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-known result concerning continuity of roots of polynomials. We also consider a more general case with polynomials of a higher degree approaching a given polynomial. We then examine the continuous dependence of solutions of linear differential 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.

The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First, every polynomial with odd degree and real coefficients has a real root. Second, every nonnegative real number has a square root. Shipman [‘Improving the fundamental theorem of algebra’, Math. Intelligencer29(4) (2007), 9–14] showed that the assumption about odd degree polynomials is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed. In this paper, we give a simpler proof of this result of Shipman.

Li introduced the normalized volume of a valuation due to its relation to K-semistability. He conjectured that over a Kawamata log terminal (klt) singularity there exists a valuation with smallest normalized volume. We prove this conjecture and give an explicit example to show that such a valuation need not be divisorial.

Let $K$ be a finitely generated extension of $\mathbb{Q}$, and let $A$ be a nonzero abelian variety over $K$. Let $\tilde{K}$ be the algebraic closure of $K$, and let $\text{Gal}(K)=\text{Gal}(\tilde{K}/K)$ be the absolute Galois group of $K$ equipped with its Haar measure. For each $\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, let $\tilde{K}(\unicode[STIX]{x1D70E})$ be the fixed field of $\unicode[STIX]{x1D70E}$ in $\tilde{K}$. We prove that for almost all $\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, there exist infinitely many prime numbers $l$ such that $A$ has a nonzero $\tilde{K}(\unicode[STIX]{x1D70E})$-rational point of order $l$. This completes the proof of a conjecture of Geyer–Jarden from 1978 in characteristic 0.

We prove a positive characteristic version of Ax’s theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group [Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math.94 (1972), 1195–1204]. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax–Schanuel type.

This is an addendum to a recent paper by Zaïmi, Bertin and Aljouiee [‘On number fields without a unit primitive element’, Bull. Aust. Math. Soc.93 (2016), 420–432], giving the answer to a question asked in that paper, together with some historical connections.

We lay the groundwork in this first installment of a series of papers aimed at developing a theory of Hrushovski–Kazhdan style motivic integration for certain types of nonarchimedean $o$-minimal fields, namely power-bounded $T$-convex valued fields, and closely related structures. The main result of the present paper is a canonical homomorphism between the Grothendieck semirings of certain categories of definable sets that are associated with the $\text{VF}$-sort and the $\text{RV}$-sort of the language ${\mathcal{L}}_{T\text{RV}}$. Many aspects of this homomorphism can be described explicitly. Since these categories do not carry volume forms, the formal groupification of the said homomorphism is understood as a universal additive invariant or a generalized Euler characteristic. It admits not just one, but two specializations to $\unicode[STIX]{x2124}$. The overall structure of the construction is modeled on that of the original Hrushovski–Kazhdan construction.

We correct some statements and proofs of K. S. Kedlaya [Local and global structure of connections on nonarchimedean curves, Compos. Math. 151 (2015), 1096–1156]. To summarize, Proposition 1.1.2 is false as written, and we provide here a corrected statement and proof (and a corresponding modification of Remark 1.1.3); the proofs of Theorem 2.3.17 and Theorem 3.8.16, which rely on Proposition 1.1.2, are corrected accordingly; some missing details in the proofs of Theorem 3.4.20 and Theorem 3.4.22 are filled in; and a few much more minor corrections are recorded.

For any integer $m\neq 0$, we prove that $f(x)=x^{9}+9mx^{6}+192m^{3}$ is irreducible over $\mathbb{Q}$ and that the Galois group of $f(x)$ over $\mathbb{Q}$ is the dihedral group of order 18. Moreover, we show that for infinitely many values of $m$, the splitting fields for $f(x)$ are distinct.

We generalize the $\mathbb{Z}/p$metabelian birational $p$-adic section conjecture for curves, as introduced and proved in Pop [On the birational$p$-adic section conjecture, Compos. Math. 146 (2010), 621–637], to all complete smooth varieties, provided $p>2$. The condition $p>2$ seems to be of technical nature only, and might be removable.

We prove analogs of the Bezout and the Bernstein–Kushnirenko–Khovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first $l$ derivatives of an $n$-tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on $n$ and $l$) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay. We illustrate the application of our estimates in two diophantine contexts: to counting transcendental lattice points on algebraic subvarieties of semi-abelian varieties, following Hrushovski and Pillay; and to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon. In both cases we obtain bounds which are singly-exponential (improving the known doubly-exponential bounds) and which exhibit the natural asymptotic growth with respect to the degrees of the equations involved.

In 2013, Weintraub gave a generalization of the classical Eisenstein irreducibility criterion in an attempt to correct a false claim made by Eisenstein. Using a different approach, we prove Weintraub's result with a weaker hypothesis in a more general setup that leads to an extension of the generalized Schönemann irreducibility criterion for polynomials with coefficients in arbitrary valued fields.

We study the question of which Henselian fields admit definable Henselian valuations (with or without parameters). We show that every field that admits a Henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) Henselian valuation. In equicharacteristic 0, we give a complete characterization of Henselian fields admitting a parameter-definable (non-trivial) Henselian valuation. We also obtain partial characterization results of fields admitting -definable (non-trivial) Henselian valuations. We then draw some Galois-theoretic conclusions from our results.