Let $f(x)=x^{6}+ax^{4}+bx^{2}+c$ be an irreducible sextic polynomial with coefficients from a field $F$ of characteristic $\neq 2$, and let $g(x)=x^{3}+ax^{2}+bx+c$. We show how to identify the conjugacy class in $S_{6}$ of the Galois group of $f$ over $F$ using only the discriminants of $f$ and $g$ and the reducibility of a related sextic polynomial. We demonstrate that our method is useful for producing one-parameter families of even sextic polynomials with a specified Galois group.

In this article, we functorially associate definable sets to $k$-analytic curves, and definable maps to analytic morphisms between them, for a large class of $k$-analytic curves. Given a $k$-analytic curve $X$, our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of $X$ and show that they satisfy a bijective relation with the radial subsets of $X$. As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of $k$-analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser’s theorem on iso-definability of curves. However, our approach can also be applied to strictly $k$-affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting.

A polynomial $f$ over a finite field $\mathbb{F}_{q}$ can be classified as a permutation polynomial by the Hermite–Dickson criterion, which consists of conditions on the powers $f^{e}$ for each $e$ from $1$ to $q-2$, as well as the existence of a unique solution to $f(x)=0$ in $\mathbb{F}_{q}$. Carlitz and Lutz gave a variant of the criterion. In this paper, we provide an alternate proof to the theorem of Carlitz and Lutz.

We give an algorithmic generalisation of Dickson’s method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson’s idea is to formulate from Hermite’s criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form.

The probability of successfully spending twice the same bitcoins is considered. A double-spending attack consists in issuing two transactions transferring the same bitcoins. The first transaction, from the fraudster to a merchant, is included in a block of the public chain. The second transaction, from the fraudster to himself, is recorded in a block that integrates a private chain, exact copy of the public chain up to substituting the fraudster-to-merchant transaction by the fraudster-to-fraudster transaction. The double-spending hack is completed once the private chain reaches the length of the public chain, in which case it replaces it. The growth of both chains are modelled by two independent counting processes. The probability distribution of the time at which the malicious chain catches up with the honest chain, or, equivalently, the time at which the two counting processes meet each other, is studied. The merchant is supposed to await the discovery of a given number of blocks after the one containing the transaction before delivering the goods. This grants a head start to the honest chain in the race against the dishonest chain.

We show that the Galois cohomology groups of $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ can be computed via the generalization of Herr’s complex to multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules. Using Tate duality and a pairing for multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules we extend this to analogues of the Iwasawa cohomology. We show that all $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ are overconvergent and, moreover, passing to overconvergent multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.

A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].

Let q be a positive integer and let (an) and (bn) be two given ℂ-valued and q-periodic sequences. First we prove that the linear recurrence in ℂ0.1

$$x_{n + 2} = a_nx_{n + 1} + b_nx_n,\quad n\in {\open Z}_+ $$

0.2

$$A_n = \left( {\matrix{ 0 & 1 \cr {b_n} & {a_n} \cr } } \right)\quad n\in {\open Z}_+ .$$

0.3

$${x}^{\prime \prime}(t) = a(t){x}^{\prime}(t) + b(t)x(t),\quad t\in {\open R},$$

0.4

$${X}^{\prime}(t) = A(t)X(t),\quad t\in {\open R},\quad X(0) = I_2,$$

0.5

$$A(t) = \left( {\matrix{ 0 & 1 \cr {b(t)} & {a(t)} \cr } } \right),\quad t\in {\open R}.$$

Suppose that $f(x)=x^{n}+A(Bx+C)^{m}\in \mathbb{Z}[x]$, with $n\geq 3$ and $1\leq m<n$, is irreducible over $\mathbb{Q}$. By explicitly calculating the discriminant of $f(x)$, we prove that, when $\gcd (n,mB)=C=1$, there exist infinitely many values of $A$ such that the set $\{1,\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{2},\ldots ,\unicode[STIX]{x1D703}^{n-1}\}$ is an integral basis for the ring of integers of $\mathbb{Q}(\unicode[STIX]{x1D703})$, where $f(\unicode[STIX]{x1D703})=0$.

Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial $f\in \mathbb{Z}[t]$ and any $\unicode[STIX]{x1D700}>0$, there are infinitely many $n\in \mathbb{N}$ for which the largest prime factor of $f(n)$ is no larger than $n^{\unicode[STIX]{x1D700}}$.

We prove that for every sufficiently large integer $n$, the polynomial $1+x+x^{2}/11+x^{3}/111+\cdots +x^{n}/111\ldots 1$ is irreducible over the rationals, where the coefficient of $x^{k}$ for $1\leqslant k\leqslant n$ is the reciprocal of the decimal number consisting of $k$ digits which are each $1$. Similar results following from the same techniques are discussed.

In this paper, we will prove that any $\mathbb{A}^{3}$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable $\mathbb{A}^{2}$-forms over a field $k$ extends to $\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing $\mathbb{Q}$.

We generalize known results about Hilbertian fields to Hilbertian rings. For example, let R be a Hilbertian ring (e.g. R is the ring of integers of a number field) with quotient field K and let A be an abelian variety over K. Then, for every extension M of K in K(Ator(Ksep)), the integral closure RM of R in M is Hilbertian.

We revisit the coordinatisation method for projective planes by considering the consequences of using finite fields to coordinatise projective planes of prime power order. This leads to some general restrictions on the form of the resulting planar ternary ring (PTR) when viewed as a trivariate polynomial over the field. We also consider how the Lenz–Barlotti type of the plane being coordinatised impacts the form of the PTR polynomial, thereby deriving further restrictions.

Let A ⊂ B be an integral ring extension of integral domains with fields of fractions K and L, respectively. The integral degree of A ⊂ B, denoted by dA(B), is defined as the supremum of the degrees of minimal integral equations of elements of B over A. It is an invariant that lies in between dK(L) and μA(B), the minimal number of generators of the A-module B. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if A ⊂ B is simple; if A ⊂ B is projective and finite and K ⊂ L is a simple algebraic field extension; or if A is integrally closed. Furthermore, d is upper-semicontinuous if A is noetherian of dimension 1 and with finite integral closure. In general, however, d is neither sub-multiplicative nor upper-semicontinuous.

In this paper, we develop a new necessary and sufficient condition for the vanishing of $4$-Massey products of elements in the modulo-$2$ Galois cohomology of a field. This new description allows us to define a splitting variety for $4$-Massey products, which is shown in the appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such $4$-Massey products vanish whenever they are defined. This provides new explicit restrictions on the structure of absolute Galois groups of number fields.

The aim of this note is to give a simple topological proof of the well-knownresult concerning continuity of roots of polynomials. We also consider amore general case with polynomials of a higher degree approaching a givenpolynomial. We then examine the continuous dependence of solutions of lineardifferential equations with constant coefficients.

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

Let K be a global field, $\mathcal{V}$ a proper subset of the set of all primes of K, $\mathcal{S}$ a finite subset of $\mathcal{V}$, and ${\tilde K}$ (resp. Ksep) a fixed algebraic (resp. separable algebraic) closure of K with $K_\mathrm{sep}\{\subseteq}{\tilde K}$. Let Gal(K) = Gal(Ksep/K) be the absolute Galois group of K. For each $\mathfrak{p}\in\mathcal{V}$, we choose a Henselian (respectively, a real or algebraic) closure $K_\mathfrak{p}$ of K at $\mathfrak{p}$ in ${\tilde K}$ if $\mathfrak{p}$ is non-archimedean (respectively, archimedean). Then, $K_{\mathrm{tot},\mathcal{S}}=\bigcap_{\mathfrak{p}\in\mathcal{S}}\bigcap_{\tau\in{\rm Gal}(K)}K_\mathfrak{p}^\tau$ is the maximal Galois extension of K in Ksep in which each $\mathfrak{p}\in\mathcal{S}$ totally splits. For each $\mathfrak{p}\in\mathcal{V}$, we choose a $\mathfrak{p}$-adic absolute value $|~|_\mathfrak{p}$ of $K_\mathfrak{p}$ and extend it in the unique possible way to ${\tilde K}$. Finally, we denote the compositum of all symmetric extensions of K by Ksymm. We consider an affine absolutely integral variety V in $\mathbb{A}_K^n$. Suppose that for each $\mathfrak{p}\in\mathcal{S}$ there exists a simple $K_\mathfrak{p}$-rational point $\mathbf{z}_\mathfrak{p}$ of V and for each $\mathfrak{p}\in\mathcal{V}\smallsetminus\mathcal{S}$ there exists $\mathbf{z}_\mathfrak{p}\in V({\tilde K})$ such that in both cases $|\mathbf{z}_\mathfrak{p}|_\mathfrak{p}\le1$ if $\mathfrak{p}$ is non-archimedean and $|\mathbf{z}_\mathfrak{p}|_\mathfrak{p}<1$ if $\mathfrak{p}$ is archimedean. Then, there exists $\mathbf{z}\in V(K_{\mathrm{tot},\mathcal{S}}\cap K_\mathrm{symm})$ such that for all $\mathfrak{p}\in\mathcal{V}$ and for all τ ∈ Gal(K), we have $|\mathbf{z}^\tau|_\mathfrak{p}\le1$ if $\mathfrak{p}$ is archimedean and $|\mathbf{z}^\tau|_\mathfrak{p}<1$ if $\mathfrak{p}$ is non-archimedean. For $\mathcal{S}=\emptyset$, we get as a corollary that the ring of integers of Ksymm is Hilbertian and Bezout.

In this paper we consider the integral functionals of the general epidemic model up to its extinction. We develop a new approach to determine the exact Laplace transform of such integrals. In particular, we obtain the Laplace transform of the duration of the epidemic T, the final susceptible size ST, the area under the trajectory of the infectives AT, and the area under the trajectory of the susceptibles BT. The method relies on the construction of a family of martingales and allows us to solve simple recursive relations for the involved parameters. The Laplace transforms are then expanded in terms of a special class of polynomials. The analysis is generalized in part to Markovian epidemic processes with arbitrary state-dependent rates.