Apollonius circles and irreducibility criteria for polynomials

We prove the irreducibility of integer polynomials $f\left(X\right)$ whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscissae $a$ and $b$, with ratio of the distances to these points depending on the canonical decomposition of $f\left(a\right)$ and $f\left(b\right)$. In particular, we obtain irreducibility criteria for the case where $f\left(a\right)$ and $f\left(b\right)$ have few prime factors, and $f$ is either an Eneström–Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.     

Divisibility of L-polynomials for a family of Artin–Schreier curves

In this paper we consider the curves $C_k^{(p,a)}:y^p?y=x^{p^k+1}+ax$ defined over ${\mathbb{F}}_p$ and give a positive answer to a conjecture about a divisibility condition on L-polynomials of the curves $C_k^{(p,a)}$. Our proof involves finding an exact formula for the number of ${\mathbb{F}}_{p^n}$-rational points on $C_k^{(p,a)}$ for all n, and uses a result we proved elsewhere about the number of rational points on supersingular curves.     

Functorial test modules

In this article we introduce a slight modification of the definition of test modules which is an additive functor τ on the category of coherent Cartier modules. We show that in many situations this modification agrees with the usual definition of test modules. Furthermore, we show that for a smooth morphism $f:X\to Y$ of F-finite schemes one has a natural isomorphism $f^{!}°\tau ?\tau °f^{!}$. If f is quasi-finite and of finite type we construct a natural transformation $\tau °f_{?}\to f_{?}°\tau$.     

Applications and homological properties of local rings with decomposable maximal ideals

We construct a local Cohen–Macaulay ring R with a prime ideal $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC). Given any positive integer n, we also construct a local Cohen–Macaulay ring R with a prime ideal $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R has exactly two non-isomorphic semidualizing modules, but the localization $R_{\mathfrak{p}}$ has $2^n$ non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen–Macaulay fiber products of finite Cohen–Macaulay type.     

The Alon–Tarsi conjecture: A perspective on the main results

The Alon–Tarsi conjecture states that if $n$ is even, then the sum of the signs of the Latin squares of order $n$ is non-zero (Alon and Tarsi, 1992). The conjecture has been proven in the cases $n=p+1$ (Drisko, 1997), and $n=p?1$ (Glynn, 2010), where $p$ is an odd prime. This paper is intended to be a concise and largely self-contained account of these results, along with streamlined, and in some cases, original proofs that should be readily accessible to a mathematician with a background in combinatorics. We also discuss the relation between the Alon–Tarsi conjecture and Rota’s basis conjecture (Huang and Rota, 1994), and present some related problems, such as Zappa’s extension of the Alon–Tarsi conjecture (Zappa, 1997), and Drisko’s proof of the extended conjecture for $n=p$ (Drisko, 1998).     

Saturation of Jacobian ideals: Some applications to nearly free curves,line arrangements and rational cuspidal plane curves

In this note we describe the minimal resolution of the ideal $I_f$, the saturation of the Jacobian ideal of a nearly free plane curve $C:f=0$. In particular, it follows that this ideal $I_f$ can be generated by at most 4 polynomials. Related general results by Hassanzadeh and Simis on the saturation of codimension 2 ideals are discussed in detail. Some applications to rational cuspidal plane curves and to line arrangements are also given.     

Basic r-symmetric tropical polynomials

We consider tropical polynomials in nr variables, divided into n blocks of r variables, and especially r-symmetric tropical polynomials, which are invariant under the action of the symmetric group $S_n$ on the blocks. We define a set of basic r-symmetric tropical polynomials and show that the basic 2-symmetric tropical polynomials give coordinates on ${\mathbb{R}}^{2n}/S_n$ more efficiently than known polynomials. Moreover, we present special cases for $r\ge 3$ where the basic polynomials separate orbits.     

Nash multiplicities and isolated points of maximum multiplicity

Let X be an algebraic variety defined over a field of characteristic zero, and let $\xi \in \underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$ be a point in the closed subset of maximum multiplicity of X. We provide a criterion, given in terms of arcs, to determine whether ξ is isolated in $\underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$. More precisely, we use invariants of arcs derived from the Nash multiplicity sequence to characterize when ξ is an isolated point in $\underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$.     

Subgroup regular sets in Cayley graphs

Let Γ be a graph with vertex set V, and let a and b be nonnegative integers. A subset C of V is called an $(a,b)$-regular set in Γ if every vertex in C has exactly a neighbors in C and every vertex in $V?C$ has exactly b neighbors in C. In particular, $(0,1)$-regular sets and $(1,1)$-regular sets in Γ are called perfect codes and total perfect codes in Γ, respectively. A subset C of a group G is said to be an $(a,b)$-regular set of G if there exists a Cayley graph of G which admits C as an $(a,b)$-regular set. In this paper we prove that, for any generalized dihedral group G or any group G of order 4p or pq for some primes p and q, if a nontrivial subgroup H of G is a $(0,1)$-regular set of G, then it must also be an $(a,b)$-regular set of G for any $0?a?|H|?1$ and $0?b?|H|$ such that a is even when $|H|$ is odd. A similar result involving $(1,1)$-regular sets of such groups is also obtained in the paper.     

A local–global principle for surjective polynomial maps

Let R be an affine domain of characteristic zero with finite quotients. We prove that a polynomial map over R is surjective if and only if it is surjective over $\stackrel{?}{R_{\mathfrak{m}}}$, the completion of R with respect to $\mathfrak{m}$, for every maximal ideal $\mathfrak{m}?R$. In fact, the completions $\stackrel{?}{R_{\mathfrak{m}}}$ may be replaced by arbitrary subrings containing R. We use this result to yield a characterization of surjective polynomial maps, and remark that there does not exist a similar principle for injective polynomial maps.     

Zero distribution of some shift polynomials

Given an entire function f of finite order ρ, let $g(f):={\sum }_{j=1}^k{b}_j(z)f(z+c_j)$ be a shift polynomial of f with small meromorphic coefficients $b_j$ in the sense of $O(r^{\lambda +\epsilon })+S(r,f)$, $\lambda <\rho$. Provided α, β, $b_0$ are similar small meromorphic functions, we consider zero distribution of $f^n{(g(f))}^s?b_0$, resp. of $g(f)?\alpha f^n?\beta$.     

Complex ball quotients from manifolds of K3[n]-type

We describe periods of irreducible holomorphic symplectic manifolds of $K{3}^{[n]}$-type with a non-symplectic automorphism of prime order $p\ge 3$. These turn out to lie on complex ball quotients and we are able to give a precise characterization of when the period map is bijective by introducing the notion of $K(T)$-generality.     

Critical loci of convex domains in the plane

Let $K$ be a bounded convex domain in ${\mathbb{R}}^2$ symmetric about the origin. The critical locus of $K$ is defined to be the (non-empty compact) set of lattices $\Lambda$ in ${\mathbb{R}}^2$ of smallest possible covolume such that $\Lambda \cap K=\left\{0\right\}$. These are classical objects in geometry of numbers; yet all previously known examples of critical loci were either finite sets or finite unions of closed curves. In this paper we give a new construction which, in particular, furnishes examples of domains having critical locus of arbitrary Hausdorff dimension between 0 and 1.     

Almost slender rings and compact rings

Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring).More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring $k^{\mathbb{N}}/k^{(\mathbb{N})}$ over a von Neumann regular ring k.For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras $R^{\mathbb{N}}\to R$ vanishing on $R^{(\mathbb{N})}$.