Integer valued polynomials on lower triangular integer matrices

Let $D$ be an integral domain with quotient field $K$ . In this paper we study the algebra of polynomials in $K[x]$ which map the set of lower triangular $n\times n$ matrices with coefficients in $D$ into itself and show that it coincides with the algebra of polynomials whose divided differences of order $k$ map $D^{k+1}$ into $D$ for every $k< n$ . Using this result we describe the polynomial closure of this set of matrices when $D$ is the ring of integers in a global field.     

On the zeros of Meixner polynomials

We investigate the zeros of a family of hypergeometric polynomials $M_n(x;\beta ,c)=(\beta )_n\,{}_2F_1(-n,-x;\beta ;1-\frac{1}{c})$ , $n\in \mathbb N ,$ known as Meixner polynomials, that are orthogonal on $(0,\infty )$ with respect to a discrete measure for $\beta >0$ and $0<c<1.$ When $\beta =-N$ , $N\in \mathbb N $ and $c=\frac{p}{p-1}$ , the polynomials $K_n(x;p,N)=(-N)_n\,{}_2F_1(-n,-x;-N;\frac{1}{p})$ , $n=0,1,\ldots , N$ , $0<p<1$ are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polynomials $M_n(x;\beta ,c)$ , $c<0$ and $n<1-\beta $ , the quasi-orthogonal polynomials $M_n(x;\beta ,c)$ , $-k<\beta <-k+1$ , $k=1,\ldots ,n-1$ and $0<c<1$ or $c>1,$ as well as the polynomials $K_{n}(x;p,N)$ with non-Hermitian orthogonality for $0<p<1$ and $n=N+1,N+2,\ldots $ . We also show that the polynomials $M_n(x;\beta ,c)$ , $\beta \in \mathbb R $ are real-rooted when $c\rightarrow 0$ .     

An Index Formula in Connection with Meromorphic Approximation

Let $\Phi $ be a continuous $n\times n$ matrix-valued function on the unit circle $\mathbb T $ such that the $(k-1)$ st singular value of the Hankel operator with symbol $\Phi $ is greater than the $k$ th singular value. In this case, it is well-known that $\Phi $ has a unique superoptimal meromorphic approximant $Q$ in $H^{\infty }_{(k)}$ ; that is, $Q$ has at most $k$ poles in the unit disc $\mathbb D $ (in the sense that the McMillan degree of $Q$ in $\mathbb D $ is at most $k$ ) and $Q$ minimizes the essential suprema of singular values $s_{j}\left((\Phi -Q)(\zeta )\right)\!, j\ge 0$ , with respect to the lexicographic ordering. For each $j\ge 0$ , the essential supremum of $s_{j}\left((\Phi -Q)(\zeta )\right)$ is called the $j$ th superoptimal singular value of degree $k$ of $\Phi $ . We prove that if $\Phi $ has $n$ non-zero superoptimal singular values of degree $k$ , then the Toeplitz operator $T_{\Phi -Q}$ with symbol $\Phi -Q$ is Fredholm and has index $$ \mathrm{ind}T_{\Phi -Q}=\dim \ker T_{\Phi -Q}=2k+\dim \mathcal E , $$ where $\mathcal E =\{ \xi \in \ker H_{Q}: \Vert H_{\Phi }\xi \Vert _{2}=\Vert (\Phi -Q)\xi \Vert _{2}\}$ and $H_{\Phi }$ denotes the Hankel operator with symbol $\Phi $ . This result can in fact be extended from continuous matrix-valued functions to the wider class of $k$ -admissible matrix-valued functions, i.e. essentially bounded $n\times n$ matrix-valued functions $\Phi $ on $\mathbb T $ for which the essential norm of the Hankel operator $H_{\Phi }$ is strictly less than the smallest non-zero superoptimal singular value of degree $k$ of $\Phi $ .     

Characterization of Siegel cusp forms by the growth of their Fourier coefficients

We characterize all Siegel cusp forms of degree $n$ and large weight $k$ by the growth of their Fourier coefficients. More precisely we prove, among other related results, that if the Fourier coefficients of a modular form on the congruence subgroup $\Gamma _0^n(N)$ of square–free level $N$ satisfy the “Hecke bound” at the cusp $\infty $ , then it must be a cusp form, provided $k >2n+1$ .     

An extension of one direction in Marty’s normality criterion

We prove the following extension of one direction in Marty’s theorem: If $k$ is a natural number, $\alpha >1$ and $\mathcal{F }$ is a family of functions meromorphic on a domain $D$ all of whose poles have multiplicity at least $\frac{k}{\alpha -1}$ , then the normality of $\mathcal{F }$ implies that the family $$\begin{aligned} \left\{ \frac{|f^{(k)}|}{1+|f|^\alpha }\,:\, f\in \mathcal{F }\right\} \end{aligned}$$ is locally uniformly bounded.     

On adding a variable to a Frobenius manifold and generalizations

Let $\pi :V\rightarrow M$ be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure $(\circ _{M},e_{M},g_{M})$ and typical fiber has the structure of a Frobenius algebra $(\circ _{V},e_{V},g_{V})$ . Using a connection $D$ on the bundle $\pi : V{\,\rightarrow \,}M$ and a morphism $\alpha :V\rightarrow TM$ , we construct an almost Frobenius structure $(\circ , e_{V},g)$ on the manifold $V$ and we study when it is Frobenius. In particular, we describe all (real) positive definite Frobenius structures on $V$ obtained in this way, when $M$ is a semisimple Frobenius manifold with non-vanishing rotation coefficients. In the holomorphic setting, we add a real structure $k_{M}$ on $M$ and a real structure $k_{V}$ on the bundle $\pi : V \rightarrow M$ . Using $k_{M}$ , $k_{V}$ and $D$ we define a real structure $k$ on the manifold $V$ . We study when $k$ , together with an almost Frobenius structure $(\circ , e_{V}, g) $ , satisfies the tt*- equations. Along the way, we prove various properties of adding variables to a Frobenius manifold, in connection with Legendre transformations and $tt^{*}$ -geometry.     

Free subalgebras of division algebras over uncountable fields

We study the existence of free subalgebras in division algebras, and prove the following general result: if $A$ is a noetherian domain which is countably generated over an uncountable algebraically closed field $k$ of characteristic $0$ , then either the quotient division algebra of $A$ contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if $k$ is an uncountable algebraically closed field and $A$ is a finitely generated $k$ -algebra that is a domain of GK-dimension strictly less than $3$ , then either $A$ satisfies a polynomial identity, or the quotient division algebra of $A$ contains a free $k$ -algebra on two generators.     

Rational matrix pseudodifferential operators

The skewfield $\mathcal{K }(\partial )$ of rational pseudodifferential operators over a differential field $\mathcal{K }$ is the skewfield of fractions of the algebra of differential operators $\mathcal{K }[\partial ]$ . In our previous paper, we showed that any $H\in \mathcal{K }(\partial )$ has a minimal fractional decomposition $H=AB^{-1}$ , where $A,B\in \mathcal{K }[\partial ],\,B\ne 0$ , and any common right divisor of $A$ and $B$ is a non-zero element of $\mathcal{K }$ . Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-zero element of $\mathcal{K }[\partial ]$ . In the present paper, we study the ring $M_n(\mathcal{K }(\partial ))$ of $n\times n$ matrices over the skewfield $\mathcal{K }(\partial )$ . We show that similarly, any $H\in M_n(\mathcal{K }(\partial ))$ has a minimal fractional decomposition $H=AB^{-1}$ , where $A,B\in M_n(\mathcal{K }[\partial ]),\,B$ is non-degenerate, and any common right divisor of $A$ and $B$ is an invertible element of the ring $M_n(\mathcal{K }[\partial ])$ . Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-degenerate element of $M_n(\mathcal{K } [\partial ])$ . We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures.     

Many Collinear k-Tuples with no k+1 Collinear Points

For every $k>3$ k > 3 , we give a construction of planar point sets with many collinear $k$ k -tuples and no collinear $(k+1)$ ( k + 1 ) -tuples. We show that there are $n_0=n_0(k)$ n 0 = n 0 ( k ) and $c=c(k)$ c = c ( k ) such that if $n\ge n_0$ n ≥ n 0 , then there exists a set of $n$ n points in the plane that does not contain $k+1$ k + 1 points on a line, but it contains at least $n^{2-({c}/{\sqrt{\log n}})}$ n 2 - ( c / log n ) collinear $k$ k -tuples of points. Thus, we significantly improve the previously best known lower bound for the largest number of collinear $k$ k -tuples in such a set, and get reasonably close to the trivial upper bound $O(n^2)$ O ( n 2 ) .     

Essential p-dimension of split simple groups of type A_n

We compute the essential $p$ -dimension of split simple groups of type $A_{n-1}$ in terms of the functor ${{\mathsf{\textit{Alg} }}}(n,m)$ of central simple algebras of degree $n$ and exponent dividing $m$ .     

Highly Incident Configurations with Chiral Symmetry

A geometric $k$ -configuration is a collection of points and straight lines in the plane so that $k$ points lie on each line and $k$ lines pass through this point. We introduce a new construction method for constructing $k$ -configurations with non-trivial dihedral or chiral (i.e., purely rotational) symmetry, for any $k \ge 3$ ; the configurations produced have $2^{k-2}$ symmetry classes of points and lines. The construction method produces the only known infinite class of symmetric geometric 7-configurations, the second known infinite class of symmetric geometric 6-configurations, and the only known 6-configurations with chiral symmetry.     

Explicit Constructions of Centrally Symmetric k-Neighborly Polytopes and Large Strictly Antipodal Sets

We present explicit constructions of centrally symmetric $2$ -neighborly $d$ -dimensional polytopes with about $3^{d/2}\approx (1.73)^d$ vertices and of centrally symmetric $k$ -neighborly $d$ -polytopes with about $2^{{3d}/{20k^2 2^k}}$ vertices. Using this result, we construct for a fixed $k\ge 2$ and arbitrarily large $d$ and $N$ , a centrally symmetric $d$ -polytope with $N$ vertices that has at least $\left( 1-k^2\cdot (\gamma _k)^d\right) \genfrac(){0.0pt}{}{N}{k}$ faces of dimension $k-1$ , where $\gamma _2=1/\sqrt{3}\approx 0.58$ and $\gamma _k = 2^{-3/{20k^2 2^k}}$ for $k\ge 3$ . Another application is a construction of a set of $3^{\lfloor d/2 -1\rfloor }-1$ points in $\mathbb R ^d$ every two of which are strictly antipodal as well as a construction of an $n$ -point set (for an arbitrarily large $n$ ) in $\mathbb R ^d$ with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively.     

Strengthening the cohomological crepant resolution conjecture for Hilbert–Chow morphisms

Given any smooth toric surface $S$ , we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov–Witten invariants of the $n$ -fold symmetric product stack $[\mathrm{Sym}^n(S)]$ of $S$ to the 3-point extremal Gromov–Witten invariants of the Hilbert scheme $\mathrm{Hilb}^n(S)$ of $n$ points on $S$ . As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan’s Cohomological Crepant Resolution Conjecture for the Hilbert–Chow morphism $\mathrm{Hilb}^n(S) \rightarrow \mathrm{Sym}^n(S)$ and yields a method of reconstructing the cup product for $\mathrm{Hilb}^n(S)$ from the orbifold invariants of $[\mathrm{Sym}^n(S)]$ .     

Counting nonsingular matrices with primitive row vectors

We give an asymptotic expression for the number of nonsingular integer $n\times n$ -matrices with primitive row vectors, determinant $k$ , and Euclidean matrix norm less than $T$ , as $T\rightarrow \infty $ . We also investigate the density of matrices with primitive rows in the space of matrices with determinant $k$ , and determine its asymptotics for large $k$ .     

Sporadic Reinhardt Polygons

Let $n$ be a positive integer, not a power of two. A Reinhardt polygon is a convex $n$ -gon that is optimal in three different geometric optimization problems: it has maximal perimeter relative to its diameter, maximal width relative to its diameter, and maximal width relative to its perimeter. For almost all $n$ , there are many Reinhardt polygons with $n$ sides, and many of them exhibit a particular periodic structure. While these periodic polygons are well understood, for certain values of $n$ , additional Reinhardt polygons exist, which do not possess this structured form. We call these polygons sporadic. We completely characterize the integers $n$ for which sporadic Reinhardt polygons exist, showing that these polygons occur precisely when $n=pqr$ with $p$ and $q$ distinct odd primes and $r\ge 2$ . We also prove that a positive proportion of the Reinhardt polygons with $n$ sides is sporadic for almost all integers $n$ , and we investigate the precise number of sporadic Reinhardt polygons that are produced for several values of $n$ by a construction that we introduce.     

True Poly-Bergman and Poly-Bergman Kernels for the Complement of a Closed Disk

Let $k$ and $j$ be positive integers. We prove that the action of the two-dimensional singular integral operators $(S_\Omega )^{j-1}$ and $(S_\Omega ^*)^{j-1}$ on a Hilbert base for the Bergman space $\mathcal{A }^2(\Omega )$ and anti-Bergman space $\mathcal{A }^2_{-1}(\Omega ),$ respectively, gives Hilbert bases $\{ \psi _{\pm j , k } \}_{ k }$ for the true poly-Bergman spaces $\mathcal{A }_{(\pm j)}^2(\Omega ),$ where $S_\Omega $ denotes the compression of the Beurling transform to the Lebesgue space $L^2(\Omega , dA).$ The functions $\psi _{\pm j,k}$ will be explicitly represented in terms of the $(2,1)$ -hypergeometric polynomials as well as by formulas of Rodrigues type. We prove explicit representations for the true poly-Bergman kernels and more transparent representations for the poly-Bergman kernels of $\Omega $ . We establish Rodrigues type formulas for the poly-Bergman kernels of $\mathbb{D }$ .     

Dimension Reduction for Finite Trees in \varvec{\ell _1}

We show that every $n$ -point tree metric admits a $(1+\varepsilon )$ -embedding into $\ell _1^{C(\varepsilon ) \log n}$ , for every $\varepsilon > 0$ , where $C(\varepsilon ) \le O\big ((\frac{1}{\varepsilon })^4 \log \frac{1}{\varepsilon })\big )$ . This matches the natural volume lower bound up to a factor depending only on $\varepsilon $ . Previously, it was unknown whether even complete binary trees on $n$ nodes could be embedded in $\ell _1^{O(\log n)}$ with $O(1)$ distortion. For complete $d$ -ary trees, our construction achieves $C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )$ .     

Planar Beurling Transform and Bergman Type Spaces

In this paper we describe the actions of the operator $S_\mathbb{D }$ or its adjoint $S_\mathbb{D }^*$ on the poly-Bergman spaces of the unit disk $\mathbb{D }.$ Let $k$ and $j$ be positive integers. We prove that $(S_\mathbb{D })^{j}$ is an isometric isomorphism between the true poly-Bergman subspace $\mathcal{A }_{(k)}^2(\mathbb{D })\ominus N_{(k),j}$ onto the true poly-Bergman space $\mathcal{A }_{(j+k)}^2(\mathbb{D }),$ where the linear space $N_{(k),j}$ have finite dimension $j.$ The action of $(S_\mathbb{D })^{j-1}$ on the canonical Hilbert base for the Bergman subspace $\mathcal{A }^2(\mathbb{D })\ominus \mathcal{P }_{j-1},$ gives a Hilbert base $\{ \phi _{ j , k } \}_{ k }$ for $\mathcal{A }_{(j)}^2(\mathbb{D }).$ It is shown that $\{ \phi _{ j , k } \}_{ j, k }$ is a Hilbert base for $L^2(\mathbb{D },d A)$ such that whenever $j$ and $k$ remain constant we obtain a Hilbert base for the true poly-Bergman space $\mathcal{A }_{(j)}^2(\mathbb{D })$ and $\mathcal{A }_{(-k)}^2(\mathbb{D }),$ respectively. The functions $\phi _{ j , k }$ are polynomials in $z$ and $\overline{z}$ and are explicitly given in terms of the $(2,1)$ -hypergeometric polynomials. We prove explicit representations for the true poly-Bergman kernels and the Koshelev representation for the poly-Bergman kernels of $\mathbb{D }.$ The action of $S_\Pi $ on the true poly-Bergman spaces of the upper half-plane $\Pi $ allows one to introduce Hilbert bases for the true poly-Bergman spaces, and to give explicit representations of the true poly-Bergman and poly-Bergman kernels.     

Towards a de Bruijn–Erdős Theorem in the L_1-Metric

A well-known theorem of de Bruijn and Erd?s states that any set of $n$ non-collinear points in the plane determines at least $n$ lines. Chen and Chvátal asked whether an analogous statement holds within the framework of finite metric spaces, with lines defined using the notion of betweenness. In this paper, we prove that the answer is affirmative for sets of $n$ points in the plane with the $L_1$ metric, provided that no two points share their $x$ - or $y$ -coordinate. In this case, either there is a line that contains all $n$ points, or $X$ induces at least $n$ distinct lines. If points of $X$ are allowed to share their coordinates, then either there is a line that contains all $n$ points, or $X$ induces at least $n/37$ distinct lines.