Dedekind–Mertens lemma and content formulas in power series rings

Let $R?X?$ be the power series ring over a commutative ring R with identity. For $f\in R?X?$, let $A_f$ denote the content ideal of f, i.e., the ideal of R generated by the coefficients of f. We show that if R is a Prüfer domain and if $g\in R?X?$ such that $A_g$ is locally finitely generated (or equivalently locally principal), then a Dedekind–Mertens type formula holds for g, namely $A_f^2{A}_g=A_f{A}_{fg}$ for all $f\in R?X?$. More generally for a Prüfer domain R, we prove the content formula ${(A_f{A}_g)}^2=(A_f{A}_g)A_{fg}$ for all $f,g\in R?X?$. As a consequence it is shown that an integral domain R is completely integrally closed if and only if ${(A_f{A}_g)}_v={(A_{fg})}_v$ for all nonzero $f,g\in R?X?$, which is a beautiful result corresponding to the well-known fact that an integral domain R is integrally closed if and only if ${(A_f{A}_g)}_v={(A_{fg})}_v$ for all nonzero $f,g\in R[X]$, where $R[X]$ is the polynomial ring over R.For a ring R and $g\in R?X?$, if $A_g$ is not locally finitely generated, then there may be no positive integer k such that $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. Assuming that the locally minimal number of generators of $A_g$ is $k+1$, Epstein and Shapiro posed a question about the validation of the formula $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. We give a negative answer to this question and show that the finiteness of the locally minimal number of special generators of $A_g$ is in fact a more suitable assumption. More precisely we prove that if the locally minimal number of special generators of $A_g$ is $k+1$, then $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$. As a consequence we show that if $A_g$ is finitely generated (in particular if $g\in R[X]$), then there exists a nonnegative integer k such that $A_f^{k+1}{A}_g=A_f^k{A}_{fg}$ for all $f\in R?X?$.     

INTERIOR ESTIMATES IN MORREY SPACES FOR SOLUTIONS OF ELLIPTIC EQUATIONS AND WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS

It is proved that, for the nondivergence elliptic equations ∑in, j=1 aijuxixj = f,if f belongs to the generalized Morrey spaces Lp,ψ(ω), then uxixj ∈ Lp,ψ(ω), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp,ψ (ω).     

Current superalgebras and unitary representations

In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is $\mathfrak{g}=A?\mathfrak{k}$, where $\mathfrak{k}$ is a compact simple Lie superalgebra and A is a supercommutative associative (super)algebra; the crucial case is when $A={\mathrm{\Lambda }}_s(\mathbb{R})$ is a Graßmann algebra. Since we are interested in projective representations, the first step consists in determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if $\mathfrak{k}$ is a simple compact Lie superalgebra with ${\mathfrak{k}}_1\ne \{0\}$, then each (projective) unitary representation of ${\mathrm{\Lambda }}_s(\mathbb{R})?\mathfrak{k}$ factors through a (projective) unitary representation of $\mathfrak{k}$ itself, and these are known by Jakobsen's classification. If ${\mathfrak{k}}_1=\{0\}$, then we likewise reduce the classification problem to semidirect products of compact Lie groups K with a Clifford–Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo and Varadarajan.     

Serial factorizations of right ideals

In a Dedekind domain D, every non-zero proper ideal A factors as a product $A=P_1^{t_1}?P_k^{t_k}$ of powers of distinct prime ideals $P_i$. For a Dedekind domain D, the D-modules $D/P_i^{t_i}$ are uniserial. We extend this property studying suitable factorizations $A=A_1\dots A_n$ of a right ideal A of an arbitrary ring R as a product of proper right ideals $A_1,\dots ,A_n$ with all the modules $R/A_i$ uniserial modules. When such factorizations exist, they are unique up to the order of the factors. Serial factorizations turn out to have connections with the theory of h-local Prüfer domains and that of semirigid commutative GCD domains.     

Non-level semi-standard graded Cohen–Macaulay domain with h-vector (h0,h1,h2)

Let $\mathbb{k}$ be an algebraically closed field of characteristic 0, and $A={?}_{i\in \mathbb{N}}{A}_i$ a Cohen–Macaulay graded domain with $A_0=\mathbb{k}$. If A is semi-standard graded (i.e., A is finitely generated as a $\mathbb{k}[A_1]$-module), it has the h-vector$(h_0,h_1,\dots ,h_s)$, which encodes the Hilbert function of A. From now on, assume that $s=2$. It is known that if A is standard graded (i.e., $A=\mathbb{k}[A_1]$), then A is level. We will show that, in the semi-standard case, if A is not level, then $h_1+1$ divides $h_2$. Conversely, for any positive integers h and n, there is a non-level A with the h-vector $(1,h,(h+1)n)$. Moreover, such examples can be constructed as Ehrhart rings (equivalently, normal toric rings).     

Sequences of binary irreducible polynomials

In this paper we construct an infinite sequence of binary irreducible polynomials starting from any irreducible polynomial $f_0\in F_2\left[x\right]$. If $f_0$ is of degree $n=2^l?m$, where $m$ is odd and $l$ is a nonnegative integer, after an initial finite sequence of polynomials $f_0,f_1,\dots ,f_s$, with $s\le l+3$, the degree of $f_{i+1}$ is twice the degree of $f_i$ for any $i\ge s$.     

All double generalized Petersen graphs are Hamiltonian

The double generalized Petersen graph $DP(n,t)$, $n\ge 3$ and $t\in {\mathbb{Z}}_n?\left\{0\right\}$, $2\le 2t<n$, has vertex-set $\{x_i,y_i,u_i,v_i\mid i\in {\mathbb{Z}}_n\}$, edge-set $\{\{x_i,x_{i+1}\},\{y_i,y_{i+1}\},\{u_i,v_{i+t}\},\{v_i,u_{i+t}\},\{x_i,u_i\},\{y_i,v_i\}\mid i\in {\mathbb{Z}}_n\}$. These graphs were first defined by Zhou and Feng as examples of vertex-transitive non-Cayley graphs. Then, Kutnar and Petecki considered the structural properties, Hamiltonicity properties, vertex-coloring and edge-coloring of $DP(n,t)$, and conjectured that all $DP(n,t)$ are Hamiltonian. In this paper, we prove this conjecture.     

Sequences of dilations and translations in function spaces

Let $f\in L^1[0,1]$ be a mean zero function and let $f_n$, $n=1,2,\dots$, be the dyadic dilations and translations of f. We investigate conditions on f, under which the linear operator $T_f$ defined by $T_f{h}_n=f_n$, $n=1,2,\dots$, where $h_n$, $n=1,2,\dots$, are mean zero Haar functions, can be continuously extended to the closed linear span $[h_n]$ in a certain function space X. Among other results we prove that $T_f$ is bounded in every symmetric space with nontrivial Boyd indices whenever $f\in BM{O}_d$ and f has “good” Haar spectral properties. In the special case of so-called Haar chaoses the above results can be essentially refined and sharpened. In particular, we find necessary and sufficient conditions, under which the operator $T_f$, generated by a Haar chaos f of order 1, is continuously invertible in $L^p$ for all $1<p<\infty$.     

List colouring of graphs and generalized Dyck paths

The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume $G$ is a graph and $f:V\left(G\right)\to N$ is a mapping. For a nonnegative integer $m$, let $f^{\left(m\right)}$ be the extension of $f$ to the graph $G$

 $\overline{K_m}$ for which $f^{\left(m\right)}\left(v\right)=\left|V\left(G\right)\right|$ for each vertex $v$ of $\overline{K_m}$. Let $m_c(G,f)$ be the minimum $m$ such that $G$

 $\overline{K_m}$ is not $f^{\left(m\right)}$-choosable and $m_p(G,f)$ be the minimum $m$ such that $G$

 $\overline{K_m}$ is not $f^{\left(m\right)}$-paintable. We study the parameter $m_c(K_n,f)$ and $m_p(K_n,f)$ for arbitrary mappings $f$. For $\overrightarrow{x}=(x_1,x_2,\dots ,x_n)$, an $\overrightarrow{x}$-dominated path ending at $(a,b)$ is a monotonic path $P$ of the $a\times b$ grid from $(0,0)$ to $(a,b)$ such that each vertex $(i,j)$ on $P$ satisfies $i\le x_{j+1}$. Let $\psi \left(\overrightarrow{x}\right)$ be the number of $\overrightarrow{x}$-dominated paths ending at $(x_n,n)$. By this definition, the Catalan number $C_n$ equals $\psi \left((0,1,\dots ,n?1)\right)$. This paper proves that if $G=K_n$ has vertices $v_1,v_2,\dots ,v_n$ and $f\left(v_1\right)\le f\left(v_2\right)\le \dots \le f\left(v_n\right)$, then $m_c(G,f)=m_p(G,f)=\psi \left(\overrightarrow{x}\left(f\right)\right)$, where $\overrightarrow{x}\left(f\right)=(x_1,x_2,\dots ,x_n)$ and $x_i=f\left(v_i\right)?i$ for $i=1,2,\dots ,n$. Therefore, if $f\left(v_i\right)=n$, then $m_c(K_n,f)=m_p(K_n,f)$ equals the Catalan number $C_n$. We also show that if $G=G_1\cup G_2\cup ?\cup G_p$ is the disjoint union of graphs $G_1,G_2,\dots ,G_p$ and $f=f_1\cup f_2\cup ?\cup f_p$, then $m_c(G,f)={\prod }_{i=1}^p{m}_c(G_i,f_i)$ and $m_p(G,f)={\prod }_{i=1}^p{m}_p(G_i,f_i)$. This generalizes a result in Carraher et al. (2014), where the case each $G_i$ is a copy of $K_1$ is considered.     

Proof of a conjecture on monomial graphs

Let e be a positive integer, p be an odd prime, $q=p^e$, and ${\mathbb{F}}_q$ be the finite field of q elements. Let $f,g\in {\mathbb{F}}_q[X,Y]$. The graph $G_q(f,g)$ is a bipartite graph with vertex partitions $P={\mathbb{F}}_q^3$ and $L={\mathbb{F}}_q^3$, and edges defined as follows: a vertex $(p)=(p_1,p_2,p_3)\in P$ is adjacent to a vertex $[l]=[l_1,l_2,l_3]\in L$ if and only if $p_2+l_2=f(p_1,l_1)$ and $p_3+l_3=g(p_1,l_1)$. If $f=XY$ and $g=X{Y}^2$, the graph $G_q(XY,X{Y}^2)$ contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs $G_q(f,g)$ containing no cycles of length less than eight and not isomorphic to the graph $G_q(XY,X{Y}^2)$, even without requiring them to be edge-transitive. So far, no such graphs $G_q(f,g)$ have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture.