Related papers: $P$-adic monomial equations and their perturbation

Bloch-Okounkov studied certain functions on partitions $f$ called shifted symmetric polynomials. They showed that certain $q$-series arising from these functions (the so-called \emph{$q$-brackets} $\left<f\right>_q$) are quasimodular forms.…

We present a novel way of realizing the Bernoulli shift on $p$ symbols on the $p$-adic integers, where $p$ is a prime. By showing that suitably small perturbations of the shift are still Bernoulli we find many "nice" maps, such as…

We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive…

We relate various approaches to coefficient systems in relative integral $p$-adic Hodge theory, working in the geometric context over the ring of integers of a perfectoid field. These include small generalised representations over…

In this brief note we present a set of equations describing the evolution of perturbed scalar fields in a cosmological spacetime with multiple scalar fields. We take into account of the simultaneously excited full metric perturbations in…

Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\dots+x_n^{2^m}=0$ over a finite field of characteristic $p\equiv\pm 3\pmod{8}$. All of the…

We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the…

Problems of stabilization of the unstable cycle of one-dimensional complex dynamical system are briefly discussed. These questions reduced to the problem of description of the ranges of polynomials $q(z) = q_1z + q_2z^2 +\dots + q_nz^n$…

The dynamical structure of the rational map $ax+1/x$ on the projective line $\P$ over the field $\mathbb{Q}\_p$ of $p$-adic numbers is described for $p\geq 3$.

We show that the coefficients of a power series occurring in $p$-adic Fourier theory for $\mathbf{Q}_{p^2}$ have valuations that are given by an intriguing formula.

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…

In this paper, we present three classes of complete permutation monomials over finite fields of odd characteristic. Meanwhile, the compositional inverses of these complete permutation polynomials are also proposed.

This paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove…

We study perturbations of linear differential equations, deriving explicit series solutions, using Dyson-type expansions. We analyze the monodromy of deformed solutions in a number of examples, and relate this to cocycles in a cohomological…

In this paper we will give a unified proof of several results on the sovability of systems of certain equations over finite fields, which were recently obtained by Fourier analytic methods. Roughly speaking, we show that almost all systems…

The most frequently asked question in the $p-$adic lattice models of statistical mechanics is that whether a root of a polynomial equation belongs to domains $\mathbb{Z}_p^{*}, \ \mathbb{Z}_p\setminus\mathbb{Z}_p^{*}, \ \mathbb{Z}_p, \…

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…

An analogue of the Gauss-Lucas theorem for polynomials over the algebraic closure $\mathbb C_p$ of the field of $p$-adic numbers is considered.

We develop a theory of $p$-adic continued fractions for a quaternion algebra $B$ over $\mathbb Q$ ramified at a rational prime $p$. Many properties holding in the commutative case can be proven also in this setting. In particular, we focus…