real and p-adic aspects of dynamical systems whose states are mapped by linear fractional transformations isomorphic to some subgroups of GL (2, Q), SL (2, Q) and SL (2, Z) groups. In…

When all ternary cubic forms over $\mathbb Z$ are ordered by the heights of their coefficients, we show that a positive proportion of them fail the Hasse principle, i.e., they have a zero over every completion of $\mathbb Q$ but no zero…

Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field $L$ has infinitely many nontrivial zeros of multiplicity at least 2 if $L$ has a subfield $K$ for which $L/K$ is a…

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronoi and…

It is proven that for any representation over a field of characteristic 0 of the non-abelian semidirect product of a cyclic group of prime order p and the group of order 3 the corresponding algebra of polynomial invariants is generated by…

For each odd prime $p$, we prove the existence of infinitely many real quadratic fields which are $p$-rational. Explicit imaginary and real bi-quadratic $p$-rational fields are also given for each prime $p$. Using a recent method developed…

A finitely based, finitely generated variety with finitely many subvarieties is a Cross variety. In the present article, it is shown that a variety of $J$-trivial monoids is Cross if and only if it excludes as subvarieties a certain list of…

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

Applying the Picard-Fuchs equation to the discontinuous differential system, we obtain the upper bounds of the number of zeros for Abelian integrals of four kinds of quadratic differential systems when it is perturbed inside all…

Let F be a field of characteristic different from 2. The u-invariant and the Hasse number of a field F are classical and important field invariants pertaining to quadratic forms. These invariants measure the suprema of dimensions of…

If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/p^{v(n)} the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)^{pke} u((kp^e)!) as e goes to…

The multi-valued quantum $j$-invariant in positive characteristic is studied at quadratic elements. For every quadratic $f$, an explicit expression for each of the values of $j^{\rm qt}(f)$ is given as a limit of rational functions of $f$.…

For a prime $p$ and an integer $x$, the $p$-adic valuation of $x$ is denoted by $\nu_{p}(x)$. For a polynomial $Q$ with integer coefficients, the sequence of valuations $\nu_{p}(Q(n))$ is shown to be either periodic or unbounded. The first…

In this note we give a closed expression for the number of points over finite fields of the Lusztig nilpotent variety associated to any quiver without edge loops, in terms of Kac's A-polynomial. We conjecture a similar result for quivers in…

The aim of this paper is to investigate the trivial zeros of the Katz $p$-adic $L$-functions by the CM congruence. We prove the existence of trivial zeros of the Katz $p$-adic $L$-functions for general CM fields and establish a first…

We prove that approximately $96.23\%$ of cubic fields, ordered by discriminant, have genus number one, and we compute the exact proportion of cubic fields with a given genus number. We also compute the average genus number. Finally, we show…